Finite Group Databasebeta

← #10697412

Subgroups — #10697412

Order 768.

468 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 79 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#83374C211yesmaximal, characteristic, complemented
2#83361C211yesmaximal, characteristic
3#79319C211yesmaximal, characteristic
4C3×#1506C211yesmaximal, characteristic, complemented
5#69440C211yesmaximal, characteristic
6#69433C211yesmaximal, characteristic, complemented
7#67758C211yesmaximal, characteristic, complemented
8#10216—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5094—21no—
15#5090—21no—
16#5087—21no—
17#5084—21no—
18#5067—21no—
19#5064—21no—
20#5052—22no—
21#5031—21no—
22#5020—21no—
23#5003—22no—
24C2×#805—22no—
25#4926C2211yescharacteristic
26#4924C2211yescharacteristic
27#4846—21no—
28#4842—21no—
29#4737—22no—
30#4713—21no—
31#4702—21no—
32C3×#328—22no—
33#4452—22no—
34#3255—31no—
35#3235—31no—
36#3235—31no—
37#1963—31no—
38#1506—31no—
39#1506D311yescharacteristic, complemented
40#1307—31no—
41C2×#235—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#216—22no—
53#930—21no—
54#930—21no—
55#926—21no—
56#924—21no—
57#924—21no—
58C2×#208D412yes—
59#875—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8D412yes—
71C3×#97—21no—
72C3×#95—21no—
73C3×#95—21no—
74#835—22no—
75#830D412yes—
76#821—22no—
77#821—22no—
78#818—22no—
79#817—22no—
80#810—22no—
81#809—22no—
82#809—22no—
83#808—22no—
84#808—22no—
85#807—22no—
86#805—44no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90#803—21no—
91#533—31no—
92#533C2×D311yescharacteristic
93#532—31no—
94#501—61no—
95#501—61no—
96#490—61no—
97#490—61no—
98#482—62no—
99#482—62no—
100#476—61no—
101#476—61no—
102#475—61no—
103#475—61no—
104#474—61no—
105#474—61no—
106#470—62no—
107#462—61no—
108#462—61no—
109#432—31no—
110#432C2×D311yescharacteristic
111#423—31no—
112C2×#104—62no—
113#376C2×D311yescharacteristic
114#328—22no—
115#328—62no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×#67—41no—
123#235—42no—
124#235—42no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×C3×C8—22no—
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×C22⋊C4—41no—
140C3×C42—22no—
141C3×C42—41no—
142C3×C42—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#217—41no—
148#216—82no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C4×Dic3—41no—
160#208—42no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×#41—61no—
164C2×Q16—62no—
165C2×M16—32no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8—31no—
169C4⋊Q8D1211yescharacteristic
170C4⋊Q8D1211yescharacteristic
171#127—31no—
172#125—21no—
173#125—61no—
174#124—61no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179#121—61no—
180#121—61no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×D4—61no—
186C4×D4—61no—
187#117—21no—
188#117—61no—
189C2×C42—21no—
190C2×C42C22×D311yescharacteristic
191#107—62no—
192#107—62no—
193#106—62no—
194#106—62no—
195#106—62no—
196C4⋊C8—21no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#104—124no—
201#103—62no—
202#103—62no—
203#102—62no—
204#102—62no—
205#102—62no—
206#102—62no—
207C22⋊C8—32no—
208C22⋊C8#6712yes—
209#97—21no—
210#97—61no—
211#95—21no—
212#95—21no—
213C23×C3#12711yescharacteristic
214C22×D3—41no—
215C3×Q8—41no—
216C3×Q8—41no—
217C3×Q8—81no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233#67—82no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237D12—42no—
238D12—42no—
239D12—42no—
240C4×D3—81no—
241C4×D3—81no—
242Dic6—42no—
243Dic6—41no—
244Dic6—41no—
245Dic6—81no—
246C3×C8—42no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249#41—122no—
250#41—122no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—121no—
260C22×C4—41no—
261C22×C4—121no—
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4C2×#6711yescharacteristic
265Q16—242no—
266M16—122no—
267C2×C8—22no—
268C2×C8—62no—
269C2×C8—62no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C22⋊C4—41no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C42—22no—
289C42—41no—
290C42—41no—
291C42—121no—
292C42C2×D1211yescharacteristic
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42D3×D411yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3#39211yescharacteristic
300C2×D3—81no—
301C2×D3—81no—
302C2×D3—81no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—82no—
309C3×C4—81no—
310C3×C4—81no—
311C3×C4#45311yescharacteristic
312C3×C4#45411yescharacteristic
313Dic3—81no—
314Dic3—81no—
315Dic3—81no—
316C23—121no—
317C23#94011yescharacteristic
318Q8—41no—
319Q8—41no—
320Q8—81no—
321Q8—122no—
322Q8—121no—
323Q8—121no—
324Q8—241no—
325D4—122no—
326D4—122no—
327D4—122no—
328D4—242no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4#88211yescharacteristic
344C2×C4#88211yescharacteristic
345C2×C4#88411yescharacteristic
346C2×C4#88411yescharacteristic
347C2×C4#93011yescharacteristic
348C2×C4#93711yescharacteristic
349C8—42no—
350C8—122no—
351C8—122no—
352C2×C3—41no—
353C2×C3#200611yescharacteristic
354C2×C3#200911yescharacteristic
355C2×C3#200911yescharacteristic
356D3—161no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—241no—
361C22—241no—
362C22—241no—
363C22#496011yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—82no—
370C4—81no—
371C4—81no—
372C4—241no—
373C4—241no—
374C4—241no—
375C4#482211yescharacteristic
376C4#482711yescharacteristic
377C3#1021611yescharacteristic, complemented, Sylow
378C2—41no—
379C2—481no—
380C2#7005311yescharacteristic
381C2#7035111yescharacteristic
382C2#7039211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1506C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1506D3
hypocenterC3#10216
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
1. #83374C2semidirect
4. C3×#1506C2semidirect
6. #69433C2semidirect
7. #67758C2semidirect
39. #1506D3semidirect
377. C3#10216semidirect

Maximal subgroups

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

subgroupindex
#833742
#833612
#793192
C3×#15062
#694402
#694332
#677582
#102163