Finite Group Databasebeta

← #10696003

Subgroups — #10696003

Order 768.

654 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 83 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#82998C211yesmaximal, characteristic
2#82949C211yesmaximal, characteristic, complemented
3#79344C211yesmaximal, characteristic, complemented
4C3×#1527C211yesmaximal, characteristic, complemented
5#69868C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#10008—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×#946—21no—
11C2×#938—21no—
12C2×C4×D12C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×C4×M16C2211yescharacteristic
16#5228—21no—
17#5218—21no—
18#5060—21no—
19#5055—21no—
20#5054—22no—
21#4960—21no—
22#4957—22no—
23#4949—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4926C2211yescharacteristic
26#4846—21no—
27#4841—21no—
28#4731—21no—
29#4722—21no—
30#4710—22no—
31C3×#329—22no—
32#4555—22no—
33#4459—22no—
34#3215—31no—
35#3110—31no—
36#2540—31no—
37#1986—31no—
38#1527D311yescharacteristic, complemented
39#1490—31no—
40#1285—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C22×#67—22no—
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C2×C3×M16—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#217—22no—
55#946—44no—
56C2×#216—22no—
57#938—44no—
58C2×#214—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—21no—
61#926—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3D411yescharacteristic
64#924—21no—
65#924—21no—
66C2×Dic3⋊C4—22no—
67C2×C4×Dic3D412yes—
68C2×#204—22no—
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×C22⋊C8—21no—
80C3×#97—21no—
81C3×#97D412yescomplemented
82C3×C4×C8—21no—
83#835—22no—
84#832—22no—
85C3⋊#95—21no—
86#830—21no—
87#821—22no—
88#818—22no—
89#816—22no—
90C4⋊(C3⋊C8)D412yescomplemented
91#803—21no—
92#533—31no—
93C2×#124C2×D311yescharacteristic
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#432—31no—
108#432C2×D311yescharacteristic
109C4×M16C2×D311yescharacteristic
110#399—61no—
111#392—61no—
112#390—62no—
113C4×C4⋊C4—31no—
114#339—62no—
115#329—22no—
116D4⋊C8—62no—
117C24×C3—21no—
118C23×D3—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—42no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—42no—
125C22×C3×C4—21no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C2×#67—44no—
130C2×#67—44no—
131C2×#67—82no—
132C22×Dic3—22no—
133C22×Dic3—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—42no—
137C2×C4×D3—42no—
138C2×C4×D3—42no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C2×Dic6—21no—
142C2×Dic6—21no—
143C3×M16—22no—
144C3×M16—41no—
145C2×C3×C8—22no—
146C2×C3×C8—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155#217—44no—
156#217—41no—
157#216—82no—
158#214—82no—
159(C2×D3)⋊C4—44no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—24no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169Dic3⋊C4—44no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172C4×Dic3—41no—
173C4×Dic3—42no—
174C2×C3⋊C8—22no—
175#204—44no—
176C22×Q8D1211yescharacteristic
177C22×D4—31no—
178C22×D4—62no—
179C22×D4D1211yescharacteristic
180C23×C4—61no—
181C2×Q16—62no—
182C2×SD16—62no—
183C2×SD16—62no—
184C2×M16—21no—
185C4⋊Q8—31no—
186C4⋊Q8—31no—
187#125—61no—
188#124—21no—
189#124—21no—
190#124—21no—
191#124—21no—
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—31no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—62no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—61no—
211C2×C42—32no—
212C2×C42C22×D311yescharacteristic
213#107—62no—
214C4⋊C8—21no—
215C4⋊C8—32no—
216#103—62no—
217#103—62no—
218#102—62no—
219#102—62no—
220C22⋊C8—21no—
221C22⋊C8—61no—
222#97—21no—
223#97—61no—
224#97#6712yescomplemented
225C4×C8—21no—
226#95—61no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3#12011yescharacteristic
233C22×D3—22no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3—41no—
240C3×Q8—42no—
241C3×Q8—81no—
242C3×D4—42no—
243C3×D4—81no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12711yescharacteristic
256#67—88no—
257C2×Dic3—24no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265D12—22no—
266D12—22no—
267D12—41no—
268D12—41no—
269C4×D3—82no—
270C4×D3—82no—
271Dic6—42no—
272Dic6—42no—
273C3×C8—42no—
274C3×C8—42no—
275C3⋊C8—42no—
276C24—21no—
277C24—61no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—42no—
281C2×Q8—61no—
282C2×Q8—61no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—42no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—122no—
289C2×D4—124no—
290C2×D4—124no—
291C2×D4—242no—
292C22×C4—21no—
293C22×C4—62no—
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—22no—
306M16—41no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—62no—
310C4⋊C4—64no—
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—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—124no—
325C22⋊C4—124no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C42—122no—
330C42—121no—
331C42#23511yescharacteristic
332C42D3×D411yescharacteristic
333C42D3×D411yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—82no—
338C22×C3—82no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3#39211yescharacteristic
343C22×C3#45511yescharacteristic
344C22×C3#45611yescharacteristic
345C2×D3—42no—
346C2×D3—42no—
347C2×D3—42no—
348C2×D3—81no—
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—81no—
358C3×C4#45312yes—
359Dic3—44no—
360Dic3—81no—
361Dic3—81no—
362Dic3—81no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—62no—
369C23—121no—
370C23—121no—
371C23—121no—
372C23—121no—
373C23—121no—
374C23—121no—
375C23#88211yescharacteristic
376Q8—42no—
377Q8—81no—
378Q8—122no—
379Q8—122no—
380D4—42no—
381D4—62no—
382D4—62no—
383D4—81no—
384D4—121no—
385D4—121no—
386D4—248no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—64no—
394C2×C4—122no—
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#88211yescharacteristic
404C2×C4#88411yescharacteristic
405C2×C4#88411yescharacteristic
406C2×C4#93011yescharacteristic
407C2×C4#93711yescharacteristic
408C2×C4#94011yescharacteristic
409C8—42no—
410C8—42no—
411C8—122no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—161no—
415C2×C3#200211yescharacteristic
416C2×C3#200711yescharacteristic
417C2×C3#201011yescharacteristic
418D3—82no—
419C22—21no—
420C22—21no—
421C22—82no—
422C22—82no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—122no—
427C22—122no—
428C22—122no—
429C22—241no—
430C22—241no—
431C22—241no—
432C22—241no—
433C22#482511yescharacteristic
434C22#482611yescharacteristic
435C22#496011yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—124no—
443C4—241no—
444C4—241no—
445C4—241no—
446C4#474712yes—
447C3#1000811yescharacteristic, complemented, Sylow
448C2—21no—
449C2—21no—
450C2—161no—
451C2—242no—
452C2#7008211yescharacteristic
453C2#7039311yescharacteristic
454C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1527C2
socleC22×C3#392
prime radical#82998C2
hypercenter#1527D3
hypocenterC3#10008
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
2. #82949C2semidirect
3. #79344C2semidirect
4. C3×#1527C2semidirect
5. #69868C2semidirect
6. #69855C2semidirect
7. #67749C2semidirect
38. #1527D3semidirect
74. #869D4semidirect
75. #869D4semidirect
81. C3×#97D4semidirect
90. C4⋊(C3⋊C8)D4semidirect
224. #97#67semidirect
447. C3#10008semidirect

Maximal subgroups

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

subgroupindex
#829982
#829492
#793442
C3×#15272
#698682
#698552
#677492
#100083