Finite Group Databasebeta

← #33245804

Subgroups — #33245804

Order 18688.

3232 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989249C211yesmaximal, characteristic
2C2×C4×#11924029C211yesmaximal, characteristic, complemented
3C2×C4×Q8×D73C211yesmaximal, characteristic, complemented
4#22613000C213yesmaximal, complemented
5#22612957C213yesmaximal, complemented
6#22612946C213yesmaximal, complemented
7#22612857C213yesmaximal, complemented
8#22612769C211yesmaximal, characteristic, complemented
9Q8×#11924005C211yesmaximal, characteristic, complemented
10#22612470C213yesmaximal, complemented
11#22612461C213yesmaximal, complemented
12#22612439C213yesmaximal, complemented
13C4×Q8×Dic73C211yesmaximal, characteristic, complemented
14C4×#14556553C213yesmaximal, complemented
15C73×#2351C211yesmaximal, characteristic, complemented
16C22×#11924029C2211yescharacteristic
17C22×Q8×D73C2211yescharacteristic
18#17989249C4116yescomplemented
19C2×#14556579C2213yes—
20C2×#14556571C2213yes—
21C2×#14556566C2213yes—
22C4×#11924029—28no—
23C2×Q8×Dic73C2211yescharacteristic
24C2×Q8×Dic73C2211yescharacteristic
25C2×C4×Dic146C2213yes—
26C2×#14556556C2213yes—
27C4×Q8×D73—28no—
28C2×C4×D292C2213yes—
29C2×C4×D292C2213yes—
30C2×#14556553C2213yes—
31C2×#14556553C2213yes—
32C2×#14556553C2213yes—
33C2×#14556549C2213yes—
34C2×C4⋊C4×D73C2213yes—
35C2×C42×D73C2213yes—
36C2×C42×D73C2213yes—
37#17988493C2213yes—
38#17988492C2213yes—
39#17988489C2213yes—
40#17988488C2213yes—
41#17988477C2216yes—
42#17988476C2216yes—
43#17988476C2216yes—
44#17988470C2216yes—
45#17988465C2216yes—
46#17988458C2216yes—
47#17988444C2213yes—
48#17988444C2213yes—
49C4⋊C4×Dic73C2213yes—
50C4⋊C4×Dic73C2213yes—
51C4⋊C4×Dic73C2213yes—
52C4⋊C4×Dic73C2213yes—
53C4×C4⋊Dic73C2213yes—
54C4×#11924005C2213yes—
55C4×#11924005C2213yes—
56#17988434C2213yes—
57#17988434C2211yescharacteristic
58#17988434C2213yes—
59#17988434C2213yes—
60#17988434C2216yes—
61C42×Dic73C2213yes—
62C2×C4×Q8×C73C2211yescharacteristic
63C2×C4×Q8×C73C2211yescharacteristic
64C2×C4×Q8×C73C2211yescharacteristic
65#389×C73C2213yes—
66#389×C73C2213yes—
67#382×C73C2213yes—
68C4×C4⋊C4×C73C2213yes—
69C2×#11924029—24no—
70C2×#11924029—24no—
71C2×#11924029C2×C418yescomplemented
72C22×Dic146C2313yes—
73C2×Q8×D73—24no—
74C2×Q8×D73—24no—
75C2×Q8×D73C2×C418yescomplemented
76C22×D292C2313yes—
77C22×D292C2313yes—
78C22×C4×D73C2313yes—
79C22×C4×D73C2313yes—
80C22×C4×D73C2313yes—
81C22×C4×D73C2313yes—
82C22×C4×D73C2313yes—
83C22×C4×D73C2311yescharacteristic
84#14556579C2×C4124yescomplemented
85#14556571C2×C4124yescomplemented
86#14556566C2×C4124yescomplemented
87Q8×Dic73—24no—
88Q8×Dic73D418yescomplemented
89C4×Dic146—212no—
90#14556556—212no—
91C4×D292—212no—
92C4×D292C2×C4124yescomplemented
93#14556553—212no—
94#14556553—212no—
95#14556553—212no—
96#14556549—212no—
97C2×C4⋊Dic73C2313yes—
98C2×C4⋊Dic73C2313yes—
99C2×C4⋊Dic73C2313yes—
100C2×#11924007C2316yes—
101C4⋊C4×D73—212no—
102C2×#11924005C2313yes—
103C2×#11924005C2313yes—
104C2×#11924005C2313yes—
105C2×#11924005C2316yes—
106C2×#11924005C2316yes—
107C2×#11924005C2311yescharacteristic
108C2×#11924005C2313yes—
109C2×#11924005C2316yes—
110C2×#11924005C2311yescharacteristic
111C2×C4×Dic73C2311yescharacteristic
112C2×C4×Dic73C2313yes—
113C2×C4×Dic73C2313yes—
114C2×C4×Dic73C2313yes—
115C2×C4×Dic73C2313yes—
116C2×C4×Dic73C2311yescharacteristic
117C2×C4×Dic73C2313yes—
118C2×C4×Dic73C2313yes—
119C2×C4×Dic73C2313yes—
120C2×C4×Dic73C2313yes—
121C42×D73—212no—
122C42×D73—212no—
123#14556498C2311yescharacteristic
124#14556498C2313yes—
125#14556498C2313yes—
126#14556498C2313yes—
127#14556498C2316yes—
128#14556498C2316yes—
129C22×Q8×C73C2311yescharacteristic
130C4×Q8×C73—24no—
131C4×Q8×C73—24no—
132C4×Q8×C73C2×C418yescomplemented
133C2×C4⋊C4×C73C2313yes—
134C2×C4⋊C4×C73C2313yes—
135C2×C4⋊C4×C73C2313yes—
136C2×C4⋊C4×C73C2313yes—
137C2×C4⋊C4×C73C2313yes—
138C2×C42×C73C2313yes—
139C2×C42×C73C2313yes—
140C2×C42×C73C2313yes—
141#95×C73C2311yescharacteristic
142#95×C73C2313yes—
143#95×C73C2316yes—
144C23×D73C2413yes—
145C23×D73C2411yescharacteristic
146#11924029—232no—
147C2×Dic146—26no—
148C2×Dic146—26no—
149C2×Dic146C22×C4112yes—
150Q8×D73—232no—
151C2×D292—26no—
152C2×D292—26no—
153C2×D292—26no—
154C2×D292—26no—
155C2×D292C22×C4112yes—
156C2×D292C22×C4112yes—
157C22×Dic73C2413yes—
158C22×Dic73C2411yescharacteristic
159C22×Dic73C2413yes—
160C22×Dic73C2413yes—
161C22×Dic73C2411yescharacteristic
162C22×Dic73C2411yescharacteristic
163C2×C4×D73—26no—
164C2×C4×D73—26no—
165C2×C4×D73—26no—
166C2×C4×D73—26no—
167C2×C4×D73—22no—
168C2×C4×D73—22no—
169C2×C4×D73—26no—
170C2×C4×D73—26no—
171C2×C4×D73—26no—
172C2×C4×D73—26no—
173C2×C4×D73—26no—
174C2×C4×D73—26no—
175C2×C4×D73—26no—
176C2×C4×D73—43no—
177C2×C4×D73—43no—
178C2×C4×D73C22×C4112yes—
179C2×C4×D73C22×C4112yes—
180C2×C4×D73#4114yescomplemented
181C4⋊Dic73—26no—
182C4⋊Dic73C22×C4112yes—
183C4⋊Dic73C2×D4112yes—
184#11924007—212no—
185#11924005—26no—
186#11924005—26no—
187#11924005—212no—
188#11924005—26no—
189#11924005—46no—
190#11924005C22×C414yes—
191#11924005C22×C4112yes—
192#11924005C22×C4124yes—
193#11924005#4114yescomplemented
194C4×Dic73—26no—
195C4×Dic73—22no—
196C4×Dic73—26no—
197C4×Dic73—26no—
198C4×Dic73—26no—
199C4×Dic73—26no—
200C4×Dic73—26no—
201C4×Dic73—22no—
202C4×Dic73—43no—
203C4×Dic73C2×D4112yes—
204C2×Q8×C73C22×C414yes—
205C2×Q8×C73C2×D414yes—
206C2×Q8×C73#4114yes—
207C22×C4×C73C2413yes—
208C22×C4×C73C2413yes—
209C22×C4×C73C2411yescharacteristic
210C22×C4×C73C2413yes—
211C22×C4×C73C2411yescharacteristic
212C22×C4×C73C2411yescharacteristic
213C22×C4×C73C2413yes—
214C4⋊C4×C73—26no—
215C4⋊C4×C73—26no—
216C4⋊C4×C73—26no—
217C4⋊C4×C73—26no—
218C4⋊C4×C73C22×C4112yes—
219C42×C73—26no—
220C42×C73—26no—
221C42×C73C22×C4112yes—
222C23×C73C2511yescharacteristic
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—21no—
227C22×D73—23no—
228C22×D73—23no—
229C22×D73—23no—
230C22×D73—23no—
231C22×D73—23no—
232C22×D73—23no—
233C22×D73C23×C416yes—
234C22×D73C23×C412yes—
235C22×D73C2×#4112yes—
236C22×D73C2×#4112yes—
237Q8×C73C4×D4116yescomplemented
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—23no—
241C2×C4×C73—23no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—23no—
247C2×C4×C73—23no—
248C2×C4×C73—23no—
249C2×C4×C73—23no—
250C2×C4×C73—23no—
251C2×C4×C73—23no—
252C2×C4×C73C23×C412yes—
253C2×C4×C73C23×C416yes—
254C2×C4×C73C23×C416yes—
255C2×C4×C73C22×D416yes—
256C2×C4×C73C2×#4112yes—
257C2×C4×C73C2×#4112yes—
258C2×C4×C73C2×#4116yes—
259C2×Dic73—21no—
260C2×Dic73—23no—
261C2×Dic73—21no—
262C2×Dic73—23no—
263C2×Dic73—21no—
264C2×Dic73—23no—
265C2×Dic73—21no—
266C2×Dic73—23no—
267C2×Dic73—23no—
268C2×Dic73—23no—
269C2×Dic73—23no—
270C2×Dic73C23×C416yes—
271C2×Dic73C23×C412yes—
272C2×Dic73C22×D416yes—
273C2×Dic73C22×D412yes—
274C2×Dic73C2×#4112yes—
275C2×Dic73C2×#4112yes—
276C2×Dic732−1+412yes—
277D292—224no—
278D292—224no—
279C4×D73—224no—
280C4×D73—224no—
281C4×D73—28no—
282C4×D73—412no—
283C4×D73—412no—
284C4×D73—412no—
285Dic146—224no—
286C22×C73C24×C411yescharacteristic
287C22×C73C23×D411yescharacteristic
288C22×C73C22×#4111yescharacteristic
289C22×C73C22×#4111yescharacteristic
290C22×C73C22×#4111yescharacteristic
291C22×C73C2×2−1+411yescharacteristic
292C22×C73#58511yescharacteristic
293C2×D73—22no—
294C2×D73—22no—
295C2×D73—22no—
296C2×D73—22no—
297C2×D73—22no—
298C2×D73—22no—
299C2×D73—26no—
300C2×D73—26no—
301C2×D73—26no—
302C2×D73—43no—
303C2×D73—43no—
304C2×D73—43no—
305C2×D73—43no—
306C2×D73C4×#4114yescomplemented
307C4×C73—26no—
308C4×C73—22no—
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—43no—
312C4×C73—43no—
313C4×C73C2×C4×D4112yes—
314Dic73—26no—
315Dic73—22no—
316Dic73—26no—
317Dic73—22no—
318Dic73—43no—
319Dic73#54814yescomplemented
320C4×#548—731nomaximal, Sylow
321C2×C73C22×C4×D411yescharacteristic
322C2×C73C2×C4×#4111yescharacteristic
323C2×C73C4×2−1+411yescharacteristic
324C2×C73C2×#54811yescharacteristic
325C2×C73D4×#4111yescharacteristic
326C2×C73#348011yescharacteristic
327C2×C73#348211yescharacteristic
328D73—24no—
329D73—46no—
330C2×#548—731no—
331C2×C4×#41—731no—
332C22×C4×Q8—731no—
333#2367—733no—
334#2362—733no—
335#2361—733no—
336#2352—733no—
337#2351D7311yescharacteristic, complemented
338#2341—731no—
339Q8×C22⋊C4—731no—
340C4×#124—733no—
341C4×C22⋊Q8—733no—
342C4×#121—733no—
343C42×Q8—731no—
344C42×D4—733no—
345C73C4×#54811yescharacteristic, complemented, Sylow
346C22×#41—731no—
347C23×Q8—731no—
348#548—7316no—
349C2×#124—733no—
350C2×C22⋊Q8—733no—
351C2×#121—733no—
352C4×#41—1468no—
353C2×C4×Q8—731no—
354C2×C4×Q8—731no—
355C2×C4×Q8—733no—
356C2×C4×Q8—733no—
357C2×C4×Q8—1468no—
358C2×C4×Q8C2×D7311yescharacteristic
359C2×C4×Q8C2×D7311yescharacteristic
360C2×C4×Q8C2×D7311yescharacteristic
361C2×C4×D4—733no—
362C2×C4×D4—733no—
363C2×C4×D4—733no—
364C2×C4×D4—733no—
365C2×C4×D4—733no—
366C2×#117—733no—
367C22×C4⋊C4—733no—
368C22×C42—733no—
369C22×C42—733no—
370#391—733no—
371#391—733no—
372#390—733no—
373#390—733no—
374#389C2×D7313yes—
375#389C2×D7313yes—
376#388—736no—
377#388—736no—
378#388—736no—
379#387—736no—
380#386—736no—
381#385—736no—
382#382C2×D7313yes—
383#380—733no—
384#380—733no—
385C4×C4⋊C4—733no—
386C4×C4⋊C4—733no—
387C4×C4⋊C4—733no—
388C4×C4⋊C4—733no—
389C4×C4⋊C4—733no—
390C4×C4⋊C4C2×D7313yes—
391C4×C22⋊C4—733no—
392C4×C22⋊C4—733no—
393C4×C22⋊C4—733no—
394C4×C22⋊C4—733no—
395C4×C22⋊C4—731no—
396C4×C22⋊C4—733no—
397C4×C22⋊C4—736no—
398C43—733no—
399C2×#41—738no—
400C2×#41—1464no—
401C2×#41—1464no—
402C22×Q8—738no—
403C22×Q8—733no—
404C22×Q8—1464no—
405C22×Q8—1464no—
406C22×Q8C22×D7311yescharacteristic
407C22×D4—733no—
408C22×D4—733no—
409C23×C4—733no—
410C23×C4—733no—
411C23×C4—733no—
412C23×C4—733no—
413C23×C4—731no—
414C23×C4—733no—
415#124—7324no—
416C22⋊Q8—7324no—
417#121—7324no—
418C4×Q8—24no—
419C4×Q8—24no—
420C4×Q8—738no—
421C4×Q8—14612no—
422C4×Q8—14612no—
423C4×Q8—1464no—
424C4×Q8C4×D7318yescomplemented
425C4×D4—7324no—
426C4×D4—14612no—
427C4×D4—14612no—
428C4×D4—14612no—
429C4×D4—14612no—
430#117—14612no—
431C2×C4⋊C4—733no—
432C2×C4⋊C4—733no—
433C2×C4⋊C4—736no—
434C2×C4⋊C4—733no—
435C2×C4⋊C4—14612no—
436C2×C4⋊C4C22×D7313yes—
437C2×C4⋊C4C22×D7313yes—
438C2×C4⋊C4C22×D7313yes—
439C2×C4⋊C4C22×D7313yes—
440C2×C4⋊C4C22×D7313yes—
441C2×C22⋊C4—736no—
442C2×C22⋊C4—731no—
443C2×C22⋊C4—731no—
444C2×C22⋊C4—733no—
445C2×C22⋊C4—736no—
446C2×C22⋊C4—736no—
447C2×C22⋊C4—733no—
448C2×C22⋊C4—733no—
449C2×C22⋊C4—733no—
450C2×C42—733no—
451C2×C42—733no—
452C2×C42—733no—
453C2×C42—733no—
454C2×C42—733no—
455C2×C42—733no—
456C2×C42—733no—
457C2×C42—733no—
458C2×C42—731no—
459C2×C42—731no—
460C2×C42—14612no—
461C2×C42—14612no—
462C2×C42C22×D7313yes—
463C2×C42C22×D7313yes—
464C2×C42C22×D7313yes—
465#95—731no—
466#95—733no—
467#95—733no—
468#95—733no—
469#95—736no—
470#95—736no—
471#95C22×D7311yescharacteristic
472#95C22×D7313yes—
473#95C22×D7316yes—
474C24—731no—
475C24—733no—
476#41—14632no—
477C2×Q8—7312no—
478C2×Q8—14632no—
479C2×Q8—1466no—
480C2×Q8—1466no—
481C2×Q8C2×C4×D7314yes—
482C2×Q8D4×D7314yes—
483C2×Q8#1192402914yes—
484C2×D4—7312no—
485C2×D4—7312no—
486C2×D4—1466no—
487C2×D4—1466no—
488C2×D4—1466no—
489C2×D4—1466no—
490C22×C4—2923no—
491C22×C4—2923no—
492C22×C4—731no—
493C22×C4—731no—
494C22×C4—731no—
495C22×C4—733no—
496C22×C4—733no—
497C22×C4—7312no—
498C22×C4—7312no—
499C22×C4—734no—
500C22×C4—733no—
501C22×C4—1466no—
502C22×C4—1466no—
503C22×C4—1466no—
504C22×C4—1466no—
505C22×C4—1466no—
506C22×C4—1466no—
507C22×C4—1462no—
508C22×C4—1462no—
509C22×C4—1466no—
510C22×C4—1466no—
511C22×C4—1466no—
512C22×C4—1466no—
513C22×C4—1466no—
514C22×C4C23×D7313yes—
515C22×C4C23×D7313yes—
516C22×C4C23×D7313yes—
517C22×C4C23×D7311yescharacteristic
518C22×C4C23×D7311yescharacteristic
519C22×C4C23×D7311yescharacteristic
520C22×C4C23×D7313yes—
521C4⋊C4—26no—
522C4⋊C4—26no—
523C4⋊C4—26no—
524C4⋊C4—26no—
525C4⋊C4—7312no—
526C4⋊C4—7312no—
527C4⋊C4—14612no—
528C4⋊C4—1466no—
529C4⋊C4C2×C4×D73112yes—
530C22⋊C4—2926no—
531C22⋊C4—734no—
532C22⋊C4—7312no—
533C22⋊C4—7324no—
534C22⋊C4—734no—
535C22⋊C4—14612no—
536C22⋊C4—1466no—
537C22⋊C4—1466no—
538C22⋊C4—1466no—
539C42—26no—
540C42—26no—
541C42—2923no—
542C42—7312no—
543C42—1462no—
544C42—1462no—
545C42—1466no—
546C42—1466no—
547C42—1466no—
548C42—1466no—
549C42—1466no—
550C42—1466no—
551C42C2×C4×D73112yes—
552C23—732no—
553C23—732no—
554C23—732no—
555C23—736no—
556C23—1463no—
557C23—1463no—
558C23—1463no—
559C23—1463no—
560C23—1463no—
561C23—1463no—
562C23—1461no—
563C23—1461no—
564C23—1461no—
565C23—1461no—
566C23C24×D7311yescharacteristic
567Q8—14624no—
568Q8#14556553116yescomplemented
569D4—14624no—
570D4—14624no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—23no—
574C2×C4—23no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—23no—
580C2×C4—23no—
581C2×C4—23no—
582C2×C4—23no—
583C2×C4—23no—
584C2×C4—23no—
585C2×C4—29212no—
586C2×C4—29212no—
587C2×C4—29212no—
588C2×C4—732no—
589C2×C4—732no—
590C2×C4—732no—
591C2×C4—736no—
592C2×C4—732no—
593C2×C4—736no—
594C2×C4—732no—
595C2×C4—1468no—
596C2×C4—1463no—
597C2×C4—1463no—
598C2×C4—1463no—
599C2×C4—1461no—
600C2×C4—1463no—
601C2×C4—1463no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1463no—
605C2×C4—1461no—
606C2×C4—1463no—
607C2×C4—14624no—
608C2×C4—14624no—
609C2×C4C22×C4×D7312yes—
610C2×C4C22×C4×D7316yes—
611C2×C4C22×C4×D7316yes—
612C2×C4C2×D4×D7316yes—
613C2×C4#41×D7312yes—
614C2×C4#41×D7312yes—
615C2×C4C2×#1192402916yes—
616C22—2923no—
617C22—2923no—
618C22—2923no—
619C22—2923no—
620C22—734no—
621C22—1466no—
622C22—1466no—
623C22—1466no—
624C22—1462no—
625C22—1462no—
626C22—1462no—
627C22—1462no—
628C22—1462no—
629C22—1462no—
630C22C23×C4×D7311yescharacteristic
631C22C22×D4×D7311yescharacteristic
632C22C2×#41×D7311yescharacteristic
633C22C2×#41×D7311yescharacteristic
634C22C22×#1192402911yescharacteristic
635C222−1+4×D7311yescharacteristic
636C22#1798942411yescharacteristic
637C4—26no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—43no—
642C4—43no—
643C4—2923no—
644C4—734no—
645C4—1462no—
646C4—1462no—
647C4—1466no—
648C4—1466no—
649C4C2×#14556553112yes—
650C2—2926no—
651C2—1464no—
652C2C22×#1455655311yescharacteristic
653C2C4×#41×D7311yescharacteristic
654C2#2262314011yescharacteristic
655C2#548×D7311yescharacteristic
656C2#2262342011yescharacteristic
657C2#2262353111yescharacteristic
658C2#2262355111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2351C2
socle, cosocleC23×C73C25
prime radicalC2×#17989249C2
hypercenter#2351D73
hypocenterC73C4×#548
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2351

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×#11924029C2semidirect
3. C2×C4×Q8×D73C2semidirect
4. #22613000C2semidirect
5. #22612957C2semidirect
6. #22612946C2semidirect
7. #22612857C2semidirect
8. #22612769C2semidirect
9. Q8×#11924005C2semidirect
10. #22612470C2semidirect
11. #22612461C2semidirect
12. #22612439C2semidirect
13. C4×Q8×Dic73C2semidirect
14. C4×#14556553C2semidirect
15. C73×#2351C2semidirect
18. #17989249C4semidirect
71. C2×#11924029C2×C4semidirect
75. C2×Q8×D73C2×C4semidirect
84. #14556579C2×C4semidirect
85. #14556571C2×C4semidirect
86. #14556566C2×C4semidirect
88. Q8×Dic73D4semidirect
92. C4×D292C2×C4semidirect
132. C4×Q8×C73C2×C4semidirect
180. C2×C4×D73#41semidirect
193. #11924005#41semidirect
237. Q8×C73C4×D4semidirect
306. C2×D73C4×#41semidirect
319. Dic73#548semidirect
337. #2351D73semidirect
345. C73C4×#548semidirect
424. C4×Q8C4×D73semidirect
568. Q8#14556553semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179892492
C2×C4×#119240292
C2×C4×Q8×D732
#226130002
#226129572
#226129462
#226128572
#226127692
Q8×#119240052
#226124702
#226124612
#226124392
C4×Q8×Dic732
C4×#145565532
C73×#23512
C4×#54873