Finite Group Databasebeta

← #10697302

Subgroups — #10697302

Order 768.

508 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 101 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#79313C211yesmaximal, characteristic, complemented
2#79027C211yesmaximal, characteristic
3#78528C211yesmaximal, characteristic
4#70789C211yesmaximal, characteristic
5#70459C211yesmaximal, characteristic, complemented
6#69868C211yesmaximal, characteristic, complemented
7C3×#1283C211yesmaximal, characteristic, complemented
8#10198—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5133—22no—
16#5114—22no—
17C4×C4⋊Dic3C2211yescharacteristic
18#4851—21no—
19#4849—22no—
20#4848—21no—
21#4842—21no—
22#4841—21no—
23#4739—22no—
24#4714—22no—
25#4699—21no—
26#4694—21no—
27#4681—21no—
28#4681—21no—
29#4555—22no—
30#4551—22no—
31#4470—22no—
32#3246—31no—
33#3209—31no—
34#3141—31no—
35#1987—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—
53C2×Dic12—24no—
54C2×#204—22no—
55#875—21no—
56#874D411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65#869D411yescharacteristic
66#869D412yes—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76C3⋊#95—21no—
77#821—24no—
78#821—22no—
79#817—22no—
80#816—24no—
81#816—24no—
82#816—22no—
83#533—31no—
84#532—31no—
85C2×C4⋊Q8—31no—
86#494—61no—
87#489—62no—
88#487—61no—
89#487—61no—
90#486—61no—
91#476—61no—
92#475—61no—
93#465—62no—
94#462—62no—
95#461—61no—
96#460—61no—
97#451—62no—
98#431C2×D311yescharacteristic
99C2×C4⋊C8C2×D311yescharacteristic
100#417—62no—
101C4×M16C2×D311yescharacteristic
102C4×C4⋊C4—31no—
103#340—62no—
104#339—62no—
105#327—62no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C2×#67—41no—
110C22×Dic3—41no—
111C22×Dic3—41no—
112#235—42no—
113#235—42no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×C4×D3—41no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6—41no—
123C2×Dic6—82no—
124C3×M16—22no—
125C3×M16—22no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—42no—
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—
137C4⋊Dic3—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—22no—
141C4⋊Dic3—21no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C4×Dic3—41no—
152C4×Dic3—42no—
153Dic12—84no—
154#204—82no—
155C2×#41—61no—
156C22×Q8—61no—
157C2×Q16—64no—
158C2×SD16—62no—
159C2×M16D1212yes—
160C22×C8D1212yes—
161C4⋊Q8—32no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168#127—31no—
169#125—61no—
170#124—61no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173#121—61no—
174#121—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×D4—61no—
178C4×D4—61no—
179#117—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185#106—62no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189#103—64no—
190#103—64no—
191#103—62no—
192#102—64no—
193#102—62no—
194C22⋊C8D1212yes—
195#97—21no—
196C4×C8—21no—
197C4×C8—21no—
198#95—61no—
199C23×C3#12711yescharacteristic
200C22×D3—41no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#13512yes—
211C2×C3×C4#13712yes—
212#67—81no—
213#67—81no—
214C2×Dic3—42no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219C2×Dic3—82no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222D12—42no—
223D12—41no—
224D12—41no—
225D12—81no—
226C4×D3—82no—
227Dic6—42no—
228Dic6—44no—
229Dic6—42no—
230Dic6—41no—
231Dic6—41no—
232Dic6—82no—
233C3×C8—42no—
234C3×C8—42no—
235C3×C8—42no—
236#41—122no—
237#41—122no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—62no—
241C2×Q8—61no—
242C2×Q8—121no—
243C2×Q8—242no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—121no—
248C22×C4—121no—
249C22×C4—121no—
250C22×C4—121no—
251C22×C4C2×D1211yescharacteristic
252C22×C4C2×D1211yescharacteristic
253C22×C4C2×D1211yescharacteristic
254Q16—244no—
255SD16—242no—
256M16—22no—
257M16—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—42no—
262C4⋊C4—62no—
263C4⋊C4—62no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—62no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C42—121no—
280C42—122no—
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42#23911yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#39211yescharacteristic
288C22×C3#49511yescharacteristic
289C22×C3#49711yescharacteristic
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4#45212yes—
297C3×C4#45511yescharacteristic
298C3×C4#45611yescharacteristic
299C3×C4#46412yes—
300Dic3—82no—
301Dic3—81no—
302Dic3—82no—
303Dic3—81no—
304Dic3—81no—
305C23—121no—
306C23#87411yescharacteristic
307Q8—122no—
308Q8—124no—
309Q8—122no—
310Q8—121no—
311Q8—121no—
312Q8—242no—
313D4—122no—
314D4—121no—
315D4—121no—
316D4—241no—
317D4—241no—
318D4—241no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—122no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—122no—
327C2×C4—242no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4—242no—
331C2×C4#88211yescharacteristic
332C2×C4#88211yescharacteristic
333C2×C4#88211yescharacteristic
334C2×C4#89511yescharacteristic
335C2×C4#89511yescharacteristic
336C2×C4#89511yescharacteristic
337C2×C4#90412yes—
338C2×C4#90912yes—
339C8—42no—
340C8—42no—
341C8—42no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3#200711yescharacteristic
345C2×C3#200811yescharacteristic
346C2×C3#201511yescharacteristic
347D3—161no—
348C22—21no—
349C22—21no—
350C22—241no—
351C22—241no—
352C22—241no—
353C22#467111yescharacteristic
354C22#469611yescharacteristic
355C22#471511yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—242no—
360C4—241no—
361C4—241no—
362C4—242no—
363C4—241no—
364C4#473312yes—
365C4#474611yescharacteristic
366C4#475211yescharacteristic
367C4#484512yes—
368C3#1019811yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—481no—
372C2#6867711yescharacteristic
373C2#6900911yescharacteristic
374C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1283D3
hypocenterC3#10198
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. #79313C2semidirect
5. #70459C2semidirect
6. #69868C2semidirect
7. C3×#1283C2semidirect
38. #1283D3semidirect
368. C3#10198semidirect

Maximal subgroups

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

subgroupindex
#793132
#790272
#785282
#707892
#704592
#698682
C3×#12832
#101983