Finite Group Databasebeta

← #10695798

Subgroups — #10695798

Order 768.

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

normal subgroups only

This group has 716 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#82741C211yesmaximal, characteristic, complemented
2#82735C211yesmaximal, characteristic
3#78640C211yesmaximal, characteristic
4C3×#1472C211yesmaximal, characteristic, complemented
5#69956C211yesmaximal, characteristic
6#69923C211yesmaximal, characteristic, complemented
7#67687C211yesmaximal, characteristic, complemented
8#9976—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#942—21no—
12C2×#924C2211yescharacteristic
13C2×#909—21no—
14C3×#407C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16#5228—21no—
17#5207C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#5133—21no—
20C2×(C2×D3)⋊C8—22no—
21#5051—22no—
22#5050—21no—
23#5050—21no—
24#5047—22no—
25#5046—21no—
26#5044—21no—
27#4970—21no—
28#4966—21no—
29#4739—21no—
30#4733—22no—
31#4731—21no—
32C3×#324—22no—
33#4469—22no—
34#3189—31no—
35#3007—31no—
36#3007—31no—
37#1780—31no—
38#1472—31no—
39#1472D311yescharacteristic, complemented
40#1293—31no—
41C22×C3×Q8D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—44no—
49C22×D12D411yescharacteristic
50C22×C4×D3—21no—
51C22×Dic6D411yescharacteristic
52C22×Dic6D411yescharacteristic
53C2×C3×M16—21no—
54C22×C3×C8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C22⋊C4—21no—
62C2×#217—21no—
63C2×#216—22no—
64C2×#216—22no—
65#942—44no—
66C2×#215—22no—
67C2×C4⋊Dic3D411yescharacteristic
68#924—21no—
69#924—21no—
70#924—21no—
71#924—21no—
72C2×Dic3⋊C4—21no—
73C2×#208—21no—
74C22×C3⋊C8D412yes—
75#909—44no—
76C2×Dic12—22no—
77C2×#204—22no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8D412yes—
82#835—22no—
83#830—21no—
84#830—21no—
85#830—21no—
86#830D412yes—
87#821—22no—
88(C2×D3)⋊C8—44no—
89#816—22no—
90#810—22no—
91#810—22no—
92#809—22no—
93#807—22no—
94C22×#41—31no—
95C2×#137—61no—
96C2×#137—61no—
97C2×C22⋊Q8—31no—
98C2×C22⋊Q8C2×D311yescharacteristic
99#484—61no—
100#484—61no—
101#467—62no—
102#452—62no—
103#452—62no—
104#451—61no—
105#451—61no—
106#451—61no—
107#451—61no—
108#450—61no—
109#450—61no—
110#417—61no—
111#417—61no—
112#407—31no—
113#407C2×D311yescharacteristic
114C2×C22⋊C8—31no—
115C2×C22⋊C8—62no—
116C2×C22⋊C8C2×D311yescharacteristic
117#324—22no—
118#324—62no—
119C24×C3#4111yescharacteristic
120C23×D3—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—42no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C2×#67—42no—
134C2×#67—42no—
135C2×#67—41no—
136C2×#67—41no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140#235—24no—
141#235—24no—
142#235—42no—
143#235—42no—
144#235—42no—
145#235—42no—
146#235—42no—
147#235—42no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—42no—
151C2×C4×D3—24no—
152C2×C4×D3—42no—
153C2×C4×D3—41no—
154C2×C4×D3—41no—
155C2×Dic6—21no—
156C2×Dic6—21no—
157C2×Dic6—21no—
158C2×Dic6—21no—
159C2×Dic6—42no—
160C2×Dic6—42no—
161C3×M16—42no—
162C2×C3×C8—22no—
163C2×C3×C8—42no—
164C2×C3×C8—41no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—41no—
169C3×C22⋊C4—41no—
170C3×C22⋊C4—41no—
171#217—41no—
172#217—41no—
173#216—44no—
174#216—82no—
175#215—44no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178Dic3⋊C4—41no—
179Dic3⋊C4—41no—
180#208—42no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—42no—
184Dic12—44no—
185#204—44no—
186C2×#41—61no—
187C2×#41—61no—
188C2×#41—61no—
189C2×#41—61no—
190C2×#41—124no—
191C22×Q8—31no—
192C22×Q8—31no—
193C22×Q8D1211yescharacteristic
194C22×D4—31no—
195C22×D4—61no—
196C23×C4—61no—
197C23×C4C22×D311yescharacteristic
198#137—124no—
199#137—124no—
200C2×Q16—62no—
201C2×Q16—62no—
202C2×Q16—62no—
203C2×SD16—62no—
204C2×SD16—62no—
205C2×M16—21no—
206C2×M16—61no—
207C22×C8—21no—
208C22×C8—32no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215C22⋊Q8—61no—
216C22⋊Q8—61no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4D1211yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—61no—
223#107—62no—
224#103—62no—
225#103—62no—
226#103—62no—
227#103—62no—
228#102—62no—
229#102—62no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—32no—
234C22⋊C8—61no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237C22⋊C8—124no—
238C22⋊C8#6712yes—
239C23×C3—22no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3#12111yescharacteristic
243C23×C3#12111yescharacteristic
244C23×C3#12111yescharacteristic
245C22×D3—22no—
246C22×D3—41no—
247C22×D3—42no—
248C22×D3—41no—
249C22×D3—42no—
250C3×Q8—42no—
251C3×Q8—42no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—81no—
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12711yescharacteristic
266#67—44no—
267#67—44no—
268C2×Dic3—22no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—42no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—81no—
275D12—22no—
276D12—22no—
277D12—41no—
278D12—41no—
279C4×D3—44no—
280C4×D3—44no—
281Dic6—22no—
282Dic6—22no—
283Dic6—41no—
284Dic6—41no—
285Dic6—42no—
286Dic6—42no—
287C3×C8—42no—
288C3×C8—81no—
289C3⋊C8—42no—
290C3⋊C8—42no—
291C24—61no—
292C24#23511yescharacteristic
293#41—64no—
294#41—64no—
295#41—122no—
296#41—122no—
297#41—122no—
298#41—122no—
299#41—122no—
300#41—122no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—42no—
304C2×Q8—61no—
305C2×Q8—61no—
306C2×Q8—61no—
307C2×Q8—61no—
308C2×Q8—122no—
309C2×Q8—122no—
310C2×D4—61no—
311C2×D4—61no—
312C2×D4—122no—
313C2×D4—122no—
314C2×D4—122no—
315C2×D4—121no—
316C2×D4—121no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—64no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—122no—
325C22×C4—121no—
326C22×C4—121no—
327C22×C4C2×D1211yescharacteristic
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4D3×D411yescharacteristic
331C22×C4D3×D411yescharacteristic
332C22×C4C2×#6711yescharacteristic
333Q16—124no—
334Q16—124no—
335Q16—242no—
336SD16—124no—
337SD16—124no—
338M16—42no—
339M16—122no—
340C2×C8—22no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—62no—
344C2×C8—62no—
345C2×C8—122no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—121no—
353C4⋊C4—121no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—121no—
357C22⋊C4—121no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—82no—
367C22×C3#39211yescharacteristic
368C22×C3#47312yes—
369C2×D3—44no—
370C2×D3—42no—
371C2×D3—82no—
372C2×D3—81no—
373C2×D3—81no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—81no—
378C3×C4—81no—
379C3×C4B(4,𝔽2)12yes—
380Dic3—42no—
381Dic3—81no—
382Dic3—81no—
383C23—22no—
384C23—41no—
385C23—41no—
386C23—62no—
387C23—121no—
388C23—122no—
389C23—121no—
390C23—122no—
391C23#88411yescharacteristic
392C23#88411yescharacteristic
393C23#93011yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—62no—
397Q8—62no—
398Q8—121no—
399Q8—121no—
400Q8—122no—
401Q8—122no—
402D4—62no—
403D4—62no—
404D4—121no—
405D4—121no—
406D4—124no—
407D4—124no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—62no—
418C2×C4—81no—
419C2×C4—124no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—124no—
423C2×C4—122no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—241no—
427C2×C4#88211yescharacteristic
428C2×C4#88211yescharacteristic
429C2×C4#93711yescharacteristic
430C2×C4#94011yescharacteristic
431C8—42no—
432C8—81no—
433C8—122no—
434C8—122no—
435C2×C3—22no—
436C2×C3—41no—
437C2×C3—41no—
438C2×C3#201711yescharacteristic
439C2×C3#202011yescharacteristic
440C2×C3#202111yescharacteristic
441D3—82no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—82no—
451C22—124no—
452C22—122no—
453C22—242no—
454C22—241no—
455C22—241no—
456C22#475512yes—
457C22#496011yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—81no—
462C4—81no—
463C4—122no—
464C4—241no—
465C4—241no—
466C4#474112yes—
467C3#997611yescharacteristic, complemented, Sylow
468C2—22no—
469C2—41no—
470C2—41no—
471C2—242no—
472C2#7001311yescharacteristic
473C2#7003711yescharacteristic
474C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1472C2
socleC22×C3#392
prime radicalC22×#235C22
hypercenter#1472D3
hypocenterC3#9976
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
1. #82741C2semidirect
4. C3×#1472C2semidirect
6. #69923C2semidirect
7. #67687C2semidirect
39. #1472D3semidirect
467. C3#9976semidirect

Maximal subgroups

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

subgroupindex
#827412
#827352
#786402
C3×#14722
#699562
#699232
#676872
#99763