Finite Group Databasebeta

← #33246930

Subgroups — #33246930

Order 18688.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989069C211yesmaximal, characteristic
2C2×#17989069C211yesmaximal, characteristic
3C2×C4×#11924029C211yesmaximal, characteristic
4#22612849C211yesmaximal, characteristic, complemented
5Q8×C4⋊Dic73C211yesmaximal, characteristic, complemented
6Q8×#11924007C211yesmaximal, characteristic, complemented
7#22612840C213yesmaximal, complemented
8#22612839C213yesmaximal, complemented
9C4⋊C4×D292C213yesmaximal, complemented
10#22612832C213yesmaximal, complemented
11#22612556C213yesmaximal, complemented
12#22612546C213yesmaximal, complemented
13#22612411C213yesmaximal, complemented
14#389×D73C213yesmaximal, complemented
15C73×#2351C211yesmaximal, characteristic, complemented
16C22×#11924029C2211yescharacteristic
17#17989069—28no—
18#17989069C4116yescomplemented
19C4×#11924029—28no—
20C2×Q8×Dic73C2211yescharacteristic
21C2×Q8×Dic73C2211yescharacteristic
22C2×Q8×Dic73C2211yescharacteristic
23C2×C4×D292C2213yes—
24C2×C4×D292C2213yes—
25C2×C4×D292C2213yes—
26C2×#14556553C2213yes—
27C2×#14556553C2213yes—
28C2×#14556553C2213yes—
29C2×#14556549C2213yes—
30C2×#14556548C2213yes—
31C2×#14556548C2213yes—
32C2×C4⋊C4×D73C2213yes—
33C2×C4⋊C4×D73C2216yes—
34C2×C4⋊C4×D73C2213yes—
35C2×C42×D73C2213yes—
36#17988486C2213yes—
37#17988486C2213yes—
38#17988485C2213yes—
39#17988485C2213yes—
40#17988484C2216yes—
41#17988484C2213yes—
42#17988484C2211yescharacteristic
43#17988483C2213yes—
44#17988483C2213yes—
45#17988482C2213yes—
46#17988482C2213yes—
47#17988461C2216yes—
48#17988461C2216yes—
49#17988461C2213yes—
50#17988461C2213yes—
51#17988458C2213yes—
52#17988458C2213yes—
53#17988457C2213yes—
54#17988456C2213yes—
55#17988452C2213yes—
56#17988446C2216yes—
57#17988444C2216yes—
58C4×#11924007C2213yes—
59C4×#11924007C2213yes—
60C4×C4⋊Dic73C2213yes—
61#95×D73C2216yes—
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×#11924029C2×C418yes—
70C2×#11924029D418yes—
71C2×#11924029Q818yes—
72C22×D292C2313yes—
73C22×C4×D73C2313yes—
74C22×C4×D73C2313yes—
75C22×C4×D73C2316yes—
76C22×C4×D73C2316yes—
77C22×C4×D73C2313yes—
78Q8×Dic73—24no—
79Q8×Dic73—24no—
80Q8×Dic73C2×C418yescomplemented
81C4×D292—212no—
82C4×D292—212no—
83C4×D292C2×C4124yescomplemented
84#14556553—212no—
85#14556553—212no—
86#14556553C2×C4124yescomplemented
87#14556549—212no—
88#14556548—212no—
89#14556548C2×C4124yescomplemented
90C2×C4⋊Dic73C2311yescharacteristic
91C2×C4⋊Dic73C2313yes—
92C2×C4⋊Dic73C2313yes—
93C2×C4⋊Dic73C2313yes—
94C2×C4⋊Dic73C2316yes—
95C2×C4⋊Dic73C2313yes—
96C2×#11924007C2311yescharacteristic
97C2×#11924007C2313yes—
98C2×#11924007C2311yescharacteristic
99C2×#11924007C2313yes—
100C2×#11924007C2313yes—
101C2×#11924007C2313yes—
102C2×#11924007C2316yes—
103C2×#11924007C2313yes—
104C2×#11924007C2311yescharacteristic
105C4⋊C4×D73—212no—
106C4⋊C4×D73—412no—
107C4⋊C4×D73C2×C4124yescomplemented
108C2×#11924005C2313yes—
109C2×#11924005C2316yes—
110C2×#11924005C2313yes—
111C2×#11924005C2313yes—
112C2×#11924005C2313yes—
113C2×C4×Dic73C2313yes—
114C2×C4×Dic73C2313yes—
115C2×C4×Dic73C2311yescharacteristic
116C2×C4×Dic73C2313yes—
117C2×C4×Dic73C2313yes—
118C42×D73—212no—
119#14556498C2313yes—
120#14556498C2313yes—
121#14556498C2316yes—
122#14556498C2313yes—
123#14556498C2311yescharacteristic
124#14556498C2313yes—
125#14556498C2313yes—
126C22×Q8×C73C2311yescharacteristic
127C4×Q8×C73—24no—
128C4×Q8×C73—24no—
129C4×Q8×C73C2×C418yescomplemented
130C2×C4⋊C4×C73C2313yes—
131C2×C4⋊C4×C73C2313yes—
132C2×C4⋊C4×C73C2313yes—
133C2×C4⋊C4×C73C2313yes—
134C2×C4⋊C4×C73C2313yes—
135C2×C42×C73C2313yes—
136C2×C42×C73C2313yes—
137C2×C42×C73C2313yes—
138#95×C73C2311yescharacteristic
139#95×C73C2313yes—
140#95×C73C2316yes—
141C23×D73C2413yes—
142#11924029C4⋊C4164yescomplemented
143C2×D292C22×C4112yes—
144C2×D292C2×D4112yes—
145C2×D292C2×Q8112yes—
146C22×Dic73C2411yescharacteristic
147C22×Dic73C2413yes—
148C22×Dic73C2413yes—
149C22×Dic73C2411yescharacteristic
150C22×Dic73C2413yes—
151C22×Dic73C2411yescharacteristic
152C22×Dic73C2411yescharacteristic
153C2×C4×D73—26no—
154C2×C4×D73—26no—
155C2×C4×D73—26no—
156C2×C4×D73—26no—
157C2×C4×D73—212no—
158C2×C4×D73—46no—
159C2×C4×D73—46no—
160C2×C4×D73—43no—
161C2×C4×D73—43no—
162C2×C4×D73—46no—
163C2×C4×D73—46no—
164C2×C4×D73C22×C4124yes—
165C2×C4×D73C22×C4112yes—
166C2×C4×D73C2×D4112yes—
167C2×C4×D73C2×Q8112yes—
168C4⋊Dic73—26no—
169C4⋊Dic73—26no—
170C4⋊Dic73—22no—
171C4⋊Dic73—46no—
172C4⋊Dic73—43no—
173C4⋊Dic73C22×C4112yes—
174#11924007—26no—
175#11924007—26no—
176#11924007—22no—
177#11924007—22no—
178#11924007—46no—
179#11924007—43no—
180#11924007—43no—
181#11924007C22×C4112yes—
182#11924007C22×C414yes—
183#11924005—26no—
184#11924005—26no—
185#11924005—212no—
186#11924005C22×C4112yes—
187#11924005C22×C4112yes—
188C4×Dic73—26no—
189C4×Dic73—26no—
190C4×Dic73—22no—
191C4×Dic73—26no—
192C4×Dic73C22×C4112yes—
193C2×Q8×C73C22×C414yes—
194C2×Q8×C73C2×D414yes—
195C2×Q8×C73C2×Q814yes—
196C22×C4×C73C2413yes—
197C22×C4×C73C2413yes—
198C22×C4×C73C2411yescharacteristic
199C22×C4×C73C2411yescharacteristic
200C22×C4×C73C2413yes—
201C22×C4×C73C2411yescharacteristic
202C22×C4×C73C2413yes—
203C4⋊C4×C73—26no—
204C4⋊C4×C73—26no—
205C4⋊C4×C73—26no—
206C4⋊C4×C73—43no—
207C4⋊C4×C73C22×C4112yes—
208C42×C73—26no—
209C42×C73—26no—
210C42×C73C22×C4112yes—
211C23×C73C2511yescharacteristic
212C22×D73—23no—
213C22×D73—23no—
214C22×D73—23no—
215C22×D73—23no—
216C22×D73C23×C416yes—
217C22×D73C22×D416yes—
218C22×D73C22×Q816yes—
219Q8×C73C2×C4⋊C4116yescomplemented
220C2×C4×C73—21no—
221C2×C4×C73—23no—
222C2×C4×C73—21no—
223C2×C4×C73—23no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—23no—
228C2×C4×C73—23no—
229C2×C4×C73—23no—
230C2×C4×C73—21no—
231C2×C4×C73—23no—
232C2×C4×C73—21no—
233C2×C4×C73—23no—
234C2×C4×C73—23no—
235C2×C4×C73C23×C412yes—
236C2×C4×C73C23×C416yes—
237C2×C4×C73C23×C416yes—
238C2×C4×C73C22×D416yes—
239C2×C4×C73C22×Q816yes—
240C2×C4×C73H(5,𝔽2)12yes—
241C2×Dic73—21no—
242C2×Dic73—23no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—23no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—23no—
249C2×Dic73—23no—
250C2×Dic73—23no—
251C2×Dic73—23no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—23no—
255C2×Dic73—23no—
256C2×Dic73C23×C412yes—
257C2×Dic73C23×C416yes—
258C2×Dic73C23×C412yes—
259C2×Dic73C22×D412yes—
260C2×Dic73C22×Q812yes—
261C2×Dic732−1+412yes—
262D292C2×C4⋊C4148yescomplemented
263C4×D73—424no—
264C4×D73—412no—
265C4×D73—412no—
266C4×D73—812no—
267C4×D73C2×C4⋊C4148yescomplemented
268C22×C73C24×C411yescharacteristic
269C22×C73C23×D411yescharacteristic
270C22×C73C23×Q811yescharacteristic
271C22×C73C2×H(5,𝔽2)11yescharacteristic
272C22×C73C2×2−1+411yescharacteristic
273C22×C73#58511yescharacteristic
274C22×C73#58511yescharacteristic
275C2×D73—26no—
276C2×D73—26no—
277C2×D73—26no—
278C2×D73—26no—
279C2×D73—26no—
280C2×D73—26no—
281C2×D73C22×C4⋊C4112yes—
282C4×C73—26no—
283C4×C73—22no—
284C4×C73—22no—
285C4×C73—43no—
286C4×C73—43no—
287C4×C73—41no—
288C4×C73C22×C4⋊C4112yes—
289Dic73—26no—
290Dic73—22no—
291Dic73—22no—
292Dic73—43no—
293Dic73—43no—
294Dic73—41no—
295Dic73C22×C4⋊C414yes—
296#34250—731nomaximal, Sylow
297C2×C73C23×C4⋊C411yescharacteristic
298C2×C73C4×H(5,𝔽2)11yescharacteristic
299C2×C73C4×2−1+411yescharacteristic
300C2×C73#348811yescharacteristic
301C2×C73#348911yescharacteristic
302C2×C73#350311yescharacteristic
303C2×C73#350411yescharacteristic
304D73—212no—
305C2×#520—731no—
306C2×#520—731no—
307C2×C4×#41—731no—
308#2351—731no—
309#2351D7311yescharacteristic, complemented
310Q8×C4⋊C4—731no—
311Q8×C4⋊C4—731no—
312#2349—733no—
313#2349—733no—
314D4×C4⋊C4—733no—
315D4×C4⋊C4—733no—
316#2317—733no—
317#2316—733no—
318#2296—733no—
319C2×#389—733no—
320C73#3425011yescharacteristic, complemented, Sylow
321C22×#41—731no—
322#520—7316no—
323#520—1468no—
324C4×#41—1468no—
325C2×C4×Q8—731no—
326C2×C4×Q8—731no—
327C2×C4×Q8—731no—
328C2×C4×Q8C2×D7311yescharacteristic
329C2×C4×Q8C2×D7311yescharacteristic
330C2×C4×Q8C2×D7311yescharacteristic
331C2×C4×D4—733no—
332C2×C4×D4—733no—
333C2×C4×D4—733no—
334C2×C4×D4—733no—
335C2×C4×D4—733no—
336C2×C4×D4—733no—
337C2×#117—733no—
338C2×#117—733no—
339C2×#117—733no—
340C22×C4⋊C4—736no—
341C22×C4⋊C4—733no—
342C22×C4⋊C4—733no—
343C22×C42—733no—
344#389—733no—
345#389—733no—
346#389—733no—
347#389—733no—
348#389—733no—
349#389—733no—
350#389—733no—
351#389—733no—
352#389—736no—
353#389—733no—
354#389—731no—
355#389C2×D7313yes—
356#389C2×D7313yes—
357#385—733no—
358#385—733no—
359#385—736no—
360#385—736no—
361#385—733no—
362#385—733no—
363#384—733no—
364#384—733no—
365#382—733no—
366#382C2×D7313yes—
367#380—736no—
368#380—736no—
369C4×C4⋊C4—733no—
370C4×C4⋊C4—733no—
371C4×C4⋊C4—733no—
372C4×C4⋊C4C2×D7313yes—
373C2×#95—736no—
374C2×#41—738no—
375C2×#41—738no—
376C2×#41—738no—
377C22×Q8C22×D7311yescharacteristic
378C22×D4—733no—
379C23×C4—736no—
380C23×C4—736no—
381C23×C4—733no—
382C23×C4—733no—
383C23×C4—733no—
384C4×Q8—24no—
385C4×Q8—24no—
386C4×Q8—738no—
387C4×Q8—1464no—
388C4×Q8—1464no—
389C4×Q8C4×D7318yescomplemented
390C4×D4—7324no—
391C4×D4—7324no—
392C4×D4—14612no—
393C4×D4—14612no—
394C4×D4—14612no—
395C4×D4—14612no—
396#117—7324no—
397#117—14612no—
398#117—14612no—
399C2×C4⋊C4—29212no—
400C2×C4⋊C4—736no—
401C2×C4⋊C4—736no—
402C2×C4⋊C4—733no—
403C2×C4⋊C4—733no—
404C2×C4⋊C4—733no—
405C2×C4⋊C4—733no—
406C2×C4⋊C4—733no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—7324no—
409C2×C4⋊C4—733no—
410C2×C4⋊C4—733no—
411C2×C4⋊C4—733no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—733no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—14612no—
417C2×C4⋊C4C22×D7313yes—
418C2×C4⋊C4C22×D7313yes—
419C2×C4⋊C4C22×D7313yes—
420C2×C4⋊C4C22×D7313yes—
421C2×C4⋊C4C22×D7313yes—
422C2×C22⋊C4—733no—
423C2×C22⋊C4—733no—
424C2×C22⋊C4—733no—
425C2×C22⋊C4—733no—
426C2×C22⋊C4—736no—
427C2×C42—733no—
428C2×C42—733no—
429C2×C42—733no—
430C2×C42—733no—
431C2×C42—731no—
432C2×C42—14612no—
433C2×C42C22×D7313yes—
434C2×C42C22×D7313yes—
435C2×C42C22×D7313yes—
436#95—733no—
437#95—733no—
438#95—736no—
439#95—731no—
440#95—733no—
441#95—733no—
442#95—733no—
443#95C22×D7316yes—
444#95C22×D7311yescharacteristic
445#95C22×D7313yes—
446C24—733no—
447#41—7364no—
448C2×Q8C2×C4×D7314yes—
449C2×Q8D4×D7314yes—
450C2×Q8Q8×D7314yes—
451C2×D4—7312no—
452C2×D4—7312no—
453C2×D4—7312no—
454C22×C4—2926no—
455C22×C4—2926no—
456C22×C4—2923no—
457C22×C4—2923no—
458C22×C4—2926no—
459C22×C4—2926no—
460C22×C4—7324no—
461C22×C4—731no—
462C22×C4—7312no—
463C22×C4—7312no—
464C22×C4—7312no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—733no—
468C22×C4—731no—
469C22×C4—733no—
470C22×C4—733no—
471C22×C4—1466no—
472C22×C4—1466no—
473C22×C4—1466no—
474C22×C4—14612no—
475C22×C4—1466no—
476C22×C4C23×D7313yes—
477C22×C4C23×D7311yescharacteristic
478C22×C4C23×D7313yes—
479C22×C4C23×D7313yes—
480C22×C4C23×D7311yescharacteristic
481C22×C4C23×D7311yescharacteristic
482C22×C4C23×D7313yes—
483C4⋊C4—26no—
484C4⋊C4—26no—
485C4⋊C4—26no—
486C4⋊C4—43no—
487C4⋊C4—2926no—
488C4⋊C4—2923no—
489C4⋊C4—2923no—
490C4⋊C4—2926no—
491C4⋊C4—2923no—
492C4⋊C4—7312no—
493C4⋊C4—734no—
494C4⋊C4—7312no—
495C4⋊C4—1462no—
496C4⋊C4—1466no—
497C4⋊C4—1462no—
498C4⋊C4—1462no—
499C4⋊C4—1466no—
500C4⋊C4—1466no—
501C4⋊C4—1466no—
502C4⋊C4C2×C4×D73112yes—
503C22⋊C4—7312no—
504C22⋊C4—7312no—
505C22⋊C4—1466no—
506C22⋊C4—14612no—
507C22⋊C4—1466no—
508C42—26no—
509C42—26no—
510C42—7312no—
511C42—1462no—
512C42—1466no—
513C42—1466no—
514C42—1466no—
515C42C2×C4×D73112yes—
516C23—736no—
517C23—736no—
518C23—736no—
519C23—1463no—
520C23—1463no—
521C23—1463no—
522C23—1463no—
523C23C24×D7311yescharacteristic
524Q8C4⋊C4×D73116yescomplemented
525D4—7348no—
526C2×C4—21no—
527C2×C4—23no—
528C2×C4—21no—
529C2×C4—23no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—23no—
534C2×C4—23no—
535C2×C4—23no—
536C2×C4—21no—
537C2×C4—23no—
538C2×C4—21no—
539C2×C4—23no—
540C2×C4—23no—
541C2×C4—29212no—
542C2×C4—29224no—
543C2×C4—29212no—
544C2×C4—58412no—
545C2×C4—732no—
546C2×C4—732no—
547C2×C4—732no—
548C2×C4—7348no—
549C2×C4—732no—
550C2×C4—736no—
551C2×C4—732no—
552C2×C4—1461no—
553C2×C4—1463no—
554C2×C4—1463no—
555C2×C4—1463no—
556C2×C4—1463no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1463no—
560C2×C4—1463no—
561C2×C4—1461no—
562C2×C4—1463no—
563C2×C4—1461no—
564C2×C4—1463no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4C22×C4×D7312yes—
568C2×C4C22×C4×D7316yes—
569C2×C4C22×C4×D7316yes—
570C2×C4C2×D4×D7316yes—
571C2×C4C2×Q8×D7316yes—
572C2×C4#1455663712yes—
573C22—7312no—
574C22—1466no—
575C22—1466no—
576C22—1466no—
577C22—1466no—
578C22—1466no—
579C22—1466no—
580C22C23×C4×D7311yescharacteristic
581C22C22×D4×D7311yescharacteristic
582C22C22×Q8×D7311yescharacteristic
583C22C2×#1455663711yescharacteristic
584C222−1+4×D7311yescharacteristic
585C22#1798942311yescharacteristic
586C22#1798942411yescharacteristic
587C4—22no—
588C4—26no—
589C4—22no—
590C4—43no—
591C4—43no—
592C4—41no—
593C4—2923no—
594C4—2923no—
595C4—2921no—
596C4—734no—
597C4—1462no—
598C4—1466no—
599C4—1462no—
600C4C2×C4⋊C4×D73112yes—
601C2—14612no—
602C2C22×C4⋊C4×D7311yescharacteristic
603C2C4×#1455663711yescharacteristic
604C2#2262314011yescharacteristic
605C2#2262359711yescharacteristic
606C2#2262361011yescharacteristic
607C2#2262374811yescharacteristic
608C2#2262375611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2351C2
socle, cosocleC23×C73C25
prime radicalC22×#11924029C22
hypercenter#2351D73
hypocenterC73#34250
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #22612849C2semidirect
5. Q8×C4⋊Dic73C2semidirect
6. Q8×#11924007C2semidirect
7. #22612840C2semidirect
8. #22612839C2semidirect
9. C4⋊C4×D292C2semidirect
10. #22612832C2semidirect
11. #22612556C2semidirect
12. #22612546C2semidirect
13. #22612411C2semidirect
14. #389×D73C2semidirect
15. C73×#2351C2semidirect
18. #17989069C4semidirect
80. Q8×Dic73C2×C4semidirect
83. C4×D292C2×C4semidirect
86. #14556553C2×C4semidirect
89. #14556548C2×C4semidirect
107. C4⋊C4×D73C2×C4semidirect
129. C4×Q8×C73C2×C4semidirect
142. #11924029C4⋊C4semidirect
219. Q8×C73C2×C4⋊C4semidirect
262. D292C2×C4⋊C4semidirect
267. C4×D73C2×C4⋊C4semidirect
309. #2351D73semidirect
320. C73#34250semidirect
389. C4×Q8C4×D73semidirect
524. Q8C4⋊C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890692
C2×C4×#119240292
#226128492
Q8×C4⋊Dic732
Q8×#119240072
#226128402
#226128392
C4⋊C4×D2922
#226128322
#226125562
#226125462
#226124112
#389×D732
C73×#23512
#3425073