Finite Group Databasebeta

← #10695813

Subgroups — #10695813

Order 768.

492 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3095C211yesmaximal, characteristic
2#82741C211yesmaximal, characteristic
3#82220C211yesmaximal, characteristic
4#71691C211yesmaximal, characteristic
5#71584C211yesmaximal, characteristic
6#69972C211yesmaximal, characteristic
7#67687C211yesmaximal, characteristic
8#9979—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#485—22no—
13C3×#484—21no—
14C3×#483—21no—
15C3×#428—21no—
16C3×#419—21no—
17C3×#407C2211yescharacteristic
18#5228—21no—
19#5207C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×#830C2211yescharacteristic
22#5051—22no—
23#5050—21no—
24#5044—21no—
25#5004—21no—
26#4996—22no—
27#4995—21no—
28#4971—21no—
29#4970—21no—
30#4970—21no—
31#4528—22no—
32#4524—22no—
33#4469—22no—
34#3189—31no—
35#3095—31no—
36#3095D311yescharacteristic
37#1818—31no—
38#1472—31no—
39#1472—31no—
40#1293—31no—
41C22×C3×Q8D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×#117—21no—
50C3×#117—21no—
51C3×#117—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42—21no—
60C2×#217—21no—
61C2×#216—22no—
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×#208—21no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8D412yes—
71C3×#107—22no—
72C3×#106—22no—
73C3×#103—22no—
74C3×#103—22no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D412yes—
77#835—22no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#810—22no—
85#810—22no—
86#810—22no—
87#808—22no—
88#807—22no—
89#807—22no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C2×#117C2×D311yescharacteristic
93#485—22no—
94#485—62no—
95#484—21no—
96#484—61no—
97#484—61no—
98#484—61no—
99#483—21no—
100#483—61no—
101#467—62no—
102#451—61no—
103#451—61no—
104#428—21no—
105#428—61no—
106#419—21no—
107#419—61no—
108#407—31no—
109#407C2×D311yescharacteristic
110C2×C22⋊C8—31no—
111C2×C22⋊C8—31no—
112#336—62no—
113#324—62no—
114#324—62no—
115C24×C3#4111yescharacteristic
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—42no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×Dic3—21no—
131C22×Dic3—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—42no—
135C3×M16—42no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42—41no—
150C3×C42—41no—
151#217—41no—
152#217—41no—
153#216—82no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158#208—42no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—42no—
162C2×C3⋊C8—42no—
163C2×C3⋊C8—41no—
164C22×Q8—31no—
165C22×Q8#6711yescharacteristic
166C23×C4C22×D311yescharacteristic
167C2×Q16—62no—
168C2×M16—21no—
169C2×M16—61no—
170C22×C8—32no—
171C22×C8—61no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180#117—21no—
181#117—21no—
182#117—21no—
183#117—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4#6711yescharacteristic
188C2×C4⋊C4#6711yescharacteristic
189C2×C4⋊C4#6711yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—61no—
193C2×C42—21no—
194#107—22no—
195#107—62no—
196#107—62no—
197#106—22no—
198#106—62no—
199#103—22no—
200#103—22no—
201#103—62no—
202#103—62no—
203#103—62no—
204#103—62no—
205C22⋊C8—21no—
206C22⋊C8—32no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8#6712yes—
213C23×C3—22no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3#12111yescharacteristic
217C23×C3#12111yescharacteristic
218C23×C3#12111yescharacteristic
219C3×Q8—42no—
220C3×Q8—42no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—81no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12711yescharacteristic
239C2×Dic3—42no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—81no—
243Dic6—42no—
244Dic6—42no—
245C3×C8—42no—
246C3⋊C8—42no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249C24#23711yescharacteristic
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—42no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8—122no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4D3×D411yescharacteristic
266C22×C4C2×#6711yescharacteristic
267C22×C4C2×#6711yescharacteristic
268C22×C4C2×#6711yescharacteristic
269Q16—242no—
270M16—42no—
271M16—122no—
272C2×C8—22no—
273C2×C8—62no—
274C2×C8—62no—
275C2×C8—122no—
276C2×C8—122no—
277C2×C8—121no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C42—41no—
297C42—41no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—82no—
307C22×C3#39211yescharacteristic
308C22×C3#47312yes—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—81no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4—81no—
316C3×C4#45812yes—
317Dic3—81no—
318Dic3—81no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23#93811yescharacteristic
323C23#93911yescharacteristic
324C23#93911yescharacteristic
325Q8—42no—
326Q8—42no—
327Q8—122no—
328Q8—122no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—122no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—241no—
347C2×C4#93711yescharacteristic
348C2×C4#93711yescharacteristic
349C2×C4#94011yescharacteristic
350C2×C4#95311yescharacteristic
351C8—42no—
352C8—122no—
353C8—122no—
354C8—241no—
355C2×C3—22no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3#201711yescharacteristic
359C2×C3#202011yescharacteristic
360C2×C3#202211yescharacteristic
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—82no—
370C22#520412yes—
371C22#522311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—241no—
380C4—241no—
381C4#505312yes—
382C3#997911yescharacteristic, complemented, Sylow
383C2—22no—
384C2—41no—
385C2—41no—
386C2#7170611yescharacteristic
387C2#7210911yescharacteristic
388C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3095C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenter#3095D3
hypocenterC3#9979
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3095

Complements

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

NH ≅ G/Ntype
382. C3#9979semidirect

Maximal subgroups

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

subgroupindex
C3×#30952
#827412
#822202
#716912
#715842
#699722
#676872
#99793