Finite Group Databasebeta

← #10696729

Subgroups — #10696729

Order 768.

470 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3165C211yesmaximal, characteristic
2#83374C211yesmaximal, characteristic
3#83224C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic
5#71789C211yesmaximal, characteristic
6#69862C211yesmaximal, characteristic
7#67669C211yesmaximal, characteristic
8#10117—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#501—22no—
13C3×#500—21no—
14C3×#498—21no—
15C3×#428—22no—
16C3×C4×M16C2211yescharacteristic
17C2×#836—22no—
18#5094—21no—
19#5093—21no—
20#5091—22no—
21#5090—21no—
22#5078—21no—
23#5052—22no—
24#5031—21no—
25#5020—21no—
26#4970—22no—
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29C4×#208C2211yescharacteristic
30#4535—22no—
31#4465—22no—
32#3255—31no—
33#3230—31no—
34#3165D311yescharacteristic
35#1966—31no—
36#1526—31no—
37#1485—31no—
38#1299—31no—
39C22×C3×Q8—21no—
40C2×C3×M16—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56C2×#216—22no—
57#926—21no—
58#924—21no—
59#924—21no—
60C2×#208—21no—
61C2×#208D412yes—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#107—24no—
68C3×#106—24no—
69C3×#97D412yes—
70C3×C4×C8—21no—
71#836—44no—
72#835—22no—
73#830—21no—
74#810—24no—
75#810—22no—
76#808—22no—
77#807—24no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#533C2×D311yescharacteristic
86C2×C4⋊Q8C2×D311yescharacteristic
87#501—22no—
88#500—21no—
89#500—61no—
90#498—21no—
91#487—61no—
92#484—62no—
93#477—62no—
94#475—61no—
95#475—61no—
96#470—62no—
97#462—61no—
98#462—61no—
99#431—31no—
100#428—22no—
101#423—31no—
102C2×#101—62no—
103C4×M16—31no—
104C4×M16C2×D311yescharacteristic
105#330—62no—
106Q8⋊C8—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—42no—
110C2×C3×Q8—42no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C22×C3×C4—22no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C2×Dic6—21no—
118C2×Dic6—21no—
119C3×M16—24no—
120C2×C3×C8—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137#217—41no—
138#216—82no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—41no—
146#208—22no—
147#208—41no—
148#208—42no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C22×Q8—21no—
153C2×Q16—62no—
154C2×M16—21no—
155C2×M16—32no—
156C2×M16—61no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163C4⋊Q8#6711yescharacteristic
164C4⋊Q8#6711yescharacteristic
165#125—21no—
166#125—61no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—61no—
174C4×Q8—61no—
175#117—21no—
176#117—61no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4#6712yes—
179C2×C42C22×D311yescharacteristic
180#107—24no—
181#107—64no—
182#106—24no—
183#106—62no—
184C4⋊C8—32no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187#103—62no—
188#103—64no—
189#103—62no—
190#101—124no—
191C22⋊C8—61no—
192#97—61no—
193#97#6712yes—
194C4×C8—21no—
195C4×C8—61no—
196C4×C8—61no—
197C23×C3#12111yescharacteristic
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—42no—
201C3×Q8—82no—
202C3×Q8—81no—
203C3×Q8—81no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—44no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218C2×Dic3—41no—
219C2×Dic3—41no—
220Dic6—41no—
221Dic6—41no—
222Dic6—42no—
223C3×C8—24no—
224C3⋊C8—42no—
225C3⋊C8—42no—
226C3⋊C8—42no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—61no—
234C2×Q8—61no—
235C22×C4—22no—
236C22×C4D3×D411yescharacteristic
237C22×C4#23711yescharacteristic
238C22×C4C2×#6711yescharacteristic
239Q16—242no—
240M16—24no—
241M16—62no—
242M16—121no—
243M16—122no—
244C2×C8—22no—
245C2×C8—62no—
246C2×C8—62no—
247C2×C8—121no—
248C4⋊C4—21no—
249C4⋊C4—22no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—41no—
265C22⋊C4—121no—
266C42—41no—
267C42—121no—
268C42D3×D411yescharacteristic
269C42D3×D411yescharacteristic
270C42C2×#6711yescharacteristic
271C42C2×#6711yescharacteristic
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3#39211yescharacteristic
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—82no—
280C3×C4—81no—
281C3×C4—81no—
282C3×C4#45311yescharacteristic
283C3×C4#45411yescharacteristic
284C3×C4#45612yes—
285C3×C4#47312yes—
286Dic3—81no—
287Dic3—81no—
288C23#93911yescharacteristic
289Q8—41no—
290Q8—41no—
291Q8—42no—
292Q8—82no—
293Q8—81no—
294Q8—81no—
295Q8—121no—
296Q8—121no—
297Q8—122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4#93711yescharacteristic
309C2×C4#93711yescharacteristic
310C2×C4#93811yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#94011yescharacteristic
313C2×C4#95311yescharacteristic
314C8—24no—
315C8—122no—
316C8—122no—
317C8—122no—
318C2×C3—41no—
319C2×C3#200511yescharacteristic
320C2×C3#200611yescharacteristic
321C2×C3#202011yescharacteristic
322C22—21no—
323C22—21no—
324C22—21no—
325C22#522311yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—82no—
330C4—81no—
331C4—81no—
332C4—241no—
333C4—241no—
334C4#509512yes—
335C4#519811yescharacteristic
336C4#520111yescharacteristic
337C4#520412yes—
338C3#1011711yescharacteristic, complemented, Sylow
339C2—41no—
340C2#7179511yescharacteristic
341C2#7209811yescharacteristic
342C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3165D3
hypocenterC3#10117
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
338. C3#10117semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833742
#832242
#718252
#717892
#698622
#676692
#101173