Finite Group Databasebeta

← #10697032

Subgroups — #10697032

Order 768.

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

normal subgroups only

This group has 578 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#83199C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic, complemented
3#79303C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic, complemented
5#70521C211yesmaximal, characteristic, complemented
6#70498C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#10160—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#874C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×#823—22no—
15#5093—21no—
16#5089—22no—
17#5088—22no—
18#5088—21no—
19#5088—22no—
20#4966—22no—
21C2×#809—22no—
22C4×#208C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4843—22no—
25#4732—22no—
26#4695—21no—
27#4694—21no—
28C3×#333—22no—
29#4483—22no—
30#3205—31no—
31#3141—31no—
32C2×#486—31no—
33#1965—31no—
34#1548D311yescharacteristic, complemented
35#1492—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C2×#235—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12D412yes—
42C2×C3×M16—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50#930—21no—
51#930—21no—
52C2×#208D412yes—
53C22×C3⋊C8—21no—
54C2×D24—24no—
55#875—21no—
56#874—21no—
57#874—21no—
58#874—21no—
59#874—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63#869D411yescharacteristic
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69#823—44no—
70#821—24no—
71#810—24no—
72#809—28no—
73#809—24no—
74#803—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8D412yes—
78#532—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×#127—31no—
81#493—61no—
82#489—62no—
83#487—61no—
84#486—61no—
85#486—62no—
86#486—61no—
87#486—62no—
88#459—62no—
89#447—62no—
90#431C2×D311yescharacteristic
91#423C2×D311yescharacteristic
92#417—62no—
93C2×#102—62no—
94C2×#100—62no—
95C4×M16—31no—
96C2×C4×C8—31no—
97#333—22no—
98D4⋊C8—62no—
99C23×D3—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—41no—
102C2×C3×Q8—82no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×#67—41no—
108#235—42no—
109#235—42no—
110C2×D12—22no—
111C2×D12—21no—
112C2×D12—22no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—44no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—42no—
121C2×D12—42no—
122C2×D12—82no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C3×M16—42no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140#208—22no—
141#208—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—42no—
145D24—84no—
146C2×#41—61no—
147C22×Q8—21no—
148C22×D4—32no—
149C22×D4—61no—
150C22×D4—61no—
151C2×D8—64no—
152C2×M16—21no—
153C2×M16—32no—
154C22×C8—61no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—31no—
159C4⋊Q8D1212yes—
160#127—31no—
161#127—61no—
162#127—61no—
163#127—61no—
164#127—61no—
165#124—61no—
166#121—61no—
167#121—61no—
168C4×D4—61no—
169C4×D4—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C42C22×D311yescharacteristic
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8#6712yes—
176#103—64no—
177#102—68no—
178#102—64no—
179#102—64no—
180#100—124no—
181C22⋊C8—21no—
182#97—61no—
183C4×C8—21no—
184C4×C8—32no—
185C4×C8—61no—
186C4×C8—61no—
187C23×C3#12111yescharacteristic
188C22×D3—41no—
189C22×D3—42no—
190C22×D3—44no—
191C22×D3—82no—
192C22×D3—82no—
193C3×Q8—82no—
194C3×Q8—82no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—42no—
198C2×C3×C4—82no—
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12711yescharacteristic
205C2×C3×C4#13512yes—
206C2×C3×C4AGL(1,ℤ/8)12yes—
207#67—82no—
208C2×Dic3—41no—
209D12—44no—
210D12—42no—
211D12—42no—
212D12—44no—
213D12—42no—
214D12—82no—
215D12—82no—
216D12—82no—
217D12—81no—
218D12—81no—
219C4×D3—81no—
220C4×D3—81no—
221Dic6—42no—
222C3×C8—42no—
223C3×C8—81no—
224C3⋊C8—42no—
225C3⋊C8—42no—
226C24—62no—
227#41—122no—
228#41—122no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—61no—
232C2×Q8—82no—
233C2×D4—61no—
234C2×D4—62no—
235C2×D4—61no—
236C2×D4—62no—
237C2×D4—61no—
238C2×D4—124no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—121no—
245C2×D4—121no—
246C2×D4—242no—
247C22×C4—41no—
248C22×C4—121no—
249C22×C4#23511yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252D8—244no—
253M16—42no—
254M16—62no—
255M16—62no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—122no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42C2×D1211yescharacteristic
270C42D3×D411yescharacteristic
271C42D3×D411yescharacteristic
272C42C2×#6711yescharacteristic
273C22×C3—21no—
274C22×C3—21no—
275C22×C3#39211yescharacteristic
276C22×C3#46312yes—
277C2×D3—82no—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—82no—
281C2×D3—81no—
282C2×D3—82no—
283C2×D3—164no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—82no—
287C3×C4#44912yes—
288C3×C4#45512yes—
289C3×C4#47212yes—
290C3×C4#49511yescharacteristic
291C3×C4#49611yescharacteristic
292Dic3—81no—
293C23—121no—
294C23—122no—
295C23—124no—
296C23—242no—
297C23—242no—
298C23#88411yescharacteristic
299Q8—82no—
300Q8—82no—
301Q8—122no—
302D4—124no—
303D4—122no—
304D4—124no—
305D4—122no—
306D4—122no—
307D4—242no—
308D4—242no—
309D4—241no—
310D4—241no—
311D4—242no—
312D4—242no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—82no—
317C2×C4—121no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4#88211yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#91412yes—
324C2×C4#91912yes—
325C2×C4#93011yescharacteristic
326C2×C4#93711yescharacteristic
327C2×C4#94011yescharacteristic
328C8—42no—
329C8—81no—
330C8—122no—
331C8—122no—
332C2×C3—22no—
333C2×C3#200411yescharacteristic
334C2×C3#201211yescharacteristic
335C2×C3#201811yescharacteristic
336D3—161no—
337D3—162no—
338C22—21no—
339C22—21no—
340C22—241no—
341C22—242no—
342C22—242no—
343C22—241no—
344C22—241no—
345C22—242no—
346C22—484no—
347C22#481912yes—
348C22#496011yescharacteristic
349C4—21no—
350C4—21no—
351C4—82no—
352C4—241no—
353C4#474912yes—
354C4#481012yes—
355C4#517611yescharacteristic
356C4#519111yescharacteristic
357C4#520212yes—
358C3#1016011yescharacteristic, complemented, Sylow
359C2—22no—
360C2—482no—
361C2—481no—
362C2#7009011yescharacteristic
363C2#7027711yescharacteristic
364C2#7034411yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radical#79303C2
hypercenter#1548D3
hypocenterC3#10160
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #83199C2semidirect
2. C2×#5088C2semidirect
4. C3×#1548C2semidirect
5. #70521C2semidirect
6. #70498C2semidirect
7. #67739C2semidirect
34. #1548D3semidirect
358. C3#10160semidirect

Maximal subgroups

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

subgroupindex
#831992
C2×#50882
#793032
C3×#15482
#705212
#704982
#677392
#101603