Finite Group Databasebeta

← #10697456

Subgroups — #10697456

Order 768.

484 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 95 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#83282C211yesmaximal, characteristic
2#83271C211yesmaximal, characteristic
3#79313C211yesmaximal, characteristic, complemented
4C3×#1517C211yesmaximal, characteristic
5#69555C211yesmaximal, characteristic, complemented
6#69546C211yesmaximal, characteristic, complemented
7#67741C211yesmaximal, characteristic, complemented
8#10223—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5233—22no—
15#5082—21no—
16#5080—21no—
17#5071—21no—
18#5070—21no—
19#5055—21no—
20#5055—21no—
21#5034—22no—
22#5008—22no—
23#4925C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4849—22no—
26#4739—22no—
27#4699—21no—
28#4681—21no—
29C3×#327—22no—
30#4529—22no—
31#4452—22no—
32#3227—31no—
33#3209—31no—
34#3135—31no—
35#1993—31no—
36#1518—31no—
37#1517D311yescharacteristic
38#1281—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×#127D411yescharacteristic
43C3×#124—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50#926—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53#924—21no—
54#924—21no—
55C22×C3⋊C8D412yes—
56C2×Dic12—24no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8D412yes—
68C3×C4×C8—21no—
69C3×#95—21no—
70C3×#95—21no—
71#835—22no—
72#835—22no—
73#832—22no—
74#832—22no—
75#832—24no—
76#830D412yes—
77#816—24no—
78#808—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#532C2×D311yescharacteristic
85C2×C4⋊Q8—31no—
86#498—61no—
87#494—61no—
88#491—61no—
89#490—61no—
90#489—61no—
91#489—61no—
92#487—61no—
93#486—61no—
94#481—62no—
95#476—62no—
96#462—62no—
97#451—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#416—62no—
101C2×#97—31no—
102#382C2×D311yescharacteristic
103#340—62no—
104#328—62no—
105#327—22no—
106C3×#41—42no—
107C3×#41—42no—
108C2×C3×Q8—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—41no—
119C2×Dic6—22no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—41no—
123C2×Dic6—82no—
124C2×C3×C8—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—42no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134#217—41no—
135C4⋊Dic3—22no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—22no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150Dic12—84no—
151C2×#41—21no—
152C22×Q8—61no—
153C2×Q16—64no—
154C22×C8—32no—
155C4⋊Q8—32no—
156C4⋊Q8—31no—
157C4⋊Q8—31no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8D1211yescharacteristic
162#127D1211yescharacteristic
163#125—61no—
164#124—21no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167#121—21no—
168#121—21no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×D4—21no—
172C4×D4—21no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#106—64no—
179C4⋊C8—21no—
180C4⋊C8—61no—
181#103—64no—
182#103—62no—
183#103—62no—
184#102—62no—
185#102—62no—
186#102—64no—
187C22⋊C8—32no—
188C22⋊C8#6712yes—
189#97—61no—
190#97—61no—
191C4×C8—21no—
192C4×C8—61no—
193#95—21no—
194#95—21no—
195C23×C3—41no—
196C23×C3#12711yescharacteristic
197C3×Q8—42no—
198C3×D4—42no—
199C3×D4—42no—
200C3×D4—81no—
201C3×D4—81no—
202C3×D4—81no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—82no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#13512yes—
217C2×C3×C4#13712yes—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—42no—
221C2×Dic3—82no—
222Dic6—42no—
223Dic6—44no—
224Dic6—82no—
225Dic6—81no—
226C3×C8—42no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229#41—42no—
230#41—42no—
231C2×Q8—21no—
232C2×Q8—61no—
233C2×Q8—62no—
234C2×Q8—61no—
235C2×Q8—121no—
236C2×Q8—242no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—121no—
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4C2×#6711yescharacteristic
247Q16—244no—
248C2×C8—22no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—62no—
252C2×C8—62no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—42no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—121no—
269C42—121no—
270C42C2×D1211yescharacteristic
271C42#23511yescharacteristic
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—81no—
279C22×C3#39211yescharacteristic
280C22×C3#49511yescharacteristic
281C22×C3#49711yescharacteristic
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—81no—
287C3×C4—161no—
288C3×C4#45212yes—
289C3×C4#46412yes—
290Dic3—82no—
291Dic3—81no—
292Dic3—81no—
293C23—41no—
294C23#94011yescharacteristic
295Q8—42no—
296Q8—122no—
297Q8—124no—
298Q8—241no—
299Q8—242no—
300D4—42no—
301D4—42no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—122no—
315C2×C4—242no—
316C2×C4#88211yescharacteristic
317C2×C4#88211yescharacteristic
318C2×C4#88411yescharacteristic
319C2×C4#88411yescharacteristic
320C2×C4#91212yes—
321C2×C4#91512yes—
322C2×C4#93011yescharacteristic
323C2×C4#93711yescharacteristic
324C8—42no—
325C8—122no—
326C8—122no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—161no—
330C2×C3#200811yescharacteristic
331C2×C3#200911yescharacteristic
332C2×C3#201511yescharacteristic
333C22—21no—
334C22—21no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22#496011yescharacteristic
339C22#515511yescharacteristic
340C22#516711yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—161no—
347C4—241no—
348C4—242no—
349C4—241no—
350C4#476712yes—
351C4#478312yes—
352C3#1022311yescharacteristic, complemented, Sylow
353C2—21no—
354C2—21no—
355C2—161no—
356C2#7005011yescharacteristic
357C2#7021511yescharacteristic
358C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1517C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1517D3
hypocenterC3#10223
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
3. #79313C2semidirect
5. #69555C2semidirect
6. #69546C2semidirect
7. #67741C2semidirect
352. C3#10223semidirect

Maximal subgroups

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

subgroupindex
#832822
#832712
#793132
C3×#15172
#695552
#695462
#677412
#102233