Finite Group Databasebeta

← #10697191

Subgroups — #10697191

Order 768.

476 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×#3209C211yesmaximal, characteristic
2#83367C211yesmaximal, characteristic
3C2×#5091C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic
5#71775C211yesmaximal, characteristic
6#70786C211yesmaximal, characteristic
7#67745C211yesmaximal, characteristic
8#10183—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#494—21no—
13C3×#487—21no—
14C3×#462—22no—
15C3×#451—22no—
16C3×#431C2211yescharacteristic
17#5094—21no—
18#5091—22no—
19#5091—21no—
20#5091—22no—
21#5063—22no—
22#5004—21no—
23#4995—21no—
24C2×#810—21no—
25C2×#807—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4547—22no—
30#4535—22no—
31#4469—22no—
32#3209D311yescharacteristic
33#3078—31no—
34C2×#477—31no—
35#1549—31no—
36#1481—31no—
37#1461—31no—
38#1301—31no—
39C22×C3×Q8—21no—
40C22×Dic6D411yescharacteristic
41C2×C3×Q16—24no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×C4×Dic3—21no—
60C2×#208—21no—
61C22×C3⋊C8—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×#103—24no—
68C3×C22⋊C8D412yes—
69C3×C4×C8—21no—
70#830—21no—
71#810—24no—
72#810—22no—
73#810—22no—
74#808—22no—
75#807—24no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4⋊(C3⋊C8)D412yes—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×C4×Q8—31no—
86#494—21no—
87#487—21no—
88#479—62no—
89#477—62no—
90#477—62no—
91#477—61no—
92#475—61no—
93#462—22no—
94#451—22no—
95#431—31no—
96#431C2×D311yescharacteristic
97#428—61no—
98C2×#107—61no—
99#423—31no—
100C2×C4⋊C8—31no—
101#419—61no—
102C2×#103—61no—
103#332—62no—
104Q8⋊C8—62no—
105#324—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6Q1612yescomplemented
123C3×Q16—84no—
124C2×C3×C8—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—41no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—41no—
146C4×Dic3—41no—
147#208—41no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C22×Q8—21no—
155C22×Q8—31no—
156C2×Q16—24no—
157C2×M16—61no—
158C22×C8—61no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8#6712yes—
163C4⋊Q8#6711yescharacteristic
164C4⋊Q8#6711yescharacteristic
165#125—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174#117—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C42—61no—
180C2×C42C22×D311yescharacteristic
181#107—64no—
182#107—62no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—32no—
186C4⋊C8—32no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—24no—
191#103—64no—
192#103—62no—
193#103—62no—
194C22⋊C8—61no—
195C22⋊C8#6712yes—
196C4×C8—21no—
197C4×C8—61no—
198C23×C3#12111yescharacteristic
199C3×Q8—42no—
200C3×Q8—44no—
201C3×Q8—42no—
202C3×Q8—82no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—82no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12711yescharacteristic
215C2×C3×C4C2×Q1611yescharacteristic
216C2×C3×C4#13511yescharacteristic
217C2×C3×C4#13711yescharacteristic
218C2×C3×C4#13711yescharacteristic
219C2×Dic3—22no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—81no—
224Dic6—22no—
225Dic6—22no—
226Dic6—41no—
227Dic6—41no—
228C3×C8—42no—
229C3⋊C8—42no—
230C3⋊C8—42no—
231C3⋊C8—81no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—21no—
235C2×Q8—32no—
236C2×Q8—41no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—82no—
243C22×C4—41no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4D3×D411yescharacteristic
247C22×C4#23711yescharacteristic
248C22×C4C2×#6711yescharacteristic
249Q16—84no—
250M16—121no—
251M16—121no—
252C2×C8—22no—
253C2×C8—62no—
254C2×C8—62no—
255C2×C8—121no—
256C2×C8—121no—
257C2×C8—121no—
258C4⋊C4—21no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—22no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C22⋊C4—41no—
273C42—41no—
274C42—121no—
275C42—121no—
276C42D3×D411yescharacteristic
277C42D3×D411yescharacteristic
278C42C2×#6711yescharacteristic
279C42C2×#6711yescharacteristic
280C22×C3—21no—
281C22×C3—21no—
282C22×C3#39211yescharacteristic
283C22×C3#46711yescharacteristic
284C22×C3#47011yescharacteristic
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—82no—
289C3×C4—81no—
290C3×C4—81no—
291C3×C4#45111yescharacteristic
292C3×C4#45111yescharacteristic
293C3×C4#45211yescharacteristic
294C3×C4#45211yescharacteristic
295C3×C4#47312yes—
296Dic3—42no—
297Dic3—81no—
298C23#93811yescharacteristic
299Q8—42no—
300Q8—44no—
301Q8—42no—
302Q8—62no—
303Q8—62no—
304Q8—82no—
305Q8—121no—
306Q8—121no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—62no—
313C2×C4—82no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—241no—
318C2×C4D3×Q1611yescharacteristic
319C2×C4#91811yescharacteristic
320C2×C4#91811yescharacteristic
321C2×C4#91911yescharacteristic
322C2×C4#93711yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93911yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#94011yescharacteristic
327C2×C4#95311yescharacteristic
328C8—42no—
329C8—122no—
330C8—122no—
331C8—241no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3#200111yescharacteristic
335C2×C3#200811yescharacteristic
336C2×C3#202011yescharacteristic
337C22—21no—
338C22—21no—
339C22#518811yescharacteristic
340C22#518911yescharacteristic
341C22#522311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—82no—
347C4—81no—
348C4—122no—
349C4—241no—
350C4#518611yescharacteristic
351C4#518611yescharacteristic
352C4#518711yescharacteristic
353C4#518711yescharacteristic
354C4#520412yes—
355C3#1018311yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2#7185811yescharacteristic
359C2#7207111yescharacteristic
360C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3209C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3209D3
hypocenterC3#10183
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
122. C2×Dic6Q16semidirect
355. C3#10183semidirect

Maximal subgroups

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

subgroupindex
C3×#32092
#833672
C2×#50912
#718242
#717752
#707862
#677452
#101833