Finite Group Databasebeta

← #10697189

Subgroups — #10697189

Order 768.

508 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 103 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#82296C211yesmaximal, characteristic
2C2×#5008C211yesmaximal, characteristic
3#79313C211yesmaximal, characteristic, complemented
4C3×#1460C211yesmaximal, characteristic
5#70780C211yesmaximal, characteristic, complemented
6#70768C211yesmaximal, characteristic, complemented
7#67745C211yesmaximal, characteristic, complemented
8#10182—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5215—21no—
15C2×#832—21no—
16#5009—22no—
17#5008—22no—
18#5008—21no—
19#5008—22no—
20#5007—21no—
21#4964—21no—
22C2×#808—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4849—22no—
26#4739—22no—
27#4699—21no—
28#4681—21no—
29C3×#323—22no—
30#4482—22no—
31#4456—22no—
32#3209—31no—
33#3077—31no—
34C2×#476—31no—
35#1549—31no—
36#1481—31no—
37#1460D311yescharacteristic
38#1301—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×Dic6—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49#926—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52#924—21no—
53#924—21no—
54C2×#208—21no—
55C22×C3⋊C8—21no—
56C2×Dic12—24no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8D412yes—
70C3×C4×C8—21no—
71C3×C4×C8—21no—
72#832—24no—
73#832—22no—
74#832—22no—
75#816—24no—
76#808—24no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4⋊(C3⋊C8)D412yes—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×C4×D4C2×D311yescharacteristic
86#494—61no—
87#487—61no—
88#478—62no—
89#476—62no—
90#476—61no—
91#476—62no—
92#474—61no—
93#462—62no—
94#451—62no—
95#431C2×D311yescharacteristic
96#431C2×D311yescharacteristic
97#428—61no—
98C2×#106—61no—
99#423—31no—
100C2×C4⋊C8—31no—
101#418—61no—
102C2×#102—61no—
103#332—62no—
104Q8⋊C8—62no—
105#323—22no—
106C24×C3—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4Q1612yescomplemented
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—82no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—41no—
121C2×Dic6—22no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—41no—
125C2×Dic6—82no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136#217—41no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—22no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147C4×Dic3—41no—
148#208—41no—
149#208—41no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154Dic12—84no—
155C22×Q8—61no—
156C22×D4D1211yescharacteristic
157C23×C4—21no—
158C2×Q16—64no—
159C2×M16—61no—
160C22×C8—61no—
161C4⋊Q8—32no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167#125—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C4×Q8—61no—
171C4×Q8—61no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—21no—
176#117—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4D1211yescharacteristic
180C2×C22⋊C4—21no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—64no—
184#106—62no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—32no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#103—64no—
192#102—64no—
193#102—62no—
194#102—62no—
195C22⋊C8—21no—
196C22⋊C8#6712yes—
197C4×C8—21no—
198C4×C8—21no—
199C23×C3—22no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—81no—
203C23×C3—81no—
204C23×C3#12111yescharacteristic
205C3×D4—22no—
206C3×D4—22no—
207C3×D4—41no—
208C3×D4—41no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—82no—
213C2×C3×C4—82no—
214C2×C3×C4—81no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221C2×C3×C4C2×Q1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228C2×Dic3—82no—
229Dic6—42no—
230Dic6—44no—
231Dic6—42no—
232Dic6—82no—
233C3×C8—42no—
234C3×C8—81no—
235C3⋊C8—42no—
236C3⋊C8—42no—
237C24—21no—
238C2×Q8—61no—
239C2×Q8—62no—
240C2×Q8—61no—
241C2×Q8—121no—
242C2×Q8—242no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4Dic1212yescomplemented
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—82no—
253C22×C4—121no—
254C22×C4C2×D1211yescharacteristic
255C22×C4#23511yescharacteristic
256C22×C4C2×#6711yescharacteristic
257Q16—244no—
258M16—121no—
259M16—121no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—121no—
265C2×C8—121no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—62no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—62no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—121no—
281C42—121no—
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C42D3×D411yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—42no—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3#39211yescharacteristic
294C22×C3#46711yescharacteristic
295C22×C3#47011yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—81no—
300C3×C4#45111yescharacteristic
301C3×C4#45111yescharacteristic
302C3×C4#45211yescharacteristic
303C3×C4#45211yescharacteristic
304C3×C4#47112yes—
305Dic3—82no—
306Dic3—81no—
307Dic3—81no—
308C23—22no—
309C23—41no—
310C23—41no—
311C23—81no—
312C23—81no—
313C23#93011yescharacteristic
314Q8—122no—
315Q8—124no—
316Q8—122no—
317Q8—242no—
318D4—22no—
319D4—22no—
320D4—41no—
321D4—41no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—242no—
332C2×C4#88211yescharacteristic
333C2×C4#88211yescharacteristic
334C2×C4#88411yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#90411yescharacteristic
337C2×C4C2×Dic1211yescharacteristic
338C2×C4#90911yescharacteristic
339C2×C4#90911yescharacteristic
340C2×C4#93711yescharacteristic
341C2×C4#94011yescharacteristic
342C8—42no—
343C8—81no—
344C8—122no—
345C8—122no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—82no—
349C2×C3#200111yescharacteristic
350C2×C3#200811yescharacteristic
351C2×C3#201911yescharacteristic
352C22—21no—
353C22—21no—
354C22—42no—
355C22—82no—
356C22—82no—
357C22—81no—
358C22—81no—
359C22#496011yescharacteristic
360C22#511611yescharacteristic
361C22#513711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—242no—
367C4—241no—
368C4—241no—
369C4#473311yescharacteristic
370C4#473311yescharacteristic
371C4#473911yescharacteristic
372C4#473911yescharacteristic
373C4#474812yes—
374C3#1018211yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—82no—
378C2#6996411yescharacteristic
379C2#7005011yescharacteristic
380C2#7009111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1460C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1460D3
hypocenterC3#10182
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #79313C2semidirect
5. #70780C2semidirect
6. #70768C2semidirect
7. #67745C2semidirect
112. C2×C3×D4Q16semidirect
248. C2×D4Dic12semidirect
374. C3#10182semidirect

Maximal subgroups

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

subgroupindex
#822962
C2×#50082
#793132
C3×#14602
#707802
#707682
#677452
#101823