Finite Group Databasebeta

← #10696600

Subgroups — #10696600

Order 768.

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

normal subgroups only

This group has 512 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#83263C211yesmaximal, characteristic
2#83198C211yesmaximal, characteristic, complemented
3#79351C211yesmaximal, characteristic
4C3×#1558C211yesmaximal, characteristic, complemented
5#70836C211yesmaximal, characteristic, complemented
6#70804C211yesmaximal, characteristic
7#67748C211yesmaximal, characteristic, complemented
8#10098—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#875C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5231—22no—
15#5225—22no—
16#5082—22no—
17#5079—21no—
18#5079—21no—
19#4992—22no—
20#4988—21no—
21#4973—21no—
22#4932C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4845—22no—
25#4739—21no—
26#4731—21no—
27#4715—21no—
28#4714—21no—
29C3×#333—22no—
30#4467—22no—
31#4464—22no—
32#3215—31no—
33#3139—31no—
34#2640—31no—
35#1558D311yescharacteristic, complemented
36#1527—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8D412yes—
40C22×D12D411yescharacteristic
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—24no—
52#926—21no—
53#924—21no—
54#924—21no—
55C2×C4×Dic3—21no—
56C2×#208—21no—
57C2×Dic12—22no—
58C2×#204—22no—
59#875—21no—
60#875—21no—
61#875—21no—
62#875—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#97—21no—
73C3×#97—21no—
74#835—24no—
75#835—24no—
76C3⋊#95—24no—
77C3⋊#95—21no—
78C3⋊#95D412yes—
79#821—22no—
80#816—22no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124—31no—
86#497—61no—
87#491—62no—
88#489—61no—
89#487—61no—
90#487—61no—
91#464—62no—
92#451—61no—
93#450—61no—
94#432C2×D311yescharacteristic
95#432C2×D311yescharacteristic
96#419—62no—
97C4×M16—31no—
98#397—62no—
99#393—62no—
100#391—61no—
101#390—61no—
102#376—31no—
103#333—22no—
104#329—62no—
105#328—62no—
106C23×D3—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8—44no—
110C22×C3×C4—22no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—22no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—42no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—42no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134#217—41no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—84no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—42no—
147C4×Dic3—41no—
148#208—42no—
149C2×C3⋊C8—22no—
150Dic12—82no—
151#204—82no—
152C22×Q8—31no—
153C22×Q8D1212yes—
154C22×D4—31no—
155C2×Q16—62no—
156C2×SD16—62no—
157C2×M16—61no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164#125—61no—
165#124—61no—
166#124—61no—
167#124—61no—
168#124—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—64no—
179C2×C42—61no—
180C2×C42C22×D311yescharacteristic
181C4⋊C8—21no—
182C4⋊C8#6712yes—
183#103—62no—
184#103—64no—
185#103—64no—
186#102—62no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189#97—21no—
190#97—21no—
191#97—32no—
192C4×C8—61no—
193#95—32no—
194#95—64no—
195#95—61no—
196C23×C3#12011yescharacteristic
197C22×D3—42no—
198C22×D3—42no—
199C22×D3—41no—
200C22×D3—42no—
201C3×Q8—44no—
202C3×Q8—82no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—44no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12711yescharacteristic
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—42no—
221D12—42no—
222D12—81no—
223Dic6—42no—
224Dic6—42no—
225Dic6—81no—
226Dic6—81no—
227C3×C8—42no—
228C3×C8—81no—
229C3⋊C8—42no—
230C24—61no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—44no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×Q8—122no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—122no—
242C22×C4—22no—
243C22×C4—62no—
244C22×C4—61no—
245C22×C4C2×D1211yescharacteristic
246C22×C4D3×D411yescharacteristic
247C22×C4D3×D411yescharacteristic
248Q16—242no—
249SD16—242no—
250M16—122no—
251C2×C8—22no—
252C2×C8—41no—
253C2×C8—62no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C22⋊C4—244no—
270C42—122no—
271C42—121no—
272C42#23511yescharacteristic
273C42D3×D411yescharacteristic
274C42D3×D411yescharacteristic
275C42C2×#6711yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#39211yescharacteristic
279C22×C3#45512yes—
280C2×D3—84no—
281C2×D3—81no—
282C2×D3—81no—
283C2×D3—81no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—82no—
290C3×C4#45512yes—
291Dic3—82no—
292Dic3—81no—
293Dic3—81no—
294Dic3—81no—
295C23—122no—
296C23—122no—
297C23—121no—
298C23—122no—
299C23#88211yescharacteristic
300Q8—44no—
301Q8—82no—
302Q8—122no—
303Q8—122no—
304Q8—241no—
305Q8—241no—
306D4—122no—
307D4—241no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—122no—
320C2×C4#88211yescharacteristic
321C2×C4#88411yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#93011yescharacteristic
324C2×C4#93711yescharacteristic
325C2×C4#94011yescharacteristic
326C8—42no—
327C8—81no—
328C8—122no—
329C2×C3—22no—
330C2×C3#200411yescharacteristic
331C2×C3#200411yescharacteristic
332C2×C3#200911yescharacteristic
333D3—161no—
334C22—21no—
335C22—21no—
336C22—244no—
337C22—241no—
338C22—241no—
339C22—241no—
340C22#474612yes—
341C22#496011yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—82no—
348C4—242no—
349C4—241no—
350C4—241no—
351C4—241no—
352C4#520212yes—
353C3#1009811yescharacteristic, complemented, Sylow
354C2—22no—
355C2—481no—
356C2#7008411yescharacteristic
357C2#7034411yescharacteristic
358C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1558C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1558D3
hypocenterC3#10098
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
2. #83198C2semidirect
4. C3×#1558C2semidirect
5. #70836C2semidirect
7. #67748C2semidirect
35. #1558D3semidirect
353. C3#10098semidirect

Maximal subgroups

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

subgroupindex
#832632
#831982
#793512
C3×#15582
#708362
#708042
#677482
#100983