Finite Group Databasebeta

← #10696771

Subgroups — #10696771

Order 768.

442 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3142C211yesmaximal, characteristic
2#83274C211yesmaximal, characteristic
3#83021C211yesmaximal, characteristic, complemented
4#71803C211yesmaximal, characteristic
5#71737C211yesmaximal, characteristic, complemented
6#69856C211yesmaximal, characteristic, complemented
7#67670C211yesmaximal, characteristic, complemented
8#10123—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#491—21no—
13C3×#490—21no—
14C3×#488—22no—
15C3×#417—21no—
16C3×#417—21no—
17C3×C4×M16C2211yescharacteristic
18#5081—21no—
19#5071—21no—
20#5069—21no—
21#5064—21no—
22#5063—22no—
23#5035—22no—
24#5008—21no—
25#5007—21no—
26#4971—21no—
27#4970—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30#4925C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32#4528—22no—
33#4456—22no—
34#3236—31no—
35#3235—31no—
36#3142D311yescharacteristic
37#1496—31no—
38#1485—31no—
39#1470—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C3×#124—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42C2311yescharacteristic
59#926—21no—
60#924—21no—
61#924—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#103—22no—
68C3×#103—22no—
69C3×#102—22no—
70C3×#102—22no—
71C3×#97—21no—
72C3×C4×C8D412yes—
73#832—22no—
74#832—22no—
75#830—21no—
76#830—21no—
77#830D412yes—
78#810—22no—
79#810—22no—
80#808—22no—
81#808—22no—
82#807—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)D412yes—
87#803—21no—
88#803—21no—
89C4×C3⋊C8—21no—
90#533—31no—
91#532C2×D311yescharacteristic
92C2×#125C2×D311yescharacteristic
93#501—61no—
94#501—61no—
95#491—21no—
96#491—61no—
97#490—21no—
98#490—61no—
99#488—22no—
100#484—61no—
101#483—61no—
102#482—62no—
103#479—62no—
104#476—61no—
105#474—61no—
106#432—31no—
107#432—31no—
108#431—31no—
109#417—21no—
110#417—21no—
111C4×M16C2×D311yescharacteristic
112Q8⋊C8—62no—
113D4⋊C8—62no—
114#324—62no—
115C3×#41—44no—
116C2×C3×Q8—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×Dic6—21no—
128C2×Dic6—21no—
129C3×M16—42no—
130C2×C3×C8—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147#217—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×#41—21no—
159C2×M16—21no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8#6711yescharacteristic
163#127#6711yescharacteristic
164#125—21no—
165#125—21no—
166#125—21no—
167#125—21no—
168#125—61no—
169#124—21no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172#121—21no—
173#121—21no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×D4—21no—
177C4×D4—21no—
178#117—61no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4#6711yescharacteristic
182C2×C4⋊C4#6711yescharacteristic
183C2×C42C22×D311yescharacteristic
184#107—62no—
185#107—62no—
186#106—62no—
187#106—62no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#103—22no—
192#103—22no—
193#103—62no—
194#103—62no—
195#102—22no—
196#102—22no—
197#102—62no—
198#102—62no—
199C22⋊C8—32no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202#97—21no—
203#97—61no—
204#97—61no—
205C4×C8—61no—
206C4×C8#6712yes—
207C23×C3—41no—
208C23×C3#12111yescharacteristic
209C3×Q8—42no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4—82no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—82no—
222C2×C3×C4—81no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12711yescharacteristic
229C2×Dic3—41no—
230C2×Dic3—41no—
231Dic6—42no—
232Dic6—42no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237#41—44no—
238C2×Q8—21no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4D3×D411yescharacteristic
249C22×C4#23711yescharacteristic
250C22×C4C2×#6711yescharacteristic
251M16—42no—
252C2×C8—22no—
253C2×C8—62no—
254C2×C8—62no—
255C2×C8—121no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—121no—
275C42—121no—
276C42D3×D411yescharacteristic
277C42D3×D411yescharacteristic
278C42C2×#6711yescharacteristic
279C42C2×#6711yescharacteristic
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3#39211yescharacteristic
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—81no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4#45412yes—
296Dic3—81no—
297Dic3—81no—
298C23—41no—
299C23#93911yescharacteristic
300Q8—42no—
301Q8—122no—
302Q8—122no—
303D4—42no—
304D4—42no—
305D4—42no—
306D4—82no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4#93711yescharacteristic
319C2×C4#93711yescharacteristic
320C2×C4#93811yescharacteristic
321C2×C4#93911yescharacteristic
322C2×C4#94011yescharacteristic
323C2×C4#95311yescharacteristic
324C8—42no—
325C8—122no—
326C8—122no—
327C8—241no—
328C2×C3—41no—
329C2×C3—161no—
330C2×C3#200311yescharacteristic
331C2×C3#200911yescharacteristic
332C2×C3#201711yescharacteristic
333C22—21no—
334C22—21no—
335C22—21no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#522311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—241no—
349C4—241no—
350C4#506212yes—
351C3#1012311yescharacteristic, complemented, Sylow
352C2—41no—
353C2—161no—
354C2#7173011yescharacteristic
355C2#7212011yescharacteristic
356C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3142C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3142D3
hypocenterC3#10123
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
3. #83021C2semidirect
5. #71737C2semidirect
6. #69856C2semidirect
7. #67670C2semidirect
351. C3#10123semidirect

Maximal subgroups

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

subgroupindex
C3×#31422
#832742
#830212
#718032
#717372
#698562
#676702
#101233