Finite Group Databasebeta

← #10696590

Subgroups — #10696590

Order 768.

542 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#3236C211yesmaximal, characteristic, complemented
2#83336C211yesmaximal, characteristic
3#83067C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic, complemented
5#71735C211yesmaximal, characteristic, complemented
6#70818C211yesmaximal, characteristic
7#67748C211yesmaximal, characteristic, complemented
8#10097—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×#874C2211yescharacteristic
12C3×#501—21no—
13C3×#491—21no—
14C3×#484—21no—
15C3×#483—21no—
16C3×#479—22no—
17C3×#432C2211yescharacteristic
18#5088—21no—
19#5088—21no—
20#5067—22no—
21#5001—22no—
22#4991—22no—
23#4991—22no—
24#4988—21no—
25#4988—21no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4543—22no—
30#4538—22no—
31#4485—22no—
32#3236D311yescharacteristic, complemented
33#3138—31no—
34#2641—31no—
35#1555—31no—
36#1528—31no—
37#1506—31no—
38#1300—31no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×(C2×D3)⋊C4—24no—
60C2×(C2×D3)⋊C4—24no—
61C2×C4×Dic3—21no—
62C2×#208—21no—
63#874—21no—
64#874—21no—
65#874—21no—
66#874—21no—
67C3×#107—22no—
68C3×#106—22no—
69C3×C4⋊C8D412yes—
70C3×#103—22no—
71C3×#103—22no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3⋊#95—21no—
75C3⋊#95D412yes—
76#830—21no—
77#809—24no—
78#809—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×#127—31no—
85C2×#125C2×D311yescharacteristic
86#501—21no—
87#491—21no—
88#490—62no—
89#486—61no—
90#486—61no—
91#484—21no—
92#483—21no—
93#479—22no—
94#432—31no—
95#432C2×D311yescharacteristic
96#418—62no—
97C4×M16—31no—
98#394—62no—
99#394—62no—
100#390—61no—
101#390—61no—
102#376—31no—
103#331—62no—
104#330—62no—
105#328—62no—
106C23×D3—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—22no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—44no—
118C2×D12—44no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—82no—
124C2×C3×C8—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145(C2×D3)⋊C4—84no—
146(C2×D3)⋊C4—84no—
147C4×Dic3—42no—
148#208—42no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C22×D4—32no—
152C22×D4—61no—
153C22×D4—61no—
154C2×M16—61no—
155C4⋊Q8#6711yescharacteristic
156C4⋊Q8#6711yescharacteristic
157#127—61no—
158#127—61no—
159#127—61no—
160#127—61no—
161#125—21no—
162#125—21no—
163#125—21no—
164#125—21no—
165#125—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170#117—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4#6711yescharacteristic
174C2×C4⋊C4#6711yescharacteristic
175C2×C22⋊C4—64no—
176C2×C22⋊C4—64no—
177C2×C42—61no—
178C2×C42C22×D311yescharacteristic
179#107—22no—
180#106—22no—
181C4⋊C8—61no—
182C4⋊C8#6712yes—
183#103—22no—
184#103—22no—
185#102—64no—
186#102—64no—
187C22⋊C8—21no—
188C22⋊C8—61no—
189#97—21no—
190#97—32no—
191#97—61no—
192C4×C8—61no—
193#95—32no—
194#95—61no—
195C23×C3#12011yescharacteristic
196C22×D3—44no—
197C22×D3—44no—
198C22×D3—42no—
199C22×D3—44no—
200C3×Q8—42no—
201C3×Q8—81no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12711yescharacteristic
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220D12—44no—
221D12—82no—
222D12—82no—
223D12—82no—
224D12—82no—
225D12—82no—
226C3×C8—42no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229C24—62no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×D4—62no—
233C2×D4—62no—
234C2×D4—124no—
235C2×D4—124no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—242no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—62no—
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4C2×#6711yescharacteristic
247M16—122no—
248C2×C8—22no—
249C2×C8—62no—
250C2×C8—121no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—244no—
267C22⋊C4—244no—
268C42—41no—
269C42—122no—
270C42D3×D411yescharacteristic
271C42#23711yescharacteristic
272C42C2×#6711yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#39211yescharacteristic
277C22×C3#45512yes—
278C2×D3—88no—
279C2×D3—82no—
280C2×D3—82no—
281C2×D3—82no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4—81no—
291C3×C4#45412yes—
292Dic3—82no—
293C23—124no—
294C23—124no—
295C23—124no—
296C23—122no—
297C23#93711yescharacteristic
298Q8—42no—
299Q8—81no—
300D4—124no—
301D4—242no—
302D4—242no—
303D4—242no—
304D4—242no—
305D4—242no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4#93711yescharacteristic
319C2×C4#93811yescharacteristic
320C2×C4#93911yescharacteristic
321C2×C4#93911yescharacteristic
322C2×C4#94011yescharacteristic
323C2×C4#95311yescharacteristic
324C8—42no—
325C8—122no—
326C8—241no—
327C2×C3—22no—
328C2×C3#200311yescharacteristic
329C2×C3#200411yescharacteristic
330C2×C3#200811yescharacteristic
331D3—162no—
332C22—21no—
333C22—21no—
334C22—248no—
335C22—242no—
336C22—242no—
337C22—242no—
338C22#520212yes—
339C22#522311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—242no—
350C4#524512yes—
351C3#1009711yescharacteristic, complemented, Sylow
352C2—22no—
353C2—482no—
354C2#7186711yescharacteristic
355C2#7210611yescharacteristic
356C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3236C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#3236D3
hypocenterC3#10097
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3236

Complements

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

NH ≅ G/Ntype
1. C3×#3236C2semidirect
4. #71836C2semidirect
5. #71735C2semidirect
7. #67748C2semidirect
32. #3236D3semidirect
351. C3#10097semidirect

Maximal subgroups

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

subgroupindex
C3×#32362
#833362
#830672
#718362
#717352
#708182
#677482
#100973