Finite Group Databasebeta

← #10695878

Subgroups — #10695878

Order 768.

826 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 87 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#82742C211yesmaximal, characteristic, complemented
2C2×#5046C211yesmaximal, characteristic
3C2×#4731C211yesmaximal, characteristic
4C3×#1472C211yesmaximal, characteristic, complemented
5#69988C211yesmaximal, characteristic
6#69972C211yesmaximal, characteristic, complemented
7#67680C211yesmaximal, characteristic, complemented
8#9990—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×#215—21no—
12C2×#924C2211yescharacteristic
13C22×#204—21no—
14C3×#407C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16#5228—21no—
17#5207C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#823—22no—
20C2×#821—21no—
21#5052—22no—
22#5050—21no—
23#5046—22no—
24#5046—22no—
25#5043—21no—
26#4971—21no—
27C2×#809—21no—
28#4731—22no—
29#4731—22no—
30C3×#324—22no—
31#4469—22no—
32#3195—31no—
33C2×#450—31no—
34C2×#450—31no—
35#1783—31no—
36#1472—31no—
37#1472D311yescharacteristic, complemented
38#1298—31no—
39C24×D3—21no—
40C22×C3×Q8D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×D12—21no—
43C22×D12—21no—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—44no—
48C22×D12D412yes—
49C22×Dic6D411yescharacteristic
50C2×C3×M16—21no—
51C22×C3×C8—21no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#216—22no—
61C2×#215—24no—
62C2×#215—44no—
63C2×C4⋊Dic3D411yescharacteristic
64#924—21no—
65#924—21no—
66#924—21no—
67#924—21no—
68C2×Dic3⋊C4—21no—
69C2×#208D412yes—
70C22×C3⋊C8—21no—
71C2×#204—24no—
72C2×#204—44no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D412yes—
77#835—22no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830D412yes—
82#823—44no—
83#821—24no—
84#810—22no—
85#809—24no—
86#808—22no—
87C23×D4—31no—
88C22×SD16—61no—
89C22×SD16—61no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92#483—61no—
93#483—61no—
94#470—62no—
95#451—61no—
96#451—61no—
97#450—62no—
98#450—62no—
99#450—62no—
100#450—62no—
101C2×#102—61no—
102C2×#102—61no—
103C2×#100—62no—
104#407—31no—
105#407C2×D311yescharacteristic
106C2×C22⋊C8—31no—
107C2×C22⋊C8C2×D311yescharacteristic
108#324—22no—
109#324—62no—
110C24×C3#4111yescharacteristic
111C23×D3—22no—
112C23×D3—44no—
113C23×D3—41no—
114C23×D3—41no—
115C23×D3—82no—
116C23×D3—82no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—42no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—21no—
131C2×D12—24no—
132C2×D12—24no—
133C2×D12—22no—
134C2×D12—22no—
135C2×D12—24no—
136C2×D12—44no—
137C2×D12—44no—
138C2×D12—44no—
139C2×D12—42no—
140C2×D12—42no—
141C2×D12—42no—
142C2×D12—41no—
143C2×D12—41no—
144C2×Dic6—21no—
145C2×Dic6—21no—
146C2×Dic6—42no—
147C3×M16—42no—
148C2×C3×C8—22no—
149C2×C3×C8—42no—
150C2×C3×C8—41no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157#217—41no—
158#217—41no—
159#216—82no—
160#215—48no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165#208—42no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—42no—
169#204—48no—
170C25—61no—
171C22×Q8—31no—
172C22×Q8D1211yescharacteristic
173C22×D4—32no—
174C22×D4—61no—
175C22×D4—61no—
176C22×D4—61no—
177C22×D4—61no—
178C22×D4—61no—
179C22×D4—124no—
180C23×C4C22×D311yescharacteristic
181C2×Q16—62no—
182C2×SD16—64no—
183C2×SD16—64no—
184C2×SD16—124no—
185C2×SD16—124no—
186C2×M16—21no—
187C2×M16—32no—
188C22×C8—21no—
189C22×C8—61no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4D1211yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—61no—
204#106—62no—
205#103—62no—
206#103—62no—
207#102—64no—
208#102—64no—
209#100—124no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—32no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8#6712yes—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3#12111yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3#12111yescharacteristic
224C22×D3—24no—
225C22×D3—44no—
226C22×D3—44no—
227C22×D3—42no—
228C22×D3—42no—
229C22×D3—82no—
230C22×D3—82no—
231C22×D3—816no—
232C22×D3—82no—
233C22×D3—82no—
234C22×D3—82no—
235C22×D3—82no—
236C3×Q8—42no—
237C3×Q8—42no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—81no—
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12711yescharacteristic
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—81no—
256D12—24no—
257D12—24no—
258D12—42no—
259D12—42no—
260D12—44no—
261D12—44no—
262Dic6—42no—
263Dic6—42no—
264C3×C8—42no—
265C3×C8—81no—
266C3⋊C8—42no—
267C3⋊C8—42no—
268C24—62no—
269C24—124no—
270C24—121no—
271C24—121no—
272C24—242no—
273C24—242no—
274C24#23511yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—42no—
278C2×Q8—61no—
279C2×Q8—61no—
280C2×Q8—122no—
281C2×D4—64no—
282C2×D4—64no—
283C2×D4—64no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—122no—
287C2×D4—122no—
288C2×D4—122no—
289C2×D4—124no—
290C2×D4—124no—
291C2×D4—124no—
292C2×D4—121no—
293C2×D4—121no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4C2×D1211yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302C22×C4D3×D411yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4C2×#6711yescharacteristic
305Q16—242no—
306SD16—128no—
307SD16—128no—
308M16—42no—
309M16—122no—
310C2×C8—22no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—62no—
314C2×C8—62no—
315C2×C8—122no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—121no—
323C4⋊C4—121no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—44no—
336C22×C3#39211yescharacteristic
337C22×C3#47112yes—
338C22×C3#47112yes—
339C2×D3—44no—
340C2×D3—88no—
341C2×D3—88no—
342C2×D3—84no—
343C2×D3—82no—
344C2×D3—82no—
345C2×D3—84no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—81no—
350C3×C4—81no—
351C3×C4B(4,𝔽2)12yes—
352Dic3—81no—
353Dic3—81no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—64no—
358C23—124no—
359C23—124no—
360C23—122no—
361C23—122no—
362C23—242no—
363C23—242no—
364C23—242no—
365C23—242no—
366C23—2416no—
367C23—242no—
368C23—242no—
369C23#88411yescharacteristic
370C23#88411yescharacteristic
371C23#93011yescharacteristic
372Q8—42no—
373Q8—42no—
374Q8—122no—
375Q8—122no—
376D4—64no—
377D4—64no—
378D4—122no—
379D4—124no—
380D4—122no—
381D4—124no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—122no—
395C2×C4—241no—
396C2×C4#88211yescharacteristic
397C2×C4#88211yescharacteristic
398C2×C4#93711yescharacteristic
399C2×C4#94011yescharacteristic
400C8—42no—
401C8—81no—
402C8—122no—
403C8—122no—
404C2×C3—22no—
405C2×C3—22no—
406C2×C3—41no—
407C2×C3#201911yescharacteristic
408C2×C3#201911yescharacteristic
409C2×C3#202111yescharacteristic
410D3—84no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—124no—
420C22—248no—
421C22—248no—
422C22—244no—
423C22—244no—
424C22—242no—
425C22—242no—
426C22#475412yes—
427C22#496011yescharacteristic
428C22#514112yes—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—81no—
433C4—81no—
434C4—241no—
435C4—241no—
436C4#474112yes—
437C3#999011yescharacteristic, complemented, Sylow
438C2—22no—
439C2—22no—
440C2—41no—
441C2—244no—
442C2#7000711yescharacteristic
443C2#7003711yescharacteristic
444C2#7011711yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1472C2
socleC22×C3#392
prime radicalC23×D12C22
hypercenter#1472D3
hypocenterC3#9990
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. #82742C2semidirect
4. C3×#1472C2semidirect
6. #69972C2semidirect
7. #67680C2semidirect
37. #1472D3semidirect
437. C3#9990semidirect

Maximal subgroups

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

subgroupindex
#827422
C2×#50462
C2×#47312
C3×#14722
#699882
#699722
#676802
#99903