Finite Group Databasebeta

← #10696147

Subgroups — #10696147

Order 768.

468 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 83 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#83373C211yesmaximal, characteristic
2#83369C211yesmaximal, characteristic
3#79356C211yesmaximal, characteristic
4C3×#1531C211yesmaximal, characteristic
5#69838C211yesmaximal, characteristic
6#69801C211yesmaximal, characteristic
7#67755C211yesmaximal, characteristic
8#10031—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14#5094—21no—
15#5091—21no—
16#5090—21no—
17#5090—21no—
18#5051—22no—
19#5050—21no—
20#5043—21no—
21#5031—21no—
22#5030—22no—
23#5021—21no—
24#4926C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4851—21no—
27#4849—21no—
28#4739—21no—
29#4721—21no—
30#4716—22no—
31C3×#330—22no—
32#4550—22no—
33#4465—22no—
34#3249—31no—
35#3113—31no—
36#3113—31no—
37#1985—31no—
38#1531—31no—
39#1531D311yescharacteristic
40#1290—31no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#216—22no—
53C2×#216—22no—
54#926—21no—
55C2×C4⋊Dic3D411yescharacteristic
56#924—21no—
57#924—21no—
58C2×Dic3⋊C4—21no—
59C2×C4×Dic3—21no—
60C22×C3⋊C8D412yes—
61C2×Dic12—22no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97—21no—
75C3×#97D412yes—
76#835—22no—
77#835—22no—
78#830—21no—
79#817—22no—
80#816—22no—
81#816—22no—
82#810—22no—
83#810—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89#803D412yes—
90#533—31no—
91#533C2×D311yescharacteristic
92C2×C4×Q8—31no—
93#483—61no—
94#483—61no—
95#477—61no—
96#477—61no—
97#475—61no—
98#475—61no—
99#467—62no—
100#462—61no—
101#462—61no—
102#462—61no—
103#462—61no—
104#451—61no—
105#451—61no—
106#441—62no—
107#441—62no—
108#432—31no—
109#432C2×D311yescharacteristic
110C2×C4⋊C8—31no—
111C2×#97C2×D311yescharacteristic
112#336—62no—
113#330—22no—
114#330—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—42no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143#217—41no—
144#216—44no—
145#216—82no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—42no—
162Dic12—44no—
163C22×Q8—31no—
164C2×Q16—62no—
165C2×Q16—62no—
166C2×Q16—62no—
167C22×C8—21no—
168C22×C8—32no—
169C4⋊Q8—31no—
170C4⋊Q8—31no—
171C4⋊Q8D1211yescharacteristic
172C4⋊Q8D1211yescharacteristic
173#125—21no—
174#125—61no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×Q8—61no—
187#117—21no—
188#117—61no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C42—61no—
192C2×C42C22×D311yescharacteristic
193#107—62no—
194#106—62no—
195#106—62no—
196C4⋊C8—21no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#103—62no—
201#103—62no—
202#103—62no—
203#103—62no—
204#103—62no—
205#103—62no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208#97—21no—
209#97—21no—
210#97—32no—
211#97#6712yes—
212C23×C3#12111yescharacteristic
213C3×Q8—42no—
214C3×Q8—41no—
215C3×Q8—41no—
216C2×C3×C4—22no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235Dic6—22no—
236Dic6—22no—
237Dic6—41no—
238Dic6—41no—
239Dic6—42no—
240Dic6—41no—
241Dic6—41no—
242C3×C8—42no—
243C3×C8—42no—
244C3⋊C8—42no—
245C3⋊C8—42no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×Q8—122no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4#23511yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258Q16—124no—
259Q16—124no—
260Q16—242no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—62no—
266C2×C8—62no—
267C2×C8—122no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C22⋊C4—41no—
285C22⋊C4—121no—
286C42—41no—
287C42—121no—
288C42—121no—
289C42—121no—
290C42C2×D1211yescharacteristic
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C42C2×#6711yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3#39211yescharacteristic
297C22×C3#47312yes—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—81no—
304C3×C4—81no—
305C3×C4#45511yescharacteristic
306C3×C4#45611yescharacteristic
307Dic3—42no—
308Dic3—81no—
309Dic3—81no—
310Dic3—81no—
311C23#88411yescharacteristic
312Q8—42no—
313Q8—41no—
314Q8—41no—
315Q8—62no—
316Q8—62no—
317Q8—121no—
318Q8—121no—
319Q8—122no—
320Q8—121no—
321Q8—121no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—62no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—122no—
335C2×C4#88211yescharacteristic
336C2×C4#88211yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4#93011yescharacteristic
339C2×C4#93711yescharacteristic
340C2×C4#94011yescharacteristic
341C8—42no—
342C8—42no—
343C8—122no—
344C8—122no—
345C2×C3—22no—
346C2×C3#200711yescharacteristic
347C2×C3#201011yescharacteristic
348C2×C3#202011yescharacteristic
349C22—21no—
350C22—21no—
351C22#475512yes—
352C22#496011yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—81no—
359C4—81no—
360C4—122no—
361C4—241no—
362C4—241no—
363C4—241no—
364C4#482511yescharacteristic
365C4#482611yescharacteristic
366C3#1003111yescharacteristic, complemented, Sylow
367C2—22no—
368C2#7012511yescharacteristic
369C2#7039311yescharacteristic
370C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1531C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1531D3
hypocenterC3#10031
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
366. C3#10031semidirect

Maximal subgroups

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

subgroupindex
#833732
#833692
#793562
C3×#15312
#698382
#698012
#677552
#100313