Finite Group Databasebeta

← #10782816

Subgroups — #10782816

Order 768.

590 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79322C211yesmaximal, characteristic, complemented
2#79296C211yesmaximal, characteristic, complemented
3#79268C211yesmaximal, characteristic, complemented
4#79028C211yesmaximal, characteristic
5C3×#1573C211yesmaximal, characteristic, complemented
6#70816C211yesmaximal, characteristic, complemented
7C2×#4481C211yesmaximal, characteristic, complemented
8#68594C211yesmaximal, characteristic, complemented
9#68589C211yesmaximal, characteristic, complemented
10#68584C211yesmaximal, characteristic, complemented
11#68579C211yesmaximal, characteristic, complemented
12#68553C211yesmaximal, characteristic, complemented
13#68548C211yesmaximal, characteristic, complemented
14#68543C211yesmaximal, characteristic, complemented
15#68538C211yesmaximal, characteristic, complemented
16#19591—31nomaximal, Sylow
17#5479C2211yescharacteristic
18#5477C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#432C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23#5115—21no—
24#5114—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4850C2211yescharacteristic
27#4850C2211yescharacteristic
28#4850C2211yescharacteristic
29#4848C2211yescharacteristic
30#4847—21no—
31#4846—21no—
32#4846—21no—
33#4844—21no—
34#4843C2211yescharacteristic
35#4842C2211yescharacteristic
36#4842C2211yescharacteristic
37#4842C2211yescharacteristic
38#4840—21no—
39#4839—21no—
40#4736—21no—
41#4736—21no—
42#4736—21no—
43#4730—21no—
44#67⋊C8—21no—
45#4721—21no—
46#4714—21no—
47#4702—21no—
48#4694C2211yescharacteristic
49#4694C2211yescharacteristic
50#4688C2211yescharacteristic
51C8×D12C2211yescharacteristic
52#4682C2211yescharacteristic
53#4682C2211yescharacteristic
54#4679C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#334C2211yescharacteristic
59C3×#334C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4481C2211yescharacteristic
66#4481C2211yescharacteristic
67#4481C2211yescharacteristic
68#3247—31no—
69#3238—31no—
70#3226—31no—
71#2994—31no—
72#1698—31no—
73#1697—31no—
74#1696—31no—
75#1695—31no—
76#1678—31no—
77#1677—31no—
78#1676—31no—
79#1675—31no—
80#1573D311yescharacteristic, complemented
81#1551—31no—
82C2×#333—31no—
83C2×#235—21no—
84C22×Dic6—21no—
85C22×C3×C8—21no—
86C2×C3×C42C2311yescharacteristic
87#930—21no—
88#930—21no—
89#930D411yescharacteristic
90#929—21no—
91C4×#67—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94#924D411yescharacteristic
95C22×C3⋊C8—21no—
96C2×D24—21no—
97C2×#204—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×D3×C8—21no—
100#876—21no—
101#875—21no—
102#875D411yescharacteristic
103#874C2311yescharacteristic
104C4×D12—21no—
105C4×D12—21no—
106C4×D12C2311yescharacteristic
107#872—21no—
108#870D411yescharacteristic
109#869—21no—
110#869—21no—
111#869—21no—
112#869C2311yescharacteristic
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115C4×Dic6C2311yescharacteristic
116C3×C4⋊C8—21no—
117C3×C4⋊C8D411yescharacteristic
118C3×C4⋊C8D411yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C22⋊C8—21no—
124C3×C22⋊C8D411yescharacteristic
125C3×C22⋊C8D411yescharacteristic
126C3×#97—21no—
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129#821—21no—
130#821—21no—
131#821—21no—
132#821—21no—
133#821—21no—
134#821—21no—
135#821—21no—
136#821—21no—
137#821—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140#818—21no—
141#818—21no—
142#818—21no—
143#818—21no—
144#818—21no—
145#818—21no—
146#818—21no—
147#817—21no—
148#817—21no—
149#817—21no—
150#816—21no—
151#816—21no—
152#816—21no—
153#815—21no—
154Dic3⋊C8—21no—
155Dic3⋊C8—21no—
156C8×Dic3—21no—
157C4⋊(C3⋊C8)—21no—
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#501—61no—
164#500—31no—
165#500—31no—
166#489—61no—
167#486—31no—
168#486—31no—
169#483—61no—
170#480—61no—
171#480—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477—31no—
176#477—31no—
177#476—31no—
178#476—31no—
179#476—31no—
180#474—61no—
181#474—61no—
182#466—61no—
183#465—61no—
184#463—61no—
185#460—31no—
186#459—31no—
187#446—31no—
188C8×Q8—31no—
189#436—31no—
190#435—61no—
191C8×D4—31no—
192C8×D4—61no—
193#433—61no—
194#432C2×D311yescharacteristic
195#431C2×D311yescharacteristic
196#424—61no—
197C2×C4⋊C8—31no—
198C2×C4⋊C8C2×D311yescharacteristic
199#335C2×D311yescharacteristic
200#335C2×D311yescharacteristic
201#334C2×D311yescharacteristic
202#334C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#333—31no—
208#333—31no—
209#331—31no—
210#331—31no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4111yescharacteristic
214C2×#67—21no—
215C2×#67—41no—
216C22×Dic3—41no—
217#235—82no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12C2×D411yescharacteristic
222C2×C4×D3—21no—
223C2×C4×D3—41no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—41no—
228C2×Dic6—82no—
229C2×Dic6C2×D411yescharacteristic
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—81no—
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42#4111yescharacteristic
242C3×C42C2411yescharacteristic
243#217—21no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—41no—
249(C2×D3)⋊C4—41no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—41no—
257C4⋊Dic3—41no—
258C4⋊Dic3C2×D411yescharacteristic
259C4⋊Dic3C2×D411yescharacteristic
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263Dic3⋊C4—21no—
264C4×Dic3—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—81no—
268D24—81no—
269#204—81no—
270ΓL(1,𝔽25)—81no—
271D3×C8—81no—
272C2×#41—61no—
273C22×Q8—61no—
274C2×SD16—61no—
275C2×D8—61no—
276C2×M16—61no—
277C22×C8—21no—
278C22×C8—61no—
279C22×C8—61no—
280C4⋊Q8—31no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283C4⋊Q8—61no—
284C4⋊Q8—61no—
285C4⋊Q8—61no—
286#127—31no—
287#126—61no—
288#125—31no—
289#124—31no—
290#124—61no—
291#123—61no—
292C22⋊Q8—31no—
293#121—31no—
294#121—61no—
295#121—61no—
296C4×Q8—31no—
297C4×D4—31no—
298C4×D4—61no—
299C4×D4—61no—
300C4×D4—61no—
301#117—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C42C22×D311yescharacteristic
305#107—61no—
306#107—61no—
307#107—61no—
308#107—61no—
309#107—61no—
310#107—61no—
311#107—61no—
312#106—61no—
313#106—61no—
314#106—61no—
315C4⋊C8—21no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8—61no—
321C4⋊C8D1211yescharacteristic
322C4⋊C8D1211yescharacteristic
323C4⋊C8C22×D311yescharacteristic
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—61no—
328#103—61no—
329#103—61no—
330#102—61no—
331#102—61no—
332#102—61no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—21no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8D1211yescharacteristic
343C22⋊C8D1211yescharacteristic
344#97—21no—
345#97—61no—
346C4×C8—61no—
347C4×C8C22×D311yescharacteristic
348C4×C8C22×D311yescharacteristic
349C23×C3#12111yescharacteristic
350C22×D3—21no—
351C22×D3—41no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4#12111yescharacteristic
358C2×C3×C4#12111yescharacteristic
359C2×C3×C4#12111yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363#67—42no—
364#67—82no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—82no—
372D12—21no—
373D12—21no—
374D12—41no—
375D12—41no—
376D12—41no—
377D12—41no—
378D12—81no—
379C4×D3—41no—
380C4×D3—41no—
381C4×D3—82no—
382Dic6—21no—
383Dic6—21no—
384Dic6—82no—
385Dic6—81no—
386Dic6—81no—
387Dic6—81no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—242no—
399C2×Q8—31no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—121no—
404C2×Q8—242no—
405C2×D4—31no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—121no—
411C22×C4—61no—
412C22×C4—121no—
413C22×C4—121no—
414C22×C4C2×D1211yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4#23911yescharacteristic
417SD16—241no—
418D8—241no—
419M16—241no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—81no—
427C2×C8—241no—
428C2×C8—241no—
429C2×C8C2×D1211yescharacteristic
430C2×C8C2×D1211yescharacteristic
431C2×C8C2×D1211yescharacteristic
432C2×C8C2×D1211yescharacteristic
433C4⋊C4—31no—
434C4⋊C4—31no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—121no—
446C4⋊C4—121no—
447C22⋊C4—61no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—61no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C42—61no—
455C42C2×D1211yescharacteristic
456C42D3×D411yescharacteristic
457C42#23911yescharacteristic
458C42C23×D311yescharacteristic
459C22×C3—21no—
460C22×C3—21no—
461C22×C3#47212yes—
462C22×C3C2×#12111yescharacteristic
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—41no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4#57711yescharacteristic
474C3×C4#57711yescharacteristic
475C3×C4#57811yescharacteristic
476C3×C4#57811yescharacteristic
477Dic3—41no—
478Dic3—41no—
479Dic3—41no—
480Dic3—81no—
481Dic3—81no—
482Dic3—81no—
483C23—61no—
484C23—121no—
485C23#89511yescharacteristic
486Q8—61no—
487Q8—61no—
488Q8—241no—
489Q8—241no—
490Q8—241no—
491Q8—242no—
492D4—61no—
493D4—61no—
494D4—122no—
495D4—121no—
496D4—121no—
497D4—121no—
498D4—121no—
499D4—241no—
500D4—242no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—242no—
515C2×C4—242no—
516C2×C4#89511yescharacteristic
517C2×C4#89511yescharacteristic
518C2×C4#89511yescharacteristic
519C2×C4C22×D1211yescharacteristic
520C2×C4C2×D3×D411yescharacteristic
521C2×C4C2×#23911yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—121no—
531C8—121no—
532C2×C3—22no—
533C2×C3#304611yescharacteristic
534C2×C3#306311yescharacteristic
535C2×C3C2×#47211yescharacteristic
536D3—81no—
537D3—161no—
538C22—21no—
539C22—21no—
540C22—121no—
541C22—121no—
542C22—121no—
543C22—241no—
544C22—241no—
545C22—241no—
546C22#489712yes—
547C22C2×#89511yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—121no—
553C4—121no—
554C4—121no—
555C4—241no—
556C4—241no—
557C4—241no—
558C4#575111yescharacteristic
559C4#575311yescharacteristic
560C4#577511yescharacteristic
561C4#577911yescharacteristic
562C3#1959111yescharacteristic, complemented, Sylow
563C2—22no—
564C2—241no—
565C2—481no—
566C2#7908911yescharacteristic
567C2#7921411yescharacteristic
568C2C2×#489711yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1573C2
socleC22×C3C2×#121
prime radical#79028C2
hypercenter#1573D3
hypocenterC3#19591
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1573

Complements

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

NH ≅ G/Ntype
1. #79322C2semidirect
2. #79296C2semidirect
3. #79268C2semidirect
5. C3×#1573C2semidirect
6. #70816C2semidirect
7. C2×#4481C2semidirect
8. #68594C2semidirect
9. #68589C2semidirect
10. #68584C2semidirect
11. #68579C2semidirect
12. #68553C2semidirect
13. #68548C2semidirect
14. #68543C2semidirect
15. #68538C2semidirect
80. #1573D3semidirect
562. C3#19591semidirect

Maximal subgroups

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

subgroupindex
#793222
#792962
#792682
#790282
C3×#15732
#708162
C2×#44812
#685942
#685892
#685842
#685792
#685532
#685482
#685432
#685382
#195913