Finite Group Databasebeta

← #10697451

Subgroups — #10697451

Order 768.

484 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3209C211yesmaximal, characteristic, complemented
2#83335C211yesmaximal, characteristic
3#83216C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#70790C211yesmaximal, characteristic
7#67741C211yesmaximal, characteristic, complemented
8#10223—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#494—21no—
13C3×#487—21no—
14C3×#462—22no—
15C3×#451—22no—
16C3×#431C2211yescharacteristic
17#5089—21no—
18#5089—21no—
19#5088—21no—
20#5084—22no—
21#5080—21no—
22#5069—21no—
23#5067—21no—
24#5002—22no—
25#4968—22no—
26#4936C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4538—22no—
30#4529—22no—
31#4470—22no—
32#3227—31no—
33#3209D311yescharacteristic, complemented
34#3135—31no—
35#1993—31no—
36#1518—31no—
37#1517—31no—
38#1281—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C2×C3×Q16—24no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57#930—21no—
58#930—21no—
59C2×C4⋊Dic3—21no—
60C22×C3⋊C8D412yes—
61#875—21no—
62#874D411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65#869D411yescharacteristic
66C3×C4⋊C8—21no—
67C3×#103—24no—
68C3×C22⋊C8D412yes—
69C3×C4×C8—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72#830D412yes—
73#810—22no—
74#810—22no—
75#809—24no—
76#809—22no—
77#809—22no—
78#808—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84#532—31no—
85C2×C4⋊Q8C2×D311yescharacteristic
86#498—61no—
87#494—21no—
88#491—61no—
89#490—61no—
90#489—61no—
91#489—61no—
92#487—21no—
93#486—61no—
94#481—62no—
95#476—62no—
96#462—22no—
97#451—22no—
98#431—31no—
99#431C2×D311yescharacteristic
100#416—62no—
101C2×#97—31no—
102#382—31no—
103#340—62no—
104#328—62no—
105#327—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C2×#67—41no—
116C22×Dic3—41no—
117#235—42no—
118#235—42no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×C4×D3—41no—
123C2×Dic6—21no—
124C3×Q16—84no—
125C2×C3×C8—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—42no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×#41—61no—
152C22×Q8—21no—
153C2×Q16—24no—
154C22×C8—32no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—31no—
159C4⋊Q8#6712yes—
160C4⋊Q8#6711yescharacteristic
161C4⋊Q8#6711yescharacteristic
162#127—31no—
163#125—21no—
164#124—61no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167#121—61no—
168#121—61no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×D4—61no—
172C4×D4—61no—
173#117—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#106—64no—
179C4⋊C8—21no—
180C4⋊C8—61no—
181#103—24no—
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—61no—
194#95—61no—
195C23×C3#12711yescharacteristic
196C22×D3—41no—
197C3×Q8—42no—
198C3×Q8—44no—
199C3×Q8—81no—
200C3×Q8—82no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—82no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#13512yes—
214C2×C3×C4#13712yes—
215#67—81no—
216#67—81no—
217C2×Dic3—41no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221D12—42no—
222D12—42no—
223D12—81no—
224C4×D3—82no—
225Dic6—42no—
226C3×C8—42no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229#41—122no—
230#41—122no—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×Q8—21no—
234C2×Q8—41no—
235C2×Q8—61no—
236C2×Q8—82no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—121no—
241C22×C4—41no—
242C22×C4—121no—
243C22×C4—121no—
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4C2×#6711yescharacteristic
247Q16—84no—
248C2×C8—22no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—62no—
252C2×C8—62no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—122no—
266C22⋊C4—41no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—41no—
270C42D3×D411yescharacteristic
271C42#23711yescharacteristic
272C42C2×#6711yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#39211yescharacteristic
277C22×C3#49511yescharacteristic
278C22×C3#49711yescharacteristic
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—82no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4#45212yes—
290C3×C4#46412yes—
291Dic3—81no—
292Dic3—161no—
293C23—121no—
294C23#94011yescharacteristic
295Q8—42no—
296Q8—44no—
297Q8—81no—
298Q8—82no—
299Q8—122no—
300D4—122no—
301D4—122no—
302D4—241no—
303D4—241no—
304D4—241no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—82no—
311C2×C4—121no—
312C2×C4—242no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4#91812yes—
317C2×C4#91912yes—
318C2×C4#93711yescharacteristic
319C2×C4#93711yescharacteristic
320C2×C4#93811yescharacteristic
321C2×C4#93911yescharacteristic
322C2×C4#93911yescharacteristic
323C2×C4#95311yescharacteristic
324C8—42no—
325C8—122no—
326C8—122no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3#200811yescharacteristic
330C2×C3#200911yescharacteristic
331C2×C3#201511yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22#519011yescharacteristic
339C22#519111yescharacteristic
340C22#522311yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—82no—
347C4—81no—
348C4—241no—
349C4—481no—
350C4#518512yes—
351C4#518712yes—
352C3#1022311yescharacteristic, complemented, Sylow
353C2—21no—
354C2—21no—
355C2—481no—
356C2#7185811yescharacteristic
357C2#7207211yescharacteristic
358C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3209C2
socleC22×C3#392
prime radical#5477C22
hypercenter#3209D3
hypocenterC3#10223
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
1. C3×#3209C2semidirect
4. #71838C2semidirect
5. #71788C2semidirect
7. #67741C2semidirect
33. #3209D3semidirect
352. C3#10223semidirect

Maximal subgroups

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

subgroupindex
C3×#32092
#833352
#832162
#718382
#717882
#707902
#677412
#102233