Finite Group Databasebeta

← #10782812

Subgroups — #10782812

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#83311C211yesmaximal, characteristic
2#83270C211yesmaximal, characteristic, complemented
3#79322C211yesmaximal, characteristic, complemented
4#79017C211yesmaximal, characteristic, complemented
5C3×#1551C211yesmaximal, characteristic, complemented
6#70797C211yesmaximal, characteristic, complemented
7C2×#4481C211yesmaximal, characteristic, complemented
8#68513C211yesmaximal, characteristic, complemented
9#68509C211yesmaximal, characteristic, complemented
10#68501C211yesmaximal, characteristic, complemented
11#68497C211yesmaximal, characteristic, complemented
12#68414C211yesmaximal, characteristic, complemented
13#68410C211yesmaximal, characteristic, complemented
14#68400C211yesmaximal, characteristic, complemented
15#68396C211yesmaximal, characteristic, complemented
16#19591—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18#5479C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#432C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22#5111—21no—
23C8×#67—21no—
24#5091C2211yescharacteristic
25#5084C2211yescharacteristic
26#5078C2211yescharacteristic
27#5078C2211yescharacteristic
28#5075C2211yescharacteristic
29#5072C2211yescharacteristic
30#5070C2211yescharacteristic
31#5070C2211yescharacteristic
32#5041—21no—
33#5038—21no—
34#5033—21no—
35#5033—21no—
36#5016—21no—
37#5009—21no—
38#5007—21no—
39#5007—21no—
40#4925C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42#4850C2211yescharacteristic
43#4850C2211yescharacteristic
44#4842C2211yescharacteristic
45#4842C2211yescharacteristic
46#4840—21no—
47#4839—21no—
48#4838C2211yescharacteristic
49#4834C2211yescharacteristic
50#4831C2211yescharacteristic
51#4830C2211yescharacteristic
52#4736—21no—
53#4721—21no—
54#4714—21no—
55#4702—21no—
56C3×#333C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#331C2211yescharacteristic
59C3×#331C2211yescharacteristic
60#4481C2211yescharacteristic
61#4481C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
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#1573—31no—
81#1551D311yescharacteristic, complemented
82C2×#333—31no—
83C2×C3×#41—21no—
84C22×Dic6—21no—
85C22×C3×C8—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×#127C2311yescharacteristic
88C3×#124—21no—
89C3×#121—21no—
90C3×#121—21no—
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#214—21no—
95C2×#213—21no—
96#930D411yescharacteristic
97#929—21no—
98C4×#67—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101#924D411yescharacteristic
102C22×C3⋊C8—21no—
103C2×ΓL(1,𝔽25)—21no—
104C2×D3×C8—21no—
105#876—21no—
106#875D411yescharacteristic
107C4×D12C2311yescharacteristic
108#872—21no—
109#870D411yescharacteristic
110#869—21no—
111#869—21no—
112#869—21no—
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115C4×Dic6C2311yescharacteristic
116C3×C4⋊C8—21no—
117C3×C4⋊C8C2311yescharacteristic
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C22⋊C8—21no—
122C3×#97—21no—
123#832—21no—
124#832—21no—
125#832—21no—
126#832—21no—
127#832—21no—
128#832—21no—
129#830D411yescharacteristic
130#830D411yescharacteristic
131#821—21no—
132#821—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135#818—21no—
136#818—21no—
137#817—21no—
138#817—21no—
139#816—21no—
140#816—21no—
141#815—21no—
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144C8×Dic3—21no—
145#810—21no—
146#809—21no—
147#808—21no—
148#807—21no—
149#807—21no—
150#807—21no—
151#807—21no—
152#807—21no—
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)D411yescharacteristic
155C4⋊(C3⋊C8)D411yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532C2×D311yescharacteristic
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#431—31no—
196#424—61no—
197C2×C4⋊C8—31no—
198C2×C4⋊C8C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#333C2×D311yescharacteristic
208#333C2×D311yescharacteristic
209#331C2×D311yescharacteristic
210#331C2×D311yescharacteristic
211C3×#41—82no—
212C2×C3×Q8—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—41no—
217C22×C3×C4—41no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4#4111yescharacteristic
221C2×#67—21no—
222C22×Dic3—41no—
223C2×D12C2×D411yescharacteristic
224C2×C4×D3—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—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C22⋊C4—41no—
238C3×C22⋊C4—41no—
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42#4111yescharacteristic
242C3×C42C2411yescharacteristic
243#217—21no—
244#214—81no—
245#213—81no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—41no—
255C4⋊Dic3—41no—
256C4⋊Dic3C2×D411yescharacteristic
257C4⋊Dic3C2×D411yescharacteristic
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262C4×Dic3—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—81no—
266C2×C3⋊C8C2×D411yescharacteristic
267C2×C3⋊C8C2×D411yescharacteristic
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270ΓL(1,𝔽25)—81no—
271D3×C8—81no—
272C2×#41—21no—
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—61no—
283C4⋊Q8—61no—
284C4⋊Q8—61no—
285C4⋊Q8C22×D311yescharacteristic
286#127C22×D311yescharacteristic
287#126—61no—
288#125—31no—
289#124—21no—
290#124—31no—
291#123—61no—
292C22⋊Q8—31no—
293#121—21no—
294#121—21no—
295#121—31no—
296C4×Q8—31no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—31no—
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—31no—
319C4⋊C8—31no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322C4⋊C8—61no—
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—31no—
341C22⋊C8—31no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—21no—
345#97—61no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—61no—
349C23×C3—41no—
350C23×C3#12111yescharacteristic
351C22×D3—21no—
352C3×Q8—81no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—41no—
356C3×D4—41no—
357C3×D4—81no—
358C3×D4—82no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—82no—
366C2×C3×C4#12111yescharacteristic
367C2×C3×C4#12111yescharacteristic
368C2×C3×C4#12111yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—42no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—82no—
379D12—21no—
380D12—21no—
381C4×D3—41no—
382C4×D3—41no—
383Dic6—21no—
384Dic6—21no—
385Dic6—82no—
386Dic6—81no—
387Dic6—81no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—82no—
399C2×Q8—21no—
400C2×Q8—31no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—121no—
404C2×Q8—242no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—31no—
409C2×D4—41no—
410C2×D4—61no—
411C22×C4—41no—
412C22×C4—61no—
413C22×C4—121no—
414C22×C4#23511yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4D3×D411yescharacteristic
417SD16—241no—
418D8—241no—
419M16—241no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—31no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—81no—
431C2×C8—241no—
432C2×C8—241no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—31no—
436C4⋊C4—31no—
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—41no—
448C22⋊C4—41no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—61no—
452C22⋊C4—61no—
453C22⋊C4—61no—
454C42—61no—
455C42#23511yescharacteristic
456C42D3×D411yescharacteristic
457C42D3×D411yescharacteristic
458C42C23×D311yescharacteristic
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3#47212yes—
465C22×C3C2×#12111yescharacteristic
466C2×D3—41no—
467C2×D3—41no—
468C2×D3—41no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—81no—
474C3×C4#57711yescharacteristic
475C3×C4#57711yescharacteristic
476C3×C4#57811yescharacteristic
477C3×C4#57811yescharacteristic
478Dic3—41no—
479Dic3—41no—
480Dic3—41no—
481Dic3—81no—
482Dic3—81no—
483C23—41no—
484C23—61no—
485C23#88411yescharacteristic
486Q8—61no—
487Q8—61no—
488Q8—81no—
489Q8—241no—
490Q8—241no—
491Q8—242no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—61no—
497D4—61no—
498D4—81no—
499D4—82no—
500D4—122no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—82no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—242no—
516C2×C4#88411yescharacteristic
517C2×C4#88411yescharacteristic
518C2×C4#88411yescharacteristic
519C2×C4C2×#23511yescharacteristic
520C2×C4C2×D3×D411yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—61no—
527C8—61no—
528C8—61no—
529C8—61no—
530C8—121no—
531C8—121no—
532C2×C3—22no—
533C2×C3—161no—
534C2×C3#304611yescharacteristic
535C2×C3#306311yescharacteristic
536C2×C3C2×#47211yescharacteristic
537D3—81no—
538C22—21no—
539C22—21no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—121no—
544C22—121no—
545C22—121no—
546C22#474912yes—
547C22C2×#88411yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—81no—
553C4—121no—
554C4—121no—
555C4—121no—
556C4—241no—
557C4—241no—
558C4#577011yescharacteristic
559C4#577111yescharacteristic
560C4#577511yescharacteristic
561C4#577911yescharacteristic
562C3#1959111yescharacteristic, complemented, Sylow
563C2—22no—
564C2—161no—
565C2—241no—
566C2C2×#474911yescharacteristic
567C2#7884111yescharacteristic
568C2#7891211yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1551C2
socleC22×C3C2×#121
prime radical#83311C2
hypercenter#1551D3
hypocenterC3#19591
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1551

Complements

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

NH ≅ G/Ntype
2. #83270C2semidirect
3. #79322C2semidirect
4. #79017C2semidirect
5. C3×#1551C2semidirect
6. #70797C2semidirect
7. C2×#4481C2semidirect
8. #68513C2semidirect
9. #68509C2semidirect
10. #68501C2semidirect
11. #68497C2semidirect
12. #68414C2semidirect
13. #68410C2semidirect
14. #68400C2semidirect
15. #68396C2semidirect
81. #1551D3semidirect
562. C3#19591semidirect

Maximal subgroups

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

subgroupindex
#833112
#832702
#793222
#790172
C3×#15512
#707972
C2×#44812
#685132
#685092
#685012
#684972
#684142
#684102
#684002
#683962
#195913