Finite Group Databasebeta

← #10829913

Subgroups — #10829913

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85819C211yesmaximal, characteristic
2C3×#2327C211yesmaximal, characteristic, complemented
3#82441C211yesmaximal, characteristic
4#82431C211yesmaximal, characteristic, complemented
5#71795C212yesmaximal, complemented
6#71637C211yesmaximal, characteristic, complemented
7#69695C212yesmaximal, complemented
8#69693C212yesmaximal, complemented
9#69562C212yesmaximal, complemented
10#69441C211yesmaximal, characteristic, complemented
11#69440C211yesmaximal, characteristic
12#24183—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16#5816C2212yes—
17C2×#942C2212yes—
18#5585—21no—
19#5584—21no—
20#5581—22no—
21#5579—21no—
22#5568—22no—
23#5567—21no—
24#5515—22no—
25#5504C2212yes—
26#5479—22no—
27#5477C2211yescharacteristic
28#5470C2211yescharacteristic
29C3×#391C2211yescharacteristic
30C3×#391C2211yescharacteristic
31C3×#387—22no—
32C3×#387—22no—
33C3×#382C2211yescharacteristic
34C3×#376C2211yescharacteristic
35C2×#836C2212yes—
36#5225C2212yes—
37#5031—21no—
38#5028—22no—
39#5027—21no—
40#5022—22no—
41#5021—21no—
42#5020—21no—
43#4966C2212yes—
44#4964C2212yes—
45C2×#805C2212yes—
46C2×#805C2212yes—
47#4924C2211yescharacteristic
48#4924C2211yescharacteristic
49#4530C2212yes—
50#4523C2212yes—
51#4523C2212yes—
52#3507—31no—
53#3078—31no—
54#3037—31no—
55#2327D311yescharacteristic, complemented
56#2038—32no—
57#2005—32no—
58#1995—31no—
59#1963—31no—
60#1882—32no—
61#1860—32no—
62#1801—31no—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×Q8C2311yescharacteristic
65C2×#235C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C3×C4×Q8—21no—
71C3×C4×Q8—21no—
72C3×C4×Q8—21no—
73C3×C4×Q8—21no—
74C3×C4×Q8—21no—
75C3×C4×Q8—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4D412yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C42—21no—
82C2×C3×C42D412yes—
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86#947—22no—
87#946—22no—
88#946—22no—
89#944—22no—
90C2×#216—22no—
91#942—24no—
92#942—42no—
93C2×#215—22no—
94#930—21no—
95#930—21no—
96#929—24no—
97C4×#67—24no—
98#926C2312yes—
99#924—21no—
100#924—21no—
101C2×#208C2312yes—
102C2×#208C2312yes—
103#898—24no—
104#896—24no—
105#894—24no—
106#893—22no—
107#892—22no—
108D3×C4⋊C4—22no—
109#889—24no—
110#887—22no—
111#876—24no—
112#875—22no—
113#875D412yes—
114#874D411yescharacteristic
115C4×D12—22no—
116C4×D12—21no—
117C4×D12—21no—
118#872—24no—
119#870—22no—
120#870D412yes—
121#869D411yescharacteristic
122#869D411yescharacteristic
123#869D411yescharacteristic
124C4×Dic6—22no—
125C4×Dic6—21no—
126C4×Dic6—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—22no—
130C3×#95—22no—
131C3×#95—22no—
132#836—42no—
133#835—22no—
134#835—22no—
135#810—22no—
136#809—22no—
137#808—22no—
138#807—22no—
139#805—24no—
140#805—42no—
141C4⋊(C3⋊C8)—21no—
142C4⋊(C3⋊C8)—21no—
143C4⋊(C3⋊C8)—21no—
144C4⋊(C3⋊C8)—21no—
145C2×#137—32no—
146#559—61no—
147#557—61no—
148#556—62no—
149#556—61no—
150#554—61no—
151#552—62no—
152#539—62no—
153#536—32no—
154#533—31no—
155#532—31no—
156C2×C4⋊Q8C2×D311yescharacteristic
157#529—62no—
158#520—32no—
159C2×C4×Q8C2×D311yescharacteristic
160C2×C4×Q8C2×D311yescharacteristic
161#479—62no—
162#477—61no—
163#475—61no—
164#464—62no—
165#462—61no—
166#460—61no—
167#428—32no—
168#423—31no—
169#423—31no—
170C2×#104—32no—
171C2×#104—32no—
172#419—32no—
173#417—32no—
174C2×#101—32no—
175#391C2×D311yescharacteristic
176#391C2×D311yescharacteristic
177#387—22no—
178#387—22no—
179#382C2×D311yescharacteristic
180#376C2×D311yescharacteristic
181#343—32no—
182#339—32no—
183#339—32no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8#4112yes—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C2×#67—22no—
206#235—22no—
207#235—22no—
208#235—22no—
209#235—22no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12C2×D412yes—
213C2×C4×D3—22no—
214C2×C4×D3—24no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6C2×D412yes—
218C3×C4⋊C4—28no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—41no—
226C3×C4⋊C4—41no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—41no—
233C3×C42—41no—
234C3×C42C2×D411yescharacteristic
235C3×C42C2×D411yescharacteristic
236C3×C42C2×D411yescharacteristic
237C3×C42C2×D411yescharacteristic
238C3×C42#4112yes—
239#217—22no—
240#216—44no—
241#215—44no—
242(C2×D3)⋊C4—24no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—22no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3C2×D412yes—
252C4⋊Dic3C2×D412yes—
253Dic3⋊C4—24no—
254Dic3⋊C4—24no—
255Dic3⋊C4—24no—
256Dic3⋊C4—22no—
257Dic3⋊C4—22no—
258C4×Dic3—24no—
259C4×Dic3—22no—
260#208—22no—
261#208—22no—
262#208—22no—
263#208—22no—
264C2×C3⋊C8—22no—
265C2×C3⋊C8—22no—
266C2×#41—32no—
267C22×Q8C22×D311yescharacteristic
268C22×Q8C22×D311yescharacteristic
269#137—64no—
270#137—122no—
271C2×Q16—62no—
272C2×SD16—62no—
273C2×M16—32no—
274C2×M16—32no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—31no—
278C4⋊Q8—31no—
279C4⋊Q8—31no—
280C4⋊Q8—62no—
281#127—31no—
282#126—64no—
283#126—64no—
284#125—32no—
285#125—62no—
286#125—64no—
287#124—32no—
288#124—62no—
289#124—62no—
290#123—64no—
291#123—64no—
292C22⋊Q8—64no—
293C22⋊Q8—62no—
294C22⋊Q8—62no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297#121—61no—
298#121—61no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—61no—
310C4×Q8—61no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—64no—
314C4×D4—61no—
315C4×D4—61no—
316#117—32no—
317#117—64no—
318#117—62no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—62no—
322C2×C4⋊C4#6712yes—
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C42—21no—
326C2×C42#6712yes—
327C2×C42C22×D311yescharacteristic
328C2×C42C22×D311yescharacteristic
329C2×C42C22×D311yescharacteristic
330#107—62no—
331#106—62no—
332C4⋊C8—61no—
333C4⋊C8—61no—
334C4⋊C8—61no—
335C4⋊C8—61no—
336#104—64no—
337#104—122no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—62no—
342#101—122no—
343#95—21no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—22no—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4#12011yescharacteristic
377C2×C3×C4#12011yescharacteristic
378C2×C3×C4#12011yescharacteristic
379C2×C3×C4#12011yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4H(5,𝔽2)11yescharacteristic
385C2×C3×C42−1+411yescharacteristic
386#67—42no—
387#67—42no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391D12—22no—
392D12—22no—
393D12—42no—
394D12—81no—
395C4×D3—42no—
396C4×D3—42no—
397C4×D3—84no—
398Dic6—22no—
399Dic6—22no—
400Dic6—42no—
401Dic6—81no—
402C3⋊C8—42no—
403C3⋊C8—42no—
404#41—62no—
405#41—62no—
406#41—62no—
407#41—62no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—32no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8#23512yes—
423C2×D4—32no—
424C2×D4—62no—
425C2×D4—61no—
426C2×D4—61no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—64no—
432C22×C4—62no—
433C22×C4C2×#6712yes—
434C22×C4C2×#6712yes—
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C23×D311yescharacteristic
438Q16—124no—
439SD16—124no—
440M16—62no—
441M16—62no—
442M16—62no—
443M16—62no—
444C2×C8—62no—
445C2×C8—62no—
446C4⋊C4—28no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—64no—
458C4⋊C4—64no—
459C4⋊C4—64no—
460C4⋊C4—62no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—62no—
465C4⋊C4—61no—
466C22⋊C4—64no—
467C22⋊C4—64no—
468C22⋊C4—64no—
469C22⋊C4—62no—
470C22⋊C4—62no—
471C22⋊C4—62no—
472C42—22no—
473C42—22no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—41no—
478C42—41no—
479C42—64no—
480C42—62no—
481C42#23512yes—
482C42D3×D411yescharacteristic
483C42D3×D411yescharacteristic
484C42D3×D411yescharacteristic
485C42D3×D411yescharacteristic
486C22×C3—21no—
487C22×C3—21no—
488C22×C3C2×#12011yescharacteristic
489C22×C3#52511yescharacteristic
490C22×C3#53711yescharacteristic
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4#54611yescharacteristic
505C3×C4#54711yescharacteristic
506C3×C4#54811yescharacteristic
507C3×C4#55011yescharacteristic
508Dic3—42no—
509Dic3—42no—
510Dic3—42no—
511C23—62no—
512C23C22×#6711yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—62no—
522Q8—62no—
523Q8—122no—
524Q8—241no—
525D4—62no—
526D4—62no—
527D4—122no—
528D4—122no—
529D4—122no—
530D4—241no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—244no—
555C2×C4#93711yescharacteristic
556C2×C4#93711yescharacteristic
557C2×C4#93711yescharacteristic
558C2×C4#93711yescharacteristic
559C2×C4C2×#23511yescharacteristic
560C2×C4C2×#23511yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4C2×D3×D411yescharacteristic
563C2×C4#100911yescharacteristic
564C2×C4#101011yescharacteristic
565C8—122no—
566C8—122no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3#240711yescharacteristic
570C2×C3#302011yescharacteristic
571C2×C3#302211yescharacteristic
572D3—82no—
573C22—21no—
574C22—21no—
575C22—122no—
576C22—122no—
577C22—122no—
578C22#579111yescharacteristic
579C22C2×#93711yescharacteristic
580C22#583311yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—122no—
592C4—122no—
593C4—122no—
594C4#555011yescharacteristic
595C4#555111yescharacteristic
596C4#555211yescharacteristic
597C4#555311yescharacteristic
598C3#2418311yescharacteristic, complemented, Sylow
599C2—21no—
600C2—21no—
601C2—242no—
602C2#8240211yescharacteristic
603C2#8341511yescharacteristic
604C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2327C2
socleC22×C3C2×#120
prime radical#5477C22
hypercenter#2327D3
hypocenterC3#24183
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
2. C3×#2327C2semidirect
4. #82431C2semidirect
5. #71795C2semidirect
6. #71637C2semidirect
7. #69695C2semidirect
8. #69693C2semidirect
9. #69562C2semidirect
10. #69441C2semidirect
55. #2327D3semidirect
598. C3#24183semidirect

Maximal subgroups

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

subgroupindex
#858192
C3×#23272
#824412
#824312
#717952
#716372
#696952
#696932
#695622
#694412
#694402
#241833