Finite Group Databasebeta

← #10696616

Subgroups — #10696616

Order 768.

512 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 85 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#79351C211yesmaximal, characteristic
2#78506C211yesmaximal, characteristic, complemented
3#73428C211yesmaximal, characteristic
4#70822C211yesmaximal, characteristic, complemented
5#69860C211yesmaximal, characteristic
6#69433C211yesmaximal, characteristic, complemented
7C3×#1300C211yesmaximal, characteristic, complemented
8#10100—31nomaximal, Sylow
9C2×#875C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×#376C2211yescharacteristic
15#5121—22no—
16#4939—21no—
17#4929—21no—
18#4926C2211yescharacteristic
19#4845—22no—
20#4739—21no—
21#4731—21no—
22#4715—21no—
23#4714—21no—
24#4702—22no—
25#4682—21no—
26#4680—21no—
27#4674—22no—
28#4673—22no—
29#4486—22no—
30#4464—22no—
31#4452—22no—
32#3215—31no—
33#3165—31no—
34#2640—31no—
35#1556—31no—
36#1527—31no—
37#1506—31no—
38#1300D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—24no—
48#926—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D412yes—
51#924—21no—
52#924—21no—
53C2×Dic12—22no—
54C2×#204—22no—
55#875—21no—
56#875—21no—
57#875—21no—
58#875—21no—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3×C4×C8—21no—
73C3×#95—21no—
74C3×#95D412yes—
75C3⋊#95—24no—
76#830—21no—
77#821—22no—
78#818—24no—
79#817—24no—
80#816—22no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×#124—31no—
86#501—62no—
87#500—61no—
88#498—61no—
89#497—61no—
90#489—61no—
91#464—62no—
92#451—61no—
93#450—61no—
94#432—31no—
95#432C2×D311yescharacteristic
96#428—62no—
97C4×M16C2×D311yescharacteristic
98#397—62no—
99#393—62no—
100#391—61no—
101#390—61no—
102#376C2×D311yescharacteristic
103#332—62no—
104#329—62no—
105#328—62no—
106C23×D3—21no—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—22no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—42no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—42no—
121C2×Dic6—41no—
122C2×Dic6—41no—
123C2×Dic6—82no—
124C3×M16—24no—
125C2×C3×C8—22no—
126C2×C3×C8—22no—
127C3×C42—42no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132#217—41no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—84no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—22no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C4×Dic3—41no—
149C2×C3⋊C8—41no—
150Dic12—82no—
151#204—82no—
152C22×Q8—31no—
153C22×Q8—61no—
154C22×D4—31no—
155C2×Q16—62no—
156C2×SD16—62no—
157C2×M16—21no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
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—32no—
175C2×C4⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—64no—
179C2×C42—21no—
180C2×C42C22×D311yescharacteristic
181#107—64no—
182#106—64no—
183C4⋊C8—61no—
184C4⋊C8D1212yes—
185#103—62no—
186#102—62no—
187C22⋊C8—21no—
188C22⋊C8—61no—
189#97—21no—
190#97—61no—
191#97D1212yes—
192C4×C8—21no—
193#95—21no—
194#95—64no—
195#95D1212yes—
196C23×C3#12011yescharacteristic
197C22×D3—42no—
198C22×D3—42no—
199C22×D3—41no—
200C22×D3—42no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12711yescharacteristic
213C2×Dic3—44no—
214C2×Dic3—42no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219D12—42no—
220D12—81no—
221Dic6—42no—
222Dic6—42no—
223Dic6—82no—
224Dic6—82no—
225Dic6—81no—
226Dic6—81no—
227C3×C8—24no—
228C3×C8—42no—
229C3⋊C8—81no—
230C24—61no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—122no—
236C2×Q8—121no—
237C2×Q8—121no—
238C2×Q8—242no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—122no—
242C22×C4—22no—
243C22×C4—61no—
244C22×C4—62no—
245C22×C4C2×D1211yescharacteristic
246C22×C4D3×D411yescharacteristic
247C22×C4D3×D411yescharacteristic
248Q16—242no—
249SD16—242no—
250M16—24no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—121no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—62no—
257C4⋊C4—61no—
258C4⋊C4—62no—
259C4⋊C4—61no—
260C4⋊C4—121no—
261C4⋊C4—121no—
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—42no—
271C42—121no—
272C42C2×D1211yescharacteristic
273C42C2×D1211yescharacteristic
274C42D3×D411yescharacteristic
275C42#23911yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#39211yescharacteristic
279C22×C3#45612yes—
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—81no—
292Dic3—81no—
293Dic3—82no—
294Dic3—81no—
295C23—122no—
296C23—122no—
297C23—121no—
298C23—122no—
299C23#88211yescharacteristic
300Q8—122no—
301Q8—122no—
302Q8—241no—
303Q8—241no—
304Q8—242no—
305Q8—242no—
306D4—122no—
307D4—241no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—124no—
315C2×C4—122no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—122no—
320C2×C4#87411yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#88211yescharacteristic
323C2×C4#89511yescharacteristic
324C2×C4#89511yescharacteristic
325C2×C4#89511yescharacteristic
326C8—24no—
327C8—42no—
328C8—241no—
329C2×C3—22no—
330C2×C3#200411yescharacteristic
331C2×C3#200511yescharacteristic
332C2×C3#200811yescharacteristic
333D3—161no—
334C22—21no—
335C22—21no—
336C22—244no—
337C22—241no—
338C22—241no—
339C22—241no—
340C22#467111yescharacteristic
341C22#482612yes—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—82no—
348C4—241no—
349C4—241no—
350C4—242no—
351C4—241no—
352C4#482512yes—
353C3#1010011yescharacteristic, complemented, Sylow
354C2—22no—
355C2—481no—
356C2#6876211yescharacteristic
357C2#6876811yescharacteristic
358C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1300D3
hypocenterC3#10100
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
2. #78506C2semidirect
4. #70822C2semidirect
6. #69433C2semidirect
7. C3×#1300C2semidirect
38. #1300D3semidirect
353. C3#10100semidirect

Maximal subgroups

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

subgroupindex
#793512
#785062
#734282
#708222
#698602
#694332
C3×#13002
#101003