Finite Group Databasebeta

← #33092524

Subgroups — #33092524

Order 18688.

954 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988826C211yesmaximal, characteristic
2C2×C4×#11924018C211yesmaximal, characteristic, complemented
3C2×#17988543C211yesmaximal, characteristic
4#22612613C211yesmaximal, characteristic
5#22610791C211yesmaximal, characteristic, complemented
6#22610661C211yesmaximal, characteristic, complemented
7#22610624C211yesmaximal, characteristic, complemented
8#22610620C211yesmaximal, characteristic, complemented
9#22610365C211yesmaximal, characteristic
10#22610362C211yesmaximal, characteristic
11#22609958C211yesmaximal, characteristic
12C4×#14556524C211yesmaximal, characteristic, complemented
13#22609764C211yesmaximal, characteristic, complemented
14#22609761C211yesmaximal, characteristic, complemented
15C73×#2323C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C22×#11924012C2211yescharacteristic
18C23×#10596831C2211yescharacteristic
19C2×#14556570C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21#17988826C418yescomplemented
22C4×#11924018—24no—
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32#17988543—24no—
33C2×#14556502C2211yescharacteristic
34C2×#14556502C2211yescharacteristic
35C2×C4×#10596831C2211yescharacteristic
36C2×C4×#10596831C2211yescharacteristic
37#17988493C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988482—21no—
40#17988481—21no—
41#17988475—21no—
42#17988468—21no—
43#17988447C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51C2×C22⋊Q8×C73C2211yescharacteristic
52C2×C4×Q8×C73C2211yescharacteristic
53C2×#117×C73C2211yescharacteristic
54#391×C73C2211yescharacteristic
55#389×C73—21no—
56#389×C73C2211yescharacteristic
57#388×C73—21no—
58#388×C73—21no—
59#387×C73—21no—
60#380×C73C2211yescharacteristic
61C4×C4⋊C4×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C2×#11924018—22no—
64C2×#11924018—22no—
65C2×#11924018C2×C414yes—
66C2×#11924012—22no—
67C2×#11924012—22no—
68C2×#11924012—22no—
69C22×#10596831—22no—
70C22×#10596831—22no—
71C22×#10596831C2×C414yes—
72C22×#10596831C2311yescharacteristic
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75C22×#10596831C2311yescharacteristic
76#14556570—22no—
77#14556570C2×C414yes—
78C4×Dic146—22no—
79C4×Dic146—22no—
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90#14556524—22no—
91#14556524—22no—
92#14556524C2×C414yes—
93#14556515—22no—
94#14556515—22no—
95#14556515C2×C414yes—
96#14556514—22no—
97#14556514—22no—
98#14556514C2×C414yes—
99#14556502—22no—
100#14556502—22no—
101C4×#10596831—22no—
102C4×#10596831D414yes—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C22×Q8×C73C2311yescharacteristic
109C23×C4×C73C2311yescharacteristic
110C22⋊Q8×C73—22no—
111C22⋊Q8×C73C2×C414yes—
112C4×Q8×C73—22no—
113C4×Q8×C73—22no—
114#117×C73—22no—
115#117×C73—22no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73D411yescharacteristic
120C2×C4⋊C4×C73D411yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C22⋊C4×C73—21no—
125C2×C22⋊C4×C73—21no—
126C2×C22⋊C4×C73D411yescharacteristic
127C2×C22⋊C4×C73D411yescharacteristic
128C2×C42×C73—21no—
129C2×C42×C73—21no—
130C2×C42×C73C2311yescharacteristic
131C2×C42×C73C2311yescharacteristic
132#95×C73—21no—
133#95×C73—21no—
134#95×C73—21no—
135#95×C73—21no—
136#95×C73—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C412yes—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147#11924018—28no—
148#11924012—28no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—28no—
156C2×#10596831C22×C412yes—
157C2×#10596831C22×C412yes—
158C2×#10596831C2×D412yes—
159C2×#10596831C2×D412yes—
160C2×#10596831#4112yes—
161C2×#10596831#4112yes—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73C22×C412yes—
168#11924007—24no—
169#11924007—42no—
170#11924007—81no—
171C22⋊Dic73—24no—
172C22⋊Dic73—81no—
173C4×Dic73—42no—
174C4×Dic73—81no—
175C24×C73C2×D411yescharacteristic
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73C22×C412yes—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73C22×C412yes—
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73#4112yes—
197C22×C4×C73C2411yescharacteristic
198C4⋊C4×C73—24no—
199C4⋊C4×C73—22no—
200C4⋊C4×C73—22no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—42no—
207C4⋊C4×C73—81no—
208C4⋊C4×C73C22×C412yes—
209C22⋊C4×C73—24no—
210C22⋊C4×C73—22no—
211C22⋊C4×C73—22no—
212C22⋊C4×C73—81no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—42no—
216C42×C73—81no—
217C42×C73C2×D412yes—
218C42×C73#4112yes—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73C4×D412yes—
226C23×C73C22×D411yescharacteristic
227Q8×C73—22no—
228Q8×C73—22no—
229Q8×C73—41no—
230Q8×C73—41no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73C4×D412yes—
260C2×C4×C73#12112yes—
261C2×C4×C73#12112yes—
262C2×C4×C73C23×C411yescharacteristic
263C2×C4×C73C22×D411yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×Dic73—22no—
269C2×Dic73—22no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280Dic146—22no—
281Dic146—22no—
282Dic146—41no—
283Dic146—41no—
284#10596831—22no—
285#10596831—22no—
286#10596831—22no—
287#10596831C4×D414yes—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73C2×C4×D411yescharacteristic
296C22×C73C2×#12111yescharacteristic
297C22×C73#53211yescharacteristic
298C22×C73#57611yescharacteristic
299C22×C73#57811yescharacteristic
300C22×C73#57811yescharacteristic
301C22×C73#57811yescharacteristic
302C4×C73—22no—
303C4×C73—22no—
304C4×C73—22no—
305C4×C73—42no—
306C4×C73—41no—
307C4×C73—41no—
308C4×C73—42no—
309C4×C73—81no—
310C4×C73—81no—
311C4×C73C2×C4×D411yescharacteristic
312C4×C73C4×#4111yescharacteristic
313C4×C73#52511yescharacteristic
314C4×C73#52911yescharacteristic
315Dic73—42no—
316Dic73—42no—
317Dic73—81no—
318Dic73—81no—
319#21525—731nomaximal, Sylow
320C2×C73—42no—
321C2×C73C4×#12111yescharacteristic
322C2×C73#295011yescharacteristic
323C2×C73#295111yescharacteristic
324C2×C73#305911yescharacteristic
325C2×C73#306111yescharacteristic
326C2×C73#317511yescharacteristic
327C2×C73#317711yescharacteristic
328C2×#467—731no—
329C2×C4×Q16—731no—
330C2×#405—731no—
331#2323—731no—
332#2323D7311yescharacteristic
333#1969—731no—
334#1942—731no—
335#1935—731no—
336#1935—731no—
337#1894—731no—
338#1894—731no—
339#1808—731no—
340C4×#107—731no—
341#1763—731no—
342#1763—731no—
343C73#2152511yescharacteristic, complemented, Sylow
344C22×Q16—731no—
345C22×M16—731no—
346C23×C8—731no—
347C2×C22⋊Q8—731no—
348C2×C22⋊Q8C2×D7311yescharacteristic
349C2×C4×Q8—731no—
350C2×C4×Q8C2×D7311yescharacteristic
351C2×#117C2×D7311yescharacteristic
352#467—738no—
353C4×Q16—1464no—
354C2×#107—731no—
355C2×#107—731no—
356C2×#107—731no—
357C2×#103—731no—
358C2×#103—731no—
359C2×#103—731no—
360C2×#103—731no—
361C2×#103—731no—
362C2×#103—731no—
363#405—1464no—
364C2×#97—731no—
365C2×#97—731no—
366C2×C4×C8—731no—
367C2×C4×C8—731no—
368#391—731no—
369#391C2×D7311yescharacteristic
370#389—21no—
371#389—731no—
372#389—1461no—
373#389C2×D7311yescharacteristic
374#388—21no—
375#388—21no—
376#388—1461no—
377#388—1461no—
378#387—21no—
379#387—1461no—
380#380—731no—
381#380C2×D7311yescharacteristic
382C4×C4⋊C4—731no—
383C4×C4⋊C4C2×D7311yescharacteristic
384#340—731no—
385#340—731no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390C22×Q8—731no—
391C22×Q8C22×D7311yescharacteristic
392C23×C4C22×D7311yescharacteristic
393C2×Q16—734no—
394C2×Q16—1462no—
395C2×Q16—1462no—
396C2×M16—1462no—
397C2×M16—1462no—
398C2×M16—1462no—
399C22×C8—731no—
400C22×C8—734no—
401C22×C8—731no—
402C22×C8—731no—
403C22×C8—731no—
404C22×C8—1462no—
405C22×C8—1462no—
406C22⋊Q8—22no—
407C22⋊Q8—734no—
408C22⋊Q8—1462no—
409C22⋊Q8C4×D7314yes—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—1462no—
413C4×Q8—1462no—
414#117—22no—
415#117—22no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4C22⋊D7311yescharacteristic
426C2×C4⋊C4C22⋊D7311yescharacteristic
427C2×C4⋊C4C22×D7311yescharacteristic
428C2×C4⋊C4C22×D7311yescharacteristic
429C2×C4⋊C4C22×D7311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1461no—
433C2×C22⋊C4—1461no—
434C2×C22⋊C4C22⋊D7311yescharacteristic
435C2×C22⋊C4C22⋊D7311yescharacteristic
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—1461no—
439C2×C42—1461no—
440C2×C42C22×D7311yescharacteristic
441C2×C42C22×D7311yescharacteristic
442#107—734no—
443#107—1462no—
444#107—1462no—
445#103—734no—
446#103—734no—
447#103—1462no—
448#103—1462no—
449#103—1462no—
450#103—1462no—
451#97—1462no—
452#97—1462no—
453C4×C8—734no—
454C4×C8—1462no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—1461no—
461#95—1461no—
462#95—1461no—
463#95—1461no—
464#95—1461no—
465C24C2×C22⋊D7311yescharacteristic
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—732no—
472C2×Q8—1461no—
473C2×Q8—1461no—
474C2×Q8—1461no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×Q8C2×C4×D7312yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4C2×C4×D7312yes—
491C22×C4C2×C22⋊D7311yescharacteristic
492C22×C4C2×C22⋊D7311yescharacteristic
493C22×C4C2×C22⋊D7311yescharacteristic
494C22×C4C2×C22⋊D7311yescharacteristic
495C22×C4C2×C22⋊D7311yescharacteristic
496C22×C4#1192402812yes—
497C22×C4C23×D7311yescharacteristic
498Q16—1468no—
499M16—1468no—
500C2×C8—732no—
501C2×C8—732no—
502C2×C8—732no—
503C2×C8—732no—
504C2×C8—732no—
505C2×C8—732no—
506C2×C8—1468no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—2922no—
524C4⋊C4—5841no—
525C4⋊C4—732no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1464no—
531C4⋊C4—1461no—
532C4⋊C4C2×C4×D7312yes—
533C22⋊C4—24no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—81no—
537C22⋊C4—5841no—
538C22⋊C4—1464no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—81no—
543C42—2922no—
544C42—5841no—
545C42D4×D7312yes—
546C42#1192402712yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23C4×C22⋊D7312yes—
554C23C22×C22⋊D7311yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—2921no—
560Q8—2921no—
561Q8—1462no—
562Q8—1462no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—1462no—
602C2×C4—1462no—
603C2×C4C4×C22⋊D7312yes—
604C2×C4#1455656312yes—
605C2×C4#1455656312yes—
606C2×C4C22×C4×D7311yescharacteristic
607C2×C4C2×D4×D7311yescharacteristic
608C2×C4#41×D7311yescharacteristic
609C2×C4#41×D7311yescharacteristic
610C2×C4C2×#1192402711yescharacteristic
611C2×C4C2×#1192402811yescharacteristic
612C8—734no—
613C8—1462no—
614C8—1462no—
615C8—1462no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D7311yescharacteristic
624C22C2×#1455656311yescharacteristic
625C22#1798913311yescharacteristic
626C22#1798939011yescharacteristic
627C22#1798940211yescharacteristic
628C22#1798940211yescharacteristic
629C22#1798940211yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—2922no—
640C4—2922no—
641C4—5841no—
642C4—5841no—
643C4C4×D4×D7311yescharacteristic
644C4C4×#1192402711yescharacteristic
645C4#1798909611yescharacteristic
646C4#1798911711yescharacteristic
647C2—42no—
648C2C4×#1455656311yescharacteristic
649C2#2261879711yescharacteristic
650C2#2261880411yescharacteristic
651C2#2261958311yescharacteristic
652C2#2261959811yescharacteristic
653C2#2262056411yescharacteristic
654C2#2262058411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2323C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#2323D73
hypocenterC73#21525
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11924018C2semidirect
5. #22610791C2semidirect
6. #22610661C2semidirect
7. #22610624C2semidirect
8. #22610620C2semidirect
12. C4×#14556524C2semidirect
13. #22609764C2semidirect
14. #22609761C2semidirect
21. #17988826C4semidirect
343. C73#21525semidirect

Maximal subgroups

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

subgroupindex
C2×#179888262
C2×C4×#119240182
C2×#179885432
#226126132
#226107912
#226106612
#226106242
#226106202
#226103652
#226103622
#226099582
C4×#145565242
#226097642
#226097612
C73×#23232
#2152573