Finite Group Databasebeta

← #10697194

Subgroups — #10697194

Order 768.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3078C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic
3C2×#5021C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic
5#71624C211yesmaximal, characteristic
6#70508C211yesmaximal, characteristic
7#67714C211yesmaximal, characteristic
8#10183—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#479—22no—
13C3×#477—21no—
14C3×#475—21no—
15C3×#428—21no—
16C3×#423C2211yescharacteristic
17C3×#419—21no—
18C2×#835—21no—
19#5093—21no—
20#5092—21no—
21#5050—22no—
22#5031—22no—
23#5021—22no—
24#5021—22no—
25C2×#807—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4535—22no—
30#4528—22no—
31#4486—22no—
32#3209—31no—
33#3078D311yescharacteristic
34C2×#477—31no—
35#1549—31no—
36#1481—31no—
37#1461—31no—
38#1301—31no—
39C22×C3×Q8D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#216—24no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C22×C3⋊C8—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D412yes—
66C3×#107—22no—
67C3×#106—22no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×#103—22no—
71C3×#103—22no—
72#835—24no—
73#830—21no—
74#830D412yes—
75#810—24no—
76#807—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×C4×Q8C2×D311yescharacteristic
86#494—61no—
87#487—61no—
88#479—22no—
89#477—21no—
90#477—62no—
91#477—62no—
92#475—21no—
93#462—62no—
94#451—62no—
95#431—31no—
96#431—31no—
97#428—21no—
98C2×#107—61no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8—31no—
101#419—21no—
102C2×#103—61no—
103#332—62no—
104Q8⋊C8—62no—
105#324—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8Q1612yescomplemented
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—41no—
120C2×Dic6—22no—
121C2×Dic6—41no—
122C2×Dic6—82no—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42—41no—
139C3×C42—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144#216—84no—
145C4⋊Dic3—22no—
146C4⋊Dic3—22no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C22×Q8—61no—
155C22×Q8#6711yescharacteristic
156C2×Q16—64no—
157C2×M16—21no—
158C22×C8—61no—
159C4⋊Q8—32no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8#6711yescharacteristic
164C4⋊Q8#6711yescharacteristic
165#125—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174#117—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4#6711yescharacteristic
179C2×C42—21no—
180C2×C42C22×D311yescharacteristic
181#107—22no—
182#107—64no—
183#106—22no—
184C4⋊C8—21no—
185C4⋊C8—32no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8#6712yes—
190#103—22no—
191#103—22no—
192#103—64no—
193#103—64no—
194C22⋊C8—32no—
195C22⋊C8—61no—
196C4×C8—61no—
197C4×C8—61no—
198C23×C3#12111yescharacteristic
199C3×Q8—22no—
200C3×Q8—22no—
201C3×Q8—42no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×Q8—42no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—81no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220C2×C3×C4C2×Q1611yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4#13711yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×Dic3—42no—
225C2×Dic3—82no—
226Dic6—44no—
227Dic6—82no—
228C3×C8—42no—
229C3⋊C8—42no—
230C3⋊C8—42no—
231C3⋊C8—81no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—62no—
240C2×Q8—121no—
241C2×Q8—242no—
242C2×Q8#21612yescomplemented
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—121no—
246C22×C4D3×D411yescharacteristic
247C22×C4#23711yescharacteristic
248C22×C4C2×#6711yescharacteristic
249Q16—244no—
250M16—41no—
251M16—41no—
252C2×C8—22no—
253C2×C8—62no—
254C2×C8—62no—
255C2×C8—121no—
256C2×C8—121no—
257C2×C8—121no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—62no—
269C4⋊C4—62no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C22⋊C4—41no—
273C42—41no—
274C42—41no—
275C42—41no—
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—42no—
289C3×C4—81no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4#45111yescharacteristic
293C3×C4#45111yescharacteristic
294C3×C4#45211yescharacteristic
295C3×C4#45211yescharacteristic
296C3×C4#47312yes—
297Dic3—82no—
298C23#93911yescharacteristic
299Q8—22no—
300Q8—22no—
301Q8—42no—
302Q8—41no—
303Q8—41no—
304Q8—42no—
305Q8—124no—
306Q8—242no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—122no—
317C2×C4—242no—
318C2×C4#93711yescharacteristic
319C2×C4#93711yescharacteristic
320C2×C4#93811yescharacteristic
321C2×C4#93911yescharacteristic
322C2×C4#94011yescharacteristic
323C2×C4#94211yescharacteristic
324C2×C4C2×#21611yescharacteristic
325C2×C4#95011yescharacteristic
326C2×C4#95111yescharacteristic
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#505111yescharacteristic
340C22#505211yescharacteristic
341C22#522311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—242no—
350C4#504711yescharacteristic
351C4#505011yescharacteristic
352C4#520412yes—
353C4#522811yescharacteristic
354C4#523911yescharacteristic
355C3#1018311yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2#7170011yescharacteristic
359C2#7210911yescharacteristic
360C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3078C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3078D3
hypocenterC3#10183
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
113. C2×C3×Q8Q16semidirect
242. C2×Q8#216semidirect
355. C3#10183semidirect

Maximal subgroups

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

subgroupindex
C3×#30782
#833702
C2×#50212
#718242
#716242
#705082
#677142
#101833