Finite Group Databasebeta

← #10697188

Subgroups — #10697188

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#83370C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3C2×#4842C211yesmaximal, characteristic
4C3×#1481C211yesmaximal, characteristic, complemented
5#70783C211yesmaximal, characteristic
6#70780C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#10182—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#821—21no—
15C2×#817—21no—
16#5093—21no—
17#5092—21no—
18#5087—21no—
19#5084—21no—
20#5066—22no—
21#5050—22no—
22#5031—22no—
23#5001—21no—
24#4995—21no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4842—22no—
28#4842—22no—
29C3×Q8⋊C8—22no—
30#4482—22no—
31#4468—22no—
32#3209—31no—
33#3077—31no—
34C2×#476—31no—
35#1549—31no—
36#1481D311yescharacteristic, complemented
37#1460—31no—
38#1301—31no—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#216—24no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×#208—21no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D412yes—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71#830D412yes—
72#821—24no—
73#817—24no—
74#810—24no—
75#809—22no—
76#809—22no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×C4×D4—31no—
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#431—31no—
96#431C2×D311yescharacteristic
97#428—61no—
98C2×#106—61no—
99#423—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101#418—61no—
102C2×#102—61no—
103#332—62no—
104Q8⋊C8—22no—
105#323—62no—
106C23×D3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—41no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12Q1612yescomplemented
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×C4×D3—82no—
123C2×Dic6—22no—
124C2×Dic6—41no—
125C2×Dic6—82no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142#216—84no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—22no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151#208—41no—
152#208—41no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C22×Q8—61no—
156C22×D4—31no—
157C23×C4—61no—
158C2×Q16—64no—
159C2×M16—61no—
160C22×C8—21no—
161C4⋊Q8—32no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8D1211yescharacteristic
166C4⋊Q8D1211yescharacteristic
167#125—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176#117—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184#106—64no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—32no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8#6712yes—
191#103—64no—
192#102—62no—
193#102—62no—
194#102—64no—
195C22⋊C8—21no—
196C22⋊C8—32no—
197C4×C8—21no—
198C4×C8—61no—
199C23×C3#12111yescharacteristic
200C22×D3—22no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—81no—
204C22×D3—81no—
205C3×Q8—42no—
206C3×Q8—42no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12711yescharacteristic
217C2×C3×C4C2×Q1611yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4#13711yescharacteristic
220C2×C3×C4#13711yescharacteristic
221C2×Dic3—42no—
222C2×Dic3—41no—
223C2×Dic3—82no—
224C2×Dic3—81no—
225D12—22no—
226D12—22no—
227D12—41no—
228D12—41no—
229C4×D3—82no—
230C4×D3—82no—
231Dic6—44no—
232Dic6—82no—
233C3×C8—42no—
234C3×C8—81no—
235C3⋊C8—42no—
236C3⋊C8—42no—
237C24—61no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—62no—
241C2×Q8—121no—
242C2×Q8—242no—
243C2×D4—32no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C22×C4—61no—
250C22×C4—121no—
251C22×C4—121no—
252C22×C4—121no—
253C22×C4—242no—
254C22×C4#23511yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257Q16—244no—
258M16—121no—
259M16—121no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—62no—
265C2×C8—62no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—62no—
274C4⋊C4—62no—
275C4⋊C4—61no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C22⋊C4—41no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C42—41no—
282C42C2×D1211yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C42C2×#6711yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3#39211yescharacteristic
289C22×C3#46711yescharacteristic
290C22×C3#47011yescharacteristic
291C2×D3—42no—
292C2×D3—82no—
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—82no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—81no—
300C3×C4—81no—
301C3×C4#45111yescharacteristic
302C3×C4#45111yescharacteristic
303C3×C4#45211yescharacteristic
304C3×C4#45211yescharacteristic
305C3×C4#47112yes—
306Dic3—82no—
307Dic3—81no—
308C23—62no—
309C23—121no—
310C23—121no—
311C23—241no—
312C23—241no—
313C23#88411yescharacteristic
314Q8—42no—
315Q8—42no—
316Q8—124no—
317Q8—242no—
318D4—62no—
319D4—62no—
320D4—121no—
321D4—121no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—121no—
327C2×C4—122no—
328C2×C4—242no—
329C2×C4—242no—
330C2×C4—242no—
331C2×C4—241no—
332C2×C4#88211yescharacteristic
333C2×C4#88211yescharacteristic
334C2×C4#88411yescharacteristic
335C2×C4#91511yescharacteristic
336C2×C4#91611yescharacteristic
337C2×C4D3×Q1611yescharacteristic
338C2×C4#91811yescharacteristic
339C2×C4#93011yescharacteristic
340C2×C4#93711yescharacteristic
341C2×C4#94011yescharacteristic
342C8—42no—
343C8—81no—
344C8—122no—
345C8—122no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3#200111yescharacteristic
349C2×C3#200811yescharacteristic
350C2×C3#201911yescharacteristic
351D3—82no—
352C22—21no—
353C22—21no—
354C22—122no—
355C22—242no—
356C22—242no—
357C22—241no—
358C22—241no—
359C22#481311yescharacteristic
360C22#481611yescharacteristic
361C22#496011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—81no—
367C4—242no—
368C4—241no—
369C4#480811yescharacteristic
370C4#480911yescharacteristic
371C4#514112yes—
372C4#517311yescharacteristic
373C4#518611yescharacteristic
374C3#1018211yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—242no—
378C2#7000711yescharacteristic
379C2#7027311yescharacteristic
380C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1481C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1481D3
hypocenterC3#10182
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
1. #83370C2semidirect
4. C3×#1481C2semidirect
6. #70780C2semidirect
7. #67707C2semidirect
36. #1481D3semidirect
119. C2×D12Q16semidirect
374. C3#10182semidirect

Maximal subgroups

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

subgroupindex
#833702
#833502
C2×#48422
C3×#14812
#707832
#707802
#677072
#101823