Finite Group Databasebeta

← #10696004

Subgroups — #10696004

Order 768.

654 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79351C211yesmaximal, characteristic, complemented
2#79344C211yesmaximal, characteristic, complemented
3#73798C211yesmaximal, characteristic
4#70804C211yesmaximal, characteristic, complemented
5#70799C211yesmaximal, characteristic, complemented
6#69493C211yesmaximal, characteristic, complemented
7C3×#1285C211yesmaximal, characteristic, complemented
8#10008—31nomaximal, Sylow
9C2×#897—21no—
10C2×#895—21no—
11C2×#875C2211yescharacteristic
12C2×C4×D12C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×#432C2211yescharacteristic
16C3×C4×C4⋊C4C2211yescharacteristic
17#4939—22no—
18C4×#208C2211yescharacteristic
19#4846—21no—
20#4845—22no—
21#4841—21no—
22#4739—21no—
23#4731—21no—
24#4731—21no—
25#4722—21no—
26#4715—21no—
27#4714—21no—
28#4710—22no—
29#4677—21no—
30#4671—21no—
31#4523—22no—
32#4483—22no—
33#4464—22no—
34#3215—31no—
35#3110—31no—
36#2540—31no—
37#1986—31no—
38#1527—31no—
39#1490—31no—
40#1285D311yescharacteristic, complemented
41C22×D12—22no—
42C22×D12D411yescharacteristic
43C22×D12D411yescharacteristic
44C22×C4×D3—21no—
45C22×Dic6D411yescharacteristic
46C2×C3×C4⋊C4—21no—
47C2×C3×C42D412yes—
48C2×C3×C42C2311yescharacteristic
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54#926—21no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3D411yescharacteristic
57#924—21no—
58#924—21no—
59C2×#208—21no—
60C2×Dic12—22no—
61C2×#204—22no—
62C2×#204—22no—
63#897—44no—
64#895—44no—
65#875—21no—
66#875—21no—
67#875—21no—
68#875—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73#870—21no—
74#869D411yescharacteristic, complemented
75#869D411yescharacteristic, complemented
76C4×Dic6—21no—
77C4×Dic6—21no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8D412yescomplemented
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×#97—21no—
83C3×#97D412yescomplemented
84C3×#95—21no—
85#821—22no—
86#821—22no—
87#818—22no—
88#816—22no—
89#816—22no—
90#803—21no—
91C4×C3⋊C8—21no—
92#533—31no—
93C2×#124—31no—
94C2×C22⋊Q8—61no—
95C2×#121—61no—
96C2×C4×D4—31no—
97#497—61no—
98#489—61no—
99#484—61no—
100#474—61no—
101#464—62no—
102#461—61no—
103#451—61no—
104#450—61no—
105#450—61no—
106#440—62no—
107#432C2×D311yescharacteristic
108#432C2×D311yescharacteristic
109C4×M16—31no—
110#399—61no—
111#392—61no—
112#390—62no—
113C4×C4⋊C4C2×D311yescharacteristic
114#339—62no—
115#329—62no—
116D4⋊C8—62no—
117C23×D3—21no—
118C23×D3—21no—
119C22×C3×C4—22no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—42no—
130C2×D12—42no—
131C2×D12—44no—
132C2×D12—44no—
133C2×D12—82no—
134C2×C4×D3—42no—
135C2×C4×D3—42no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×Dic6—42no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C3×C4⋊C4—24no—
146C3×C42—42no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#217—41no—
152(C2×D3)⋊C4—44no—
153(C2×D3)⋊C4—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—44no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169C4×Dic3—41no—
170#208—22no—
171#208—41no—
172C2×C3⋊C8—41no—
173Dic12—82no—
174#204—44no—
175#204—82no—
176C22×Q8—31no—
177C22×D4—31no—
178C22×D4—31no—
179C22×D4—62no—
180C23×C4—61no—
181C2×Q16—62no—
182C2×SD16—62no—
183C2×SD16—62no—
184C2×M16—61no—
185C4⋊Q8—31no—
186C4⋊Q8—31no—
187#125—61no—
188#124—61no—
189#124—61no—
190#124—61no—
191#124—61no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—124no—
195#121—124no—
196C4×Q8—61no—
197C4×Q8—61no—
198C4×D4—61no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202#117—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—62no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C42D1212yes—
212C2×C42C22×D311yescharacteristic
213#107—62no—
214C4⋊C8—21no—
215C4⋊C8D1212yescomplemented
216#103—62no—
217#103—62no—
218#102—62no—
219#102—62no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222#97—21no—
223#97—61no—
224#97D1212yescomplemented
225C4×C8—61no—
226#95—21no—
227C23×C3#12011yescharacteristic
228C22×D3—22no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—42no—
236C22×D3—42no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3—41no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—24no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12711yescharacteristic
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260D12—22no—
261D12—22no—
262D12—41no—
263D12—41no—
264D12—42no—
265D12—81no—
266D12—88no—
267C4×D3—82no—
268C4×D3—82no—
269Dic6—42no—
270Dic6—42no—
271Dic6—42no—
272Dic6—81no—
273C3×C8—42no—
274C3×C8—42no—
275C3⋊C8—42no—
276C24—61no—
277C24—61no—
278C2×Q8—61no—
279C2×Q8—61no—
280C2×Q8—61no—
281C2×Q8—61no—
282C2×Q8—122no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—122no—
288C2×D4—124no—
289C2×D4—122no—
290C2×D4—124no—
291C2×D4—242no—
292C22×C4—22no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—122no—
296C22×C4—122no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4C2×D1211yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302Q16—242no—
303SD16—124no—
304SD16—242no—
305M16—62no—
306M16—121no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—121no—
310C4⋊C4—24no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—61no—
314C4⋊C4—61no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—124no—
318C4⋊C4—121no—
319C4⋊C4—121no—
320C22⋊C4—124no—
321C22⋊C4—124no—
322C22⋊C4—121no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C42—42no—
330C42—121no—
331C42C2×D1211yescharacteristic
332C42C2×D1211yescharacteristic
333C42D3×D411yescharacteristic
334C42#23911yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3#39211yescharacteristic
338C22×C3#45511yescharacteristic
339C22×C3#45611yescharacteristic
340C2×D3—42no—
341C2×D3—42no—
342C2×D3—42no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C2×D3—81no—
347C2×D3—82no—
348C2×D3—82no—
349C2×D3—81no—
350C2×D3—81no—
351C2×D3—81no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—44no—
358C3×C4#45312yes—
359Dic3—81no—
360Dic3—81no—
361Dic3—81no—
362Dic3—81no—
363C23—62no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23—122no—
368C23—122no—
369C23—121no—
370C23—121no—
371C23—121no—
372C23—121no—
373C23—121no—
374C23—121no—
375C23#88211yescharacteristic
376Q8—122no—
377Q8—122no—
378Q8—122no—
379Q8—241no—
380D4—62no—
381D4—62no—
382D4—121no—
383D4—121no—
384D4—122no—
385D4—241no—
386D4—248no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—242no—
402C2×C4—242no—
403C2×C4#87411yescharacteristic
404C2×C4#88211yescharacteristic
405C2×C4#88211yescharacteristic
406C2×C4#89511yescharacteristic
407C2×C4#89511yescharacteristic
408C2×C4#89511yescharacteristic
409C8—42no—
410C8—42no—
411C8—122no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3#200211yescharacteristic
415C2×C3#200711yescharacteristic
416C2×C3#201011yescharacteristic
417D3—82no—
418D3—161no—
419C22—21no—
420C22—21no—
421C22—122no—
422C22—122no—
423C22—122no—
424C22—241no—
425C22—241no—
426C22—242no—
427C22—242no—
428C22—241no—
429C22—241no—
430C22—241no—
431C22—241no—
432C22—241no—
433C22#467111yescharacteristic
434C22#474611yescharacteristic
435C22#475211yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—44no—
442C4—241no—
443C4—241no—
444C4—241no—
445C4—241no—
446C4#482212yes—
447C3#1000811yescharacteristic, complemented, Sylow
448C2—21no—
449C2—21no—
450C2—242no—
451C2—481no—
452C2#6870311yescharacteristic
453C2#6876111yescharacteristic
454C2#6900911yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#392
prime radical#73798C2
hypercenter#1285D3
hypocenterC3#10008
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
1. #79351C2semidirect
2. #79344C2semidirect
4. #70804C2semidirect
5. #70799C2semidirect
6. #69493C2semidirect
7. C3×#1285C2semidirect
40. #1285D3semidirect
74. #869D4semidirect
75. #869D4semidirect
79. C3×C4⋊C8D4semidirect
83. C3×#97D4semidirect
215. C4⋊C8D12semidirect
224. #97D12semidirect
447. C3#10008semidirect

Maximal subgroups

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

subgroupindex
#793512
#793442
#737982
#708042
#707992
#694932
C3×#12852
#100083