Finite Group Databasebeta

← #10692213

Subgroups — #10692213

Order 768.

478 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 95 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#82750C211yesmaximal, characteristic
2#82742C211yesmaximal, characteristic
3C3×#1508C211yesmaximal, characteristic
4#69981C211yesmaximal, characteristic
5#69457C211yesmaximal, characteristic
6#67684C211yesmaximal, characteristic
7#67616C211yesmaximal, characteristic
8#9342—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5229—21no—
14#5228—21no—
15#5212C2211yescharacteristic
16#5211C2211yescharacteristic
17#5207C2211yescharacteristic
18#5052—22no—
19#5050C412yescomplemented
20#5043C412yescomplemented
21#4984—21no—
22#4980—21no—
23#4980—21no—
24#4971—21no—
25#4951—21no—
26#4951—21no—
27C3×#328—22no—
28#4527—21no—
29#4527—21no—
30#4467—21no—
31#4467—21no—
32#3195—31no—
33#2378—31no—
34#1996—31no—
35#1511—31no—
36#1508D311yescharacteristic
37#1305—31no—
38#1284—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×Dic6C2×C411yescharacteristic
42C2×C3×M16—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8C2×C412yes—
46C3×C22⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C42—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217D411yescharacteristic
55C2×#217D411yescharacteristic
56C2×#216—22no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3C2×C411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#208—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72#835—22no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95D411yescharacteristic
80C3⋊#95D411yescharacteristic
81#830—21no—
82#830C2×C412yes—
83#810—22no—
84#808—22no—
85C2×C22⋊Q8—31no—
86C2×C22⋊Q8C2×D311yescharacteristic
87#483—32no—
88#483—61no—
89#470—62no—
90#451—32no—
91#451—61no—
92#407—31no—
93#407C2×D311yescharacteristic
94#391—61no—
95#389—61no—
96#388—61no—
97#388—61no—
98#387—61no—
99#387—61no—
100#381—31no—
101#380—31no—
102C4×C22⋊C4C2×D311yescharacteristic
103#342—61no—
104#342—61no—
105#328—22no—
106#328—61no—
107#328—61no—
108C24×C3C2×D411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—81no—
111C2×C3×Q8C22⋊C412yes—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4—81no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—81no—
131C3×M16—81no—
132C2×C3×C8—41no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4C22⋊C412yes—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—41no—
141C3×C42—81no—
142#217—41no—
143#217—41no—
144#217—41no—
145#217—41no—
146#217—81no—
147#216—82no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—81no—
151C4⋊Dic3—81no—
152C4⋊Dic3—81no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—81no—
156C4×Dic3—81no—
157#208—81no—
158C2×C3⋊C8—22no—
159C22×Q8—31no—
160C22×Q8D1211yescharacteristic
161C23×C4C22×D311yescharacteristic
162C2×Q16—62no—
163C2×M16—21no—
164C2×M16—61no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8C4×D312yes—
172C22⋊Q8#6712yes—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4D1211yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C42—21no—
187C2×C42—61no—
188C2×C42—61no—
189#106—62no—
190#103—62no—
191#103—62no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—32no—
195C22⋊C8—61no—
196#95—21no—
197#95—31no—
198#95—31no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3C4×D411yescharacteristic
210C23×C3#12011yescharacteristic
211C23×C3#12111yescharacteristic
212C3×Q8—42no—
213C3×Q8—81no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×C3×C4C2×C22⋊C411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12311yescharacteristic
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—81no—
248C2×Dic3—81no—
249Dic6—42no—
250Dic6—81no—
251C3×C8—81no—
252C3⋊C8—42no—
253C24D3×D411yescharacteristic
254C2×Q8—21no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—81no—
258C2×Q8—241no—
259C2×Q8(C2×D3)⋊C412yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—81no—
270C22×C4C2×C4×D311yescharacteristic
271C22×C4C2×D1211yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4#23711yescharacteristic
274C22×C4#23711yescharacteristic
275C22×C4C2×#6711yescharacteristic
276Q16—242no—
277M16—81no—
278M16—241no—
279C2×C8—41no—
280C2×C8—62no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4—241no—
289C4⋊C4—241no—
290C4⋊C4—241no—
291C4⋊C4(C2×D3)⋊C412yes—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—41no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—241no—
301C42—81no—
302C42—241no—
303C42—241no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3#38611yescharacteristic
315C22×C3#45511yescharacteristic
316C22×C3#45611yescharacteristic
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—42no—
322C3×C4—42no—
323C3×C4—82no—
324Dic3—81no—
325Dic3—81no—
326Dic3—81no—
327Dic3—81no—
328Dic3—81no—
329Dic3—81no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23#67⋊C411yescharacteristic
335C23#88211yescharacteristic
336C23#93911yescharacteristic
337Q8—42no—
338Q8—81no—
339Q8—122no—
340Q8—241no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—241no—
371C2×C4—241no—
372C2×C4#67⋊C411yescharacteristic
373C2×C4#88611yescharacteristic
374C2×C4C2×(C2×D3)⋊C411yescharacteristic
375C2×C4#93911yescharacteristic
376C8—81no—
377C8—122no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3#188211yescharacteristic
383C2×C3#190511yescharacteristic
384C2×C3#191311yescharacteristic
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—81no—
394C22—81no—
395C22#482511yescharacteristic
396C22#482611yescharacteristic
397C22#495811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—82no—
405C4—241no—
406C4—241no—
407C4—241no—
408C4—241no—
409C4—241no—
410C4—241no—
411C3#934211yescharacteristic, complemented, Sylow
412C2—21no—
413C2—21no—
414C2—41no—
415C2—41no—
416C2#7002411yescharacteristic
417C2#7033411yescharacteristic
418C2#7036511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1508C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1508D3
hypocenterC3#9342
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
19. #5050C4semidirect
20. #5043C4semidirect
411. C3#9342semidirect

Maximal subgroups

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

subgroupindex
#827502
#827422
C3×#15082
#699812
#694572
#676842
#676162
#93423