Finite Group Databasebeta

← #10697358

Subgroups — #10697358

Order 768.

560 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 109 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#79271C211yesmaximal, characteristic, complemented
2#79027C211yesmaximal, characteristic
3#78538C211yesmaximal, characteristic
4#70789C211yesmaximal, characteristic
5#70460C211yesmaximal, characteristic, complemented
6#69868C211yesmaximal, characteristic, complemented
7C3×#1283C211yesmaximal, characteristic, complemented
8#10206—31nomaximal, Sylow
9C2×#909—22no—
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C3×C4×M16C2211yescharacteristic
16#5114—22no—
17C4×C4⋊Dic3C2211yescharacteristic
18#4851—22no—
19#4851—21no—
20#4848—21no—
21#4842—21no—
22#4841—21no—
23#4721—22no—
24#4715—22no—
25#4699—21no—
26#4693—21no—
27#4681—21no—
28#4680—21no—
29#4555—22no—
30#4551—22no—
31#4470—22no—
32#3246—31no—
33#3229—31no—
34#3152—31no—
35#1988—31no—
36#1986—31no—
37#1502—31no—
38#1283D311yescharacteristic, complemented
39C2×#235—21no—
40C22×Dic6—21no—
41C2×C3×M16D412yes—
42C22×C3×C8D412yes—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46#926—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50#924—21no—
51#924—21no—
52C2×C4×Dic3—21no—
53#909—44no—
54#909—44no—
55C2×Dic12—24no—
56C2×#204—24no—
57C2×#204—22no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#870—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68#869D411yescharacteristic
69#869D412yes—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C22⋊C8D412yes—
76C3×#97—21no—
77C3×C4×C8—21no—
78C3×C4×C8—21no—
79C3⋊#95—21no—
80#821—22no—
81#817—24no—
82#817—22no—
83#816—24no—
84#816—22no—
85C2×#137—62no—
86#533—31no—
87#532—31no—
88C2×C4⋊Q8—31no—
89#498—61no—
90#497—62no—
91#494—61no—
92#492—61no—
93#487—61no—
94#483—62no—
95#476—61no—
96#475—62no—
97#475—61no—
98#465—62no—
99#461—61no—
100#460—61no—
101#431C2×D311yescharacteristic
102C2×C4⋊C8C2×D311yescharacteristic
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4—31no—
105#340—62no—
106#339—62no—
107#327—62no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C2×#67—41no—
112C22×Dic3—41no—
113C22×Dic3—41no—
114#235—42no—
115#235—42no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×C4×D3—41no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—22no—
123C2×Dic6—21no—
124C2×Dic6—42no—
125C2×Dic6—42no—
126C2×Dic6—41no—
127C3×M16—22no—
128C3×M16SD1614yescomplemented
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—42no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#217—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—22no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155C4×Dic3—42no—
156Dic12—48no—
157#204—48no—
158#204—82no—
159C2×#41—61no—
160C22×Q8—61no—
161#137—124no—
162#137—124no—
163C2×Q16—64no—
164C2×SD16—64no—
165C2×SD16—62no—
166C2×M16D1212yes—
167C22×C8D1212yes—
168C4⋊Q8—32no—
169C4⋊Q8—31no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172C4⋊Q8—61no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175#127—31no—
176#125—61no—
177#124—61no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180#121—61no—
181#121—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—61no—
185C4×D4—61no—
186#117—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C42—61no—
191C2×C42C22×D311yescharacteristic
192#106—62no—
193#106—64no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197#103—62no—
198#103—64no—
199#102—62no—
200C22⋊C8D1212yes—
201#97—21no—
202C4×C8—21no—
203C4×C8—21no—
204#95—61no—
205C23×C3#12711yescharacteristic
206C22×D3—41no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—41no—
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×C4C2×SD1612yes—
217C2×C3×C4AGL(1,ℤ/8)12yes—
218#67—81no—
219#67—81no—
220C2×Dic3—42no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228C2×Dic3—81no—
229D12—42no—
230D12—41no—
231D12—41no—
232D12—81no—
233C4×D3—82no—
234Dic6—42no—
235Dic6—44no—
236Dic6—42no—
237Dic6—41no—
238Dic6—41no—
239Dic6—82no—
240C3×C8—24no—
241C3×C8—42no—
242C3×C8—42no—
243#41—122no—
244#41—122no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—62no—
249C2×Q8—121no—
250C2×Q8—122no—
251C2×Q8—122no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4C2×D1211yescharacteristic
260C22×C4C2×D1211yescharacteristic
261C22×C4C2×D1211yescharacteristic
262Q16—128no—
263SD16—128no—
264SD16—242no—
265M16—22no—
266M16#20414yescomplemented
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—42no—
271C4⋊C4—62no—
272C4⋊C4—61no—
273C4⋊C4—62no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—62no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C42—121no—
289C42—122no—
290C42D3×D411yescharacteristic
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C42#23911yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3#39211yescharacteristic
297C22×C3#49211yescharacteristic
298C22×C3#49611yescharacteristic
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—81no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4#45012yes—
306C3×C4#45511yescharacteristic
307C3×C4#45611yescharacteristic
308C3×C4#46012yes—
309Dic3—82no—
310Dic3—81no—
311Dic3—81no—
312Dic3—82no—
313Dic3—81no—
314C23—121no—
315C23#87411yescharacteristic
316Q8—122no—
317Q8—124no—
318Q8—122no—
319Q8—121no—
320Q8—121no—
321Q8—242no—
322D4—122no—
323D4—121no—
324D4—121no—
325D4—241no—
326D4—241no—
327D4—241no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—122no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—122no—
336C2×C4—241no—
337C2×C4—241no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4—242no—
341C2×C4#88211yescharacteristic
342C2×C4#88211yescharacteristic
343C2×C4#88211yescharacteristic
344C2×C4#89511yescharacteristic
345C2×C4#89511yescharacteristic
346C2×C4#89511yescharacteristic
347C2×C4C2×#20412yes—
348C2×C4#90812yes—
349C8—24no—
350C8—42no—
351C8—42no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3#200711yescharacteristic
355C2×C3#200911yescharacteristic
356C2×C3#201411yescharacteristic
357D3—161no—
358C22—21no—
359C22—21no—
360C22—241no—
361C22—241no—
362C22—241no—
363C22#467111yescharacteristic
364C22#469311yescharacteristic
365C22#471211yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—242no—
370C4—241no—
371C4—242no—
372C4—241no—
373C4—241no—
374C4#473112yes—
375C4#474611yescharacteristic
376C4#475211yescharacteristic
377C4#484812yes—
378C3#1020611yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—481no—
382C2#6867611yescharacteristic
383C2#6900911yescharacteristic
384C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1283D3
hypocenterC3#10206
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
1. #79271C2semidirect
5. #70460C2semidirect
6. #69868C2semidirect
7. C3×#1283C2semidirect
38. #1283D3semidirect
128. C3×M16SD16semidirect
266. M16#204semidirect
378. C3#10206semidirect

Maximal subgroups

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

subgroupindex
#792712
#790272
#785382
#707892
#704602
#698682
C3×#12832
#102063