Finite Group Databasebeta

← #10696734

Subgroups — #10696734

Order 768.

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

normal subgroups only

This group has 470 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#83374C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic
4C3×#1526C211yesmaximal, characteristic
5#69877C211yesmaximal, characteristic
6#69856C211yesmaximal, characteristic
7#67669C211yesmaximal, characteristic
8#10117—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C2×#824—22no—
15#5121—22no—
16#5094—21no—
17#5090—21no—
18#5079—21no—
19#5078—21no—
20#5052—22no—
21#5044—22no—
22#5031—21no—
23#5021—22no—
24#5020—21no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4702—22no—
28#4682—21no—
29#4680—21no—
30C3×#330—22no—
31#4456—22no—
32#3255—31no—
33#3230—31no—
34#3165—31no—
35#1966—31no—
36#1526D311yescharacteristic
37#1485—31no—
38#1299—31no—
39C22×Dic6—21no—
40C2×C3×M16—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×#117—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—22no—
52#926—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3D412yes—
55#924—21no—
56#924—21no—
57C2×#208D412yes—
58#870—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#97—21no—
68C3×#97D412yes—
69C3×C4×C8—21no—
70C3×C4×C8—21no—
71#835—22no—
72#835—24no—
73#830—21no—
74#824—44no—
75#818—24no—
76#817—24no—
77#810—22no—
78#808—22no—
79#807—24no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84#533—31no—
85#533C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87#501—62no—
88#500—61no—
89#500—61no—
90#498—61no—
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—62no—
101#423—31no—
102C2×#101—62no—
103C4×M16C2×D311yescharacteristic
104C4×M16C2×D311yescharacteristic
105#330—22no—
106Q8⋊C8—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×Dic3—22no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—42no—
116C2×Dic6—42no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C3×M16—24no—
120C3×M16—22no—
121C3×M16—41no—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136#217—41no—
137#216—82no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C4×Dic3—41no—
149#208—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C22×Q8—61no—
153C2×Q16—62no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—32no—
157C4⋊Q8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8D1211yescharacteristic
164C4⋊Q8D1211yescharacteristic
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—32no—
178C2×C4⋊C4—61no—
179C2×C42C22×D311yescharacteristic
180#107—64no—
181#107—64no—
182#106—62no—
183#106—64no—
184C4⋊C8—32no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187#103—62no—
188#103—62no—
189#103—64no—
190#101—124no—
191C22⋊C8—61no—
192#97—21no—
193#97#6712yes—
194C4×C8—21no—
195C4×C8—21no—
196C4×C8—61no—
197C23×C3#12111yescharacteristic
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—42no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12711yescharacteristic
213C2×Dic3—44no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217Dic6—41no—
218Dic6—41no—
219Dic6—42no—
220Dic6—82no—
221Dic6—81no—
222Dic6—81no—
223C3×C8—24no—
224C3×C8—42no—
225C3⋊C8—42no—
226C3⋊C8—42no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—122no—
232C2×Q8—122no—
233C2×Q8—121no—
234C2×Q8—121no—
235C22×C4—62no—
236C22×C4#23511yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4D3×D411yescharacteristic
239Q16—242no—
240M16—24no—
241M16—22no—
242M16—41no—
243M16—122no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—62no—
247C2×C8—62no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—61no—
255C4⋊C4—62no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—62no—
259C4⋊C4—61no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—41no—
265C22⋊C4—121no—
266C42—41no—
267C42—121no—
268C42C2×D1211yescharacteristic
269C42D3×D411yescharacteristic
270C42D3×D411yescharacteristic
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—81no—
280C3×C4—81no—
281C3×C4#45311yescharacteristic
282C3×C4#45411yescharacteristic
283C3×C4#45612yes—
284C3×C4#47312yes—
285Dic3—81no—
286Dic3—82no—
287Dic3—81no—
288C23#88411yescharacteristic
289Q8—41no—
290Q8—42no—
291Q8—41no—
292Q8—122no—
293Q8—121no—
294Q8—121no—
295Q8—242no—
296Q8—241no—
297Q8—241no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—124no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—122no—
308C2×C4#88211yescharacteristic
309C2×C4#88211yescharacteristic
310C2×C4#88411yescharacteristic
311C2×C4#93011yescharacteristic
312C2×C4#93711yescharacteristic
313C2×C4#94011yescharacteristic
314C8—24no—
315C8—42no—
316C8—122no—
317C8—122no—
318C2×C3—41no—
319C2×C3#200511yescharacteristic
320C2×C3#200611yescharacteristic
321C2×C3#202011yescharacteristic
322C22—21no—
323C22—21no—
324C22—21no—
325C22#496011yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—81no—
330C4—81no—
331C4—241no—
332C4—242no—
333C4—241no—
334C4#475512yes—
335C4#482211yescharacteristic
336C4#482612yes—
337C4#482711yescharacteristic
338C3#1011711yescharacteristic, complemented, Sylow
339C2—41no—
340C2#7012511yescharacteristic
341C2#7038911yescharacteristic
342C2#7039211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1526C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1526D3
hypocenterC3#10117
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1526

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
#833742
#833242
#785062
C3×#15262
#698772
#698562
#676692
#101173