Finite Group Databasebeta

← #10829869

Subgroups — #10829869

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85819C211yesmaximal, characteristic
2C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#82441C211yesmaximal, characteristic
4#82431C211yesmaximal, characteristic, complemented
5#71795C212yesmaximal, complemented
6#71637C211yesmaximal, characteristic, complemented
7#69695C212yesmaximal, complemented
8#69693C212yesmaximal, complemented
9#69592C212yesmaximal, complemented
10#69396C211yesmaximal, characteristic, complemented
11#69395C211yesmaximal, characteristic
12#24179—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×#389—22no—
32C3×#384C2211yescharacteristic
33C3×C4×C4⋊C4—22no—
34C3×C43C2211yescharacteristic
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—
55C4×C4⋊Q8D311yescharacteristic, complemented
56#2038—32no—
57#2005—32no—
58#1994—31no—
59#1962—31no—
60#1882—32no—
61#1859—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—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C42—21no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42D412yes—
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#947—22no—
90#946—22no—
91#946—22no—
92#944—22no—
93C2×#216—22no—
94#942—24no—
95#942—42no—
96C2×#215—22no—
97#930—21no—
98#930—21no—
99#929—24no—
100C4×#67—24no—
101#926C2312yes—
102#924—21no—
103#924—21no—
104C2×#208C2312yes—
105C2×#208C2312yes—
106#898—24no—
107#896—24no—
108#894—24no—
109#893—22no—
110#892—22no—
111D3×C4⋊C4—22no—
112#889—24no—
113#887—22no—
114#876—24no—
115#875—22no—
116#875D412yes—
117#874D411yescharacteristic
118C4×D12—22no—
119C4×D12—21no—
120C4×D12—21no—
121#872—24no—
122#870—22no—
123#870D412yes—
124#869D411yescharacteristic
125#869D411yescharacteristic
126#869D411yescharacteristic
127C4×Dic6—22no—
128C4×Dic6—21no—
129C4×Dic6—21no—
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#389—22no—
178#384C2×D311yescharacteristic
179C4×C4⋊C4—22no—
180C43C2×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×D411yescharacteristic
202C22×C3×C4C2×D412yes—
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—44no—
226C3×C4⋊C4—41no—
227C3×C4⋊C4—41no—
228C3×C42—24no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—42no—
235C3×C42—41no—
236C3×C42—41no—
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42#4112yes—
242#217—22no—
243#216—44no—
244#215—44no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—22no—
249(C2×D3)⋊C4—22no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3C2×D412yes—
255C4⋊Dic3C2×D412yes—
256Dic3⋊C4—24no—
257Dic3⋊C4—24no—
258Dic3⋊C4—24no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261C4×Dic3—24no—
262C4×Dic3—22no—
263#208—22no—
264#208—22no—
265#208—22no—
266#208—22no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269C2×#41—32no—
270C22×Q8C22×D311yescharacteristic
271C22×Q8C22×D311yescharacteristic
272#137—64no—
273#137—122no—
274C2×Q16—62no—
275C2×SD16—62no—
276C2×M16—32no—
277C2×M16—32no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—31no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283C4⋊Q8—62no—
284#127—31no—
285#126—64no—
286#126—64no—
287#125—32no—
288#125—62no—
289#125—64no—
290#124—32no—
291#124—62no—
292#124—62no—
293#123—64no—
294#123—64no—
295C22⋊Q8—64no—
296C22⋊Q8—62no—
297C22⋊Q8—62no—
298C22⋊Q8—61no—
299C22⋊Q8—61no—
300#121—61no—
301#121—61no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×Q8—61no—
313C4×Q8—61no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—64no—
317C4×D4—61no—
318C4×D4—61no—
319#117—32no—
320#117—64no—
321#117—62no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—62no—
326C2×C4⋊C4#6712yes—
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42#6712yes—
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335C2×C42C22×D311yescharacteristic
336#107—62no—
337#106—62no—
338C4⋊C8—61no—
339C4⋊C8—61no—
340C4⋊C8—61no—
341C4⋊C8—61no—
342#104—64no—
343#104—122no—
344#103—62no—
345#103—62no—
346#103—62no—
347#102—62no—
348#101—122no—
349#95—22no—
350#95—22no—
351C23×C3C22×D411yescharacteristic
352C22×D3—22no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—41no—
379C2×C3×C4#12011yescharacteristic
380C2×C3×C4#12011yescharacteristic
381C2×C3×C4#12011yescharacteristic
382C2×C3×C4#12011yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389#67—42no—
390#67—42no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394D12—22no—
395D12—22no—
396D12—42no—
397D12—81no—
398C4×D3—42no—
399C4×D3—42no—
400C4×D3—84no—
401Dic6—22no—
402Dic6—22no—
403Dic6—42no—
404Dic6—81no—
405C3⋊C8—42no—
406C3⋊C8—42no—
407#41—62no—
408#41—62no—
409#41—62no—
410#41—62no—
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—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—32no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8#23512yes—
426C2×D4—32no—
427C2×D4—62no—
428C2×D4—61no—
429C2×D4—61no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—64no—
435C22×C4—62no—
436C22×C4C2×#6712yes—
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6712yes—
439C22×C4C2×#6711yescharacteristic
440C22×C4C23×D311yescharacteristic
441Q16—124no—
442SD16—124no—
443M16—62no—
444M16—62no—
445M16—62no—
446M16—62no—
447C2×C8—62no—
448C2×C8—62no—
449C4⋊C4—28no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—32no—
457C4⋊C4—32no—
458C4⋊C4—44no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—64no—
462C4⋊C4—64no—
463C4⋊C4—64no—
464C4⋊C4—62no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—62no—
469C4⋊C4—61no—
470C22⋊C4—64no—
471C22⋊C4—64no—
472C22⋊C4—64no—
473C22⋊C4—62no—
474C22⋊C4—62no—
475C22⋊C4—62no—
476C42—24no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—42no—
483C42—41no—
484C42—41no—
485C42—64no—
486C42—62no—
487C42#23512yes—
488C42D3×D411yescharacteristic
489C42D3×D411yescharacteristic
490C42D3×D411yescharacteristic
491C42D3×D411yescharacteristic
492C22×C3—21no—
493C22×C3—21no—
494C22×C3C2×#12011yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#53711yescharacteristic
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C3×C4—24no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—42no—
507C3×C4—42no—
508C3×C4—81no—
509C3×C4#45612yes—
510C3×C4#54611yescharacteristic
511C3×C4#54711yescharacteristic
512C3×C4#54811yescharacteristic
513C3×C4#55011yescharacteristic
514Dic3—42no—
515Dic3—42no—
516Dic3—42no—
517C23—62no—
518C23C22×#6711yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—62no—
528Q8—62no—
529Q8—122no—
530Q8—241no—
531D4—62no—
532D4—62no—
533D4—122no—
534D4—122no—
535D4—122no—
536D4—241no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—244no—
561C2×C4#93711yescharacteristic
562C2×C4#93711yescharacteristic
563C2×C4#93711yescharacteristic
564C2×C4#93711yescharacteristic
565C2×C4C2×#23511yescharacteristic
566C2×C4C2×#23511yescharacteristic
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4#100911yescharacteristic
570C2×C4#101011yescharacteristic
571C8—122no—
572C8—122no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3#240711yescharacteristic
576C2×C3C2×#45611yescharacteristic
577C2×C3#301811yescharacteristic
578D3—82no—
579C22—21no—
580C22—21no—
581C22—122no—
582C22—122no—
583C22—122no—
584C22#579111yescharacteristic
585C22C2×#93711yescharacteristic
586C22#583311yescharacteristic
587C4—24no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—81no—
596C4—122no—
597C4—122no—
598C4—122no—
599C4#509512yes—
600C4#555011yescharacteristic
601C4#555111yescharacteristic
602C4#555211yescharacteristic
603C4#555311yescharacteristic
604C3#2417911yescharacteristic, complemented, Sylow
605C2—21no—
606C2—21no—
607C2—242no—
608C2#8240211yescharacteristic
609C2C2×#509511yescharacteristic
610C2#8332211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊Q8C2
socleC22×C3C2×#120
prime radical#5477C22
hypercenterC4×C4⋊Q8D3
hypocenterC3#24179
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×C4×C4⋊Q8C2semidirect
4. #82431C2semidirect
5. #71795C2semidirect
6. #71637C2semidirect
7. #69695C2semidirect
8. #69693C2semidirect
9. #69592C2semidirect
10. #69396C2semidirect
55. C4×C4⋊Q8D3semidirect
604. C3#24179semidirect

Maximal subgroups

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

subgroupindex
#858192
C3×C4×C4⋊Q82
#824412
#824312
#717952
#716372
#696952
#696932
#695922
#693962
#693952
#241793