Finite Group Databasebeta

← #10696824

Subgroups — #10696824

Order 768.

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

normal subgroups only

This group has 520 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#83019C211yesmaximal, characteristic, complemented
2C2×#5067C211yesmaximal, characteristic, complemented
3#79303C211yesmaximal, characteristic
4C3×#1470C211yesmaximal, characteristic, complemented
5#70818C211yesmaximal, characteristic, complemented
6#70806C211yesmaximal, characteristic, complemented
7#67742C211yesmaximal, characteristic, complemented
8#10130—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#874C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5067—22no—
15#5067—21no—
16#5067—22no—
17#5066—22no—
18#5064—21no—
19#4968—21no—
20#4967—21no—
21C2×#809—21no—
22C2×#809—21no—
23#4926C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4843—22no—
26#4732—22no—
27#4695—21no—
28#4694—21no—
29C3×#324—22no—
30#4485—22no—
31#4464—22no—
32#3234—31no—
33#3205—31no—
34C2×#490—31no—
35#1555—31no—
36#1530—31no—
37#1470D311yescharacteristic, complemented
38#1302—31no—
39C2×#235—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52#930—21no—
53#930—21no—
54C22×C3⋊C8—21no—
55C2×D24—24no—
56#875—21no—
57#874—21no—
58#874—21no—
59#874—21no—
60#874D411yescharacteristic
61#874D412yes—
62C4×D12—21no—
63C4×D12—21no—
64#869D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8D412yes—
69C3×#97—21no—
70C3×C4×C8—21no—
71#830—21no—
72#821—24no—
73#809—24no—
74#809—24no—
75#809—22no—
76#809—22no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803—21no—
82#803D412yes—
83#532—31no—
84C2×#127—31no—
85C2×#125C2×D311yescharacteristic
86#501—61no—
87#493—61no—
88#490—62no—
89#490—62no—
90#490—61no—
91#486—61no—
92#481—61no—
93#480—61no—
94#478—62no—
95#459—62no—
96#447—62no—
97#432—31no—
98#432C2×D311yescharacteristic
99#431C2×D311yescharacteristic
100C2×#102—61no—
101C2×#102—61no—
102C2×#97—31no—
103#331—62no—
104#329—62no—
105#324—22no—
106C23×D3—41no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—41no—
113#235—82no—
114C2×D12—21no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—21no—
118C2×D12—22no—
119C2×D12—21no—
120C2×D12—44no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—82no—
125C2×D12—82no—
126C2×C4×D3—41no—
127C2×Dic6—21no—
128C2×C3×C8—22no—
129C2×C3×C8—41no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—42no—
149D24—84no—
150C2×#41—61no—
151C22×D4—61no—
152C22×D4—61no—
153C22×D4—61no—
154C2×D8—64no—
155C22×C8—61no—
156C4⋊Q8—31no—
157#127—32no—
158#127—31no—
159#127—61no—
160#127—61no—
161#127—61no—
162#125—21no—
163#125—21no—
164#125—21no—
165#125—21no—
166#124—61no—
167#121—61no—
168#121—61no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4D1211yescharacteristic
174C2×C4⋊C4D1211yescharacteristic
175C2×C42C22×D311yescharacteristic
176#107—62no—
177#106—62no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—32no—
181#102—64no—
182#102—64no—
183#102—62no—
184#102—62no—
185#102—64no—
186C22⋊C8—21no—
187C22⋊C8—61no—
188C22⋊C8#6712yes—
189#97—21no—
190#97—32no—
191#97—61no—
192#97—61no—
193C4×C8—21no—
194C23×C3#12111yescharacteristic
195C22×D3—41no—
196C22×D3—42no—
197C22×D3—82no—
198C22×D3—82no—
199C22×D3—82no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—81no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12711yescharacteristic
213#67—82no—
214C2×Dic3—41no—
215D12—42no—
216D12—44no—
217D12—44no—
218D12—42no—
219D12—82no—
220D12—81no—
221D12—82no—
222D12—82no—
223D12—82no—
224C4×D3—82no—
225Dic6—81no—
226C3×C8—42no—
227C3×C8—81no—
228C3⋊C8—42no—
229C3⋊C8—42no—
230C24—121no—
231#41—242no—
232C2×Q8—61no—
233C2×D4—61no—
234C2×D4—62no—
235C2×D4—61no—
236C2×D4—62no—
237C2×D4—62no—
238C2×D4—61no—
239C2×D4—124no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—121no—
244C2×D4—242no—
245C2×D4—242no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—121no—
249C22×C4C2×D1211yescharacteristic
250C22×C4#23511yescharacteristic
251C22×C4C2×#6711yescharacteristic
252D8—244no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—62no—
256C2×C8—62no—
257C2×C8—122no—
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—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C42D3×D411yescharacteristic
271C42D3×D411yescharacteristic
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#39211yescharacteristic
277C22×C3#47212yes—
278C2×D3—82no—
279C2×D3—81no—
280C2×D3—82no—
281C2×D3—81no—
282C2×D3—81no—
283C2×D3—82no—
284C2×D3—164no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4#45412yes—
293Dic3—81no—
294C23—121no—
295C23—122no—
296C23—242no—
297C23—242no—
298C23—242no—
299C23#93011yescharacteristic
300Q8—241no—
301D4—122no—
302D4—124no—
303D4—124no—
304D4—122no—
305D4—242no—
306D4—241no—
307D4—242no—
308D4—242no—
309D4—242no—
310D4—242no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—121no—
319C2×C4—242no—
320C2×C4#88211yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#88411yescharacteristic
324C2×C4#93711yescharacteristic
325C2×C4#94011yescharacteristic
326C8—42no—
327C8—81no—
328C8—122no—
329C8—122no—
330C2×C3—22no—
331C2×C3#200311yescharacteristic
332C2×C3#200811yescharacteristic
333C2×C3#201811yescharacteristic
334D3—161no—
335D3—162no—
336C22—21no—
337C22—21no—
338C22—241no—
339C22—242no—
340C22—242no—
341C22—241no—
342C22—242no—
343C22—241no—
344C22—484no—
345C22#496011yescharacteristic
346C22#514212yes—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—81no—
353C4—81no—
354C4—241no—
355C4#475312yes—
356C3#1013011yescharacteristic, complemented, Sylow
357C2—22no—
358C2—482no—
359C2—481no—
360C2#7000611yescharacteristic
361C2#7005011yescharacteristic
362C2#7011611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1470C2
socleC22×C3#392
prime radical#79303C2
hypercenter#1470D3
hypocenterC3#10130
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1470

Complements

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

NH ≅ G/Ntype
1. #83019C2semidirect
2. C2×#5067C2semidirect
4. C3×#1470C2semidirect
5. #70818C2semidirect
6. #70806C2semidirect
7. #67742C2semidirect
37. #1470D3semidirect
356. C3#10130semidirect

Maximal subgroups

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

subgroupindex
#830192
C2×#50672
#793032
C3×#14702
#708182
#708062
#677422
#101303