Finite Group Databasebeta

← #10829992

Subgroups — #10829992

Order 768.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2C3×#2582C211yesmaximal, characteristic, complemented
3C2×#5027C211yesmaximal, characteristic
4C2×C4×#216C211yesmaximal, characteristic, complemented
5#71634C211yesmaximal, characteristic, complemented
6#69689C212yesmaximal, complemented
7#69687C212yesmaximal, complemented
8#69628C212yesmaximal, complemented
9#69625C212yesmaximal, complemented
10#69588C211yesmaximal, characteristic, complemented
11#69580C211yesmaximal, characteristic
12#24190—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#947—21no—
16C22×#216C2211yescharacteristic
17C22×#215C2212yes—
18#5585—24no—
19#5585—24no—
20C2×#894—22no—
21C2×#893—21no—
22C2×#875C2212yes—
23C2×#874C2211yescharacteristic
24C2×C4×D12C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26C3×#398—22no—
27C3×#391—21no—
28C3×#391C2212yes—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#384C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C4×#216—24no—
34#5027—24no—
35C2×#810C2211yescharacteristic
36C2×#810C2212yes—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#559—31no—
47C2×#460—31no—
48C2×C4×Q16—31no—
49#2582D311yescharacteristic, complemented
50#2042—32no—
51#2015—32no—
52#1984—31no—
53#1957—31no—
54#1877—32no—
55#1862—32no—
56#1803—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×D12—21no—
60C22×D12—21no—
61C22×D12C2312yes—
62C22×C4×D3—22no—
63C22×Dic6C2311yescharacteristic
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4D412yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C42—21no—
77C2×C3×C42C2312yes—
78C2×C3×C42C2311yescharacteristic
79#947—28no—
80C2×#216—22no—
81C2×#216—22no—
82C2×#216—22no—
83C2×#215—24no—
84C2×#215—24no—
85C2×#215—42no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—21no—
91C2×C4×Dic3—21no—
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94C22×C3⋊C8C2311yescharacteristic
95#894—216no—
96#893—28no—
97#875—24no—
98#875—24no—
99#874—22no—
100#874D414yes—
101C4×D12—24no—
102C4×D12—24no—
103C4×Dic6—22no—
104C4×Dic6D414yes—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—22no—
108#835—22no—
109#810—22no—
110#810—42no—
111#809—24no—
112#809—24no—
113#809—42no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116C4×C3⋊C8—22no—
117C22×Q16—31no—
118C22×SD16—32no—
119#559—64no—
120#559—64no—
121C2×#127—31no—
122C2×#124—32no—
123C2×#124—61no—
124C2×#123—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130#460—64no—
131C4×Q16—64no—
132C2×#107—31no—
133C2×C4⋊C8—31no—
134C2×#103—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×C4×C8—31no—
141#398—22no—
142#391—21no—
143#391C2×D312yes—
144#389—21no—
145#389C2×D312yes—
146#384C2×D311yescharacteristic
147#340—32no—
148#336—32no—
149#336—32no—
150C23×D3—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8#4112yes—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—21no—
169C22×Dic3—21no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—24no—
175C2×D12—22no—
176C2×D12—24no—
177C2×D12—42no—
178C2×D12—42no—
179C2×D12—81no—
180C2×D12—81no—
181C2×D12C2×D414yes—
182C2×C4×D3—28no—
183C2×C4×D3—82no—
184C2×C4×D3—82no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6C2×D412yes—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4—81no—
200C3×C4⋊C4—82no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—42no—
204C3×C42C2×D412yes—
205C3×C42C2×D412yes—
206#216—28no—
207#215—48no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—28no—
210(C2×D3)⋊C4—28no—
211C4⋊Dic3—21no—
212C4⋊Dic3C2×D412yes—
213Dic3⋊C4—24no—
214C4×Dic3—24no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8#4112yes—
224C22×Q8—31no—
225C22×Q8#6711yescharacteristic
226C22×Q8C22×D311yescharacteristic
227C22×D4—32no—
228C22×D4—61no—
229C22×D4—61no—
230C23×C4—62no—
231C2×Q16—62no—
232C2×Q16—62no—
233C2×Q16—62no—
234C2×SD16—64no—
235C2×SD16—64no—
236C2×SD16—122no—
237C22×C8—31no—
238C22×C8—32no—
239C22×C8—31no—
240#127—34no—
241#127—62no—
242#124—64no—
243#124—64no—
244#124—68no—
245#123—616no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—34no—
251C4×Q8—62no—
252C4×D4—64no—
253C4×D4—64no—
254C4×D4—68no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4#6711yescharacteristic
262C2×C4⋊C4#6712yes—
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C42—21no—
269C2×C42—61no—
270C2×C42C22×D312yes—
271C2×C42C22×D311yescharacteristic
272#107—62no—
273C4⋊C8—62no—
274#103—62no—
275#103—62no—
276#103—122no—
277#102—64no—
278#102—64no—
279#102—122no—
280C4×C8—62no—
281#95—22no—
282#95—22no—
283#95—22no—
284C23×C3C22×D411yescharacteristic
285C22×D3—24no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—41no—
295C3×Q8—41no—
296C3×Q8—44no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4#13512yes—
317C2×C3×C4#13712yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330D12—24no—
331D12—24no—
332D12—42no—
333D12—42no—
334D12—48no—
335D12—84no—
336C4×D3—88no—
337Dic6—22no—
338Dic6—22no—
339Dic6—41no—
340Dic6—41no—
341C3⋊C8—22no—
342C3⋊C8—41no—
343C3⋊C8—42no—
344C24—62no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—32no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×Q8#23912yes—
360C2×D4—34no—
361C2×D4—64no—
362C2×D4—64no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—241no—
371C2×D4—241no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—68no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—242no—
379C22×C4—242no—
380C22×C4C2×#6712yes—
381C22×C4C2×#6712yes—
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C23×D311yescharacteristic
385Q16—68no—
386SD16—128no—
387C2×C8—32no—
388C2×C8—61no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8—62no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—32no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—64no—
406C4⋊C4—61no—
407C4⋊C4—81no—
408C4⋊C4—82no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—68no—
412C42—22no—
413C42—22no—
414C42—42no—
415C42—64no—
416C42D3×D412yes—
417C42D3×D412yes—
418C22×C3C2×#12011yescharacteristic
419C22×C3#54811yescharacteristic
420C22×C3#55011yescharacteristic
421C22×C3C2×#13511yescharacteristic
422C22×C3C2×#13711yescharacteristic
423C22×C3#57611yescharacteristic
424C22×C3#57711yescharacteristic
425C2×D3—44no—
426C2×D3—44no—
427C2×D3—44no—
428C2×D3—82no—
429C2×D3—82no—
430C2×D3—82no—
431C2×D3—82no—
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—82no—
437C3×C4—81no—
438C3×C4#45214yes—
439C3×C4#52511yescharacteristic
440C3×C4#53711yescharacteristic
441C3×C4#54611yescharacteristic
442C3×C4#54711yescharacteristic
443Dic3—42no—
444Dic3—42no—
445C23—64no—
446C23—122no—
447C23—122no—
448C23—122no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23C22×#6711yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—44no—
456Q8—41no—
457Q8—41no—
458Q8—62no—
459Q8—62no—
460Q8—121no—
461Q8—121no—
462D4—64no—
463D4—64no—
464D4—128no—
465D4—122no—
466D4—122no—
467D4—244no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—248no—
492C2×C4#93712yes—
493C2×C4#93712yes—
494C2×C4#94212yes—
495C2×C4#95012yes—
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×D3×D411yescharacteristic
498C2×C4C2×#23911yescharacteristic
499C2×C4#100711yescharacteristic
500C2×C4D3×#4111yescharacteristic
501C2×C4#100911yescharacteristic
502C8—62no—
503C8—121no—
504C8—122no—
505C2×C3#240711yescharacteristic
506C2×C3C2×#45211yescharacteristic
507C2×C3#301211yescharacteristic
508C2×C3#336511yescharacteristic
509C2×C3#337511yescharacteristic
510C2×C3#339611yescharacteristic
511C2×C3#340011yescharacteristic
512D3—84no—
513C22—124no—
514C22—124no—
515C22—124no—
516C22—242no—
517C22—242no—
518C22—242no—
519C22—242no—
520C22C2×#93711yescharacteristic
521C22C2×#94211yescharacteristic
522C22#581711yescharacteristic
523C22C2×#95011yescharacteristic
524C22#583411yescharacteristic
525C22#583511yescharacteristic
526C22#583711yescharacteristic
527C4—22no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—82no—
532C4—81no—
533C4—122no—
534C4—122no—
535C4#504714yes—
536C4#562711yescharacteristic
537C4#562911yescharacteristic
538C4#563011yescharacteristic
539C4#563111yescharacteristic
540C3#2419011yescharacteristic, complemented, Sylow
541C2—244no—
542C2C2×#504711yescharacteristic
543C2#8281211yescharacteristic
544C2#8342411yescharacteristic
545C2#8368711yescharacteristic
546C2#8369711yescharacteristic
547C2#8370811yescharacteristic
548C2#8372011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2582C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2582D3
hypocenterC3#24190
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2582

Complements

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

NH ≅ G/Ntype
2. C3×#2582C2semidirect
4. C2×C4×#216C2semidirect
5. #71634C2semidirect
6. #69689C2semidirect
7. #69687C2semidirect
8. #69628C2semidirect
9. #69625C2semidirect
10. #69588C2semidirect
49. #2582D3semidirect
540. C3#24190semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C3×#25822
C2×#50272
C2×C4×#2162
#716342
#696892
#696872
#696282
#696252
#695882
#695802
#241903