Finite Group Databasebeta

← #10697905

Subgroups — #10697905

Order 768.

640 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 113 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×#2858C211yesmaximal, characteristic
2#83223C211yesmaximal, characteristic, complemented
3#83169C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#10294—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×C3×#124—22no—
12#5470C2211yescharacteristic
13C3×#390—22no—
14C3×#390—22no—
15C3×C43C2211yescharacteristic
16#5218—22no—
17#5094—22no—
18#5093—21no—
19#5080—21no—
20#5076—21no—
21#5075—22no—
22#5070—21no—
23#4971—22no—
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4547—22no—
28#4544—22no—
29#4523—22no—
30#3229—31no—
31#3208—31no—
32#2858D311yescharacteristic
33#1994—31no—
34#1546—31no—
35#1545—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#124—44no—
50C3×#124—44no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C22⋊C4—28no—
54C2×C3×C42—21no—
55C2×C3×C42—21no—
56C2×C3×C42D412yes—
57C2×C3×C42C2311yescharacteristic
58C2×#214—24no—
59#926—21no—
60#924—21no—
61#924—21no—
62C2×#208—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68#832—24no—
69#830—21no—
70#830—21no—
71#810—24no—
72#808—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79#533—31no—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×#127C2×D311yescharacteristic
82C2×#124—22no—
83#498—61no—
84#492—61no—
85#487—61no—
86#486—61no—
87#483—62no—
88#475—62no—
89#460—62no—
90#450—62no—
91#431—31no—
92#431—31no—
93#423—31no—
94#390—22no—
95#390—22no—
96C43C2×D311yescharacteristic
97#339—62no—
98#332—62no—
99#331—62no—
100C24×C3—22no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—42no—
104C2×C3×Q8—42no—
105C2×C3×D4—22no—
106C2×C3×D4—22no—
107C2×C3×D4—44no—
108C2×C3×D4—44no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—42no—
114C2×C3×D4—42no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C2×Dic6—21no—
121C2×Dic6—21no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C22⋊C4—416no—
127C3×C42—22no—
128C3×C42—42no—
129C3×C42—42no—
130C3×C42SD1614yescomplemented
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#217—41no—
136#214—84no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143C4×Dic3—41no—
144#208—42no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C22×Q8—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4#6712yes—
152C2×SD16—64no—
153C2×M16—61no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160#127—21no—
161#127—21no—
162#127—21no—
163#127—21no—
164#125—61no—
165#124—44no—
166#124—44no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171#117—61no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4#6712yes—
174C2×C22⋊C4—28no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42#6712yes—
178C2×C42C22×D311yescharacteristic
179#106—64no—
180C4⋊C8—32no—
181C4⋊C8—32no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184#103—64no—
185#102—64no—
186C22⋊C8—61no—
187C22⋊C8—61no—
188C4×C8—61no—
189C4×C8—61no—
190C23×C3—44no—
191C23×C3—44no—
192C23×C3—42no—
193C23×C3—44no—
194C23×C3#12011yescharacteristic
195C3×Q8—82no—
196C3×Q8—81no—
197C3×Q8—81no—
198C3×D4—44no—
199C3×D4—82no—
200C3×D4—82no—
201C3×D4—82no—
202C3×D4—82no—
203C3×D4—81no—
204C3×D4—81no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—24no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—44no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218C2×C3×C4C2×SD1612yes—
219C2×C3×C4AGL(1,ℤ/8)12yes—
220C2×Dic3—41no—
221C2×Dic3—41no—
222Dic6—42no—
223Dic6—42no—
224C3⋊C8—42no—
225C3⋊C8—42no—
226C3⋊C8—81no—
227C24—22no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4C2×#6711yescharacteristic
247C22×C4C2×#6711yescharacteristic
248C22×C4C2×#6711yescharacteristic
249SD16—244no—
250M16—122no—
251C2×C8—62no—
252C2×C8—62no—
253C2×C8—121no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—416no—
265C22⋊C4—121no—
266C42—22no—
267C42—42no—
268C42—42no—
269C42—121no—
270C42#21414yescomplemented
271C42D3×D411yescharacteristic
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
274C42#23711yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—88no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3—82no—
281C22×C3#39211yescharacteristic
282C22×C3#45012yes—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—44no—
286C3×C4—82no—
287C3×C4#45412yes—
288C3×C4#45512yes—
289C3×C4#46012yes—
290C3×C4#49211yescharacteristic
291C3×C4#49611yescharacteristic
292Dic3—81no—
293Dic3—81no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23—42no—
298C23#95311yescharacteristic
299Q8—81no—
300Q8—82no—
301Q8—81no—
302Q8—122no—
303Q8—122no—
304D4—44no—
305D4—82no—
306D4—82no—
307D4—82no—
308D4—81no—
309D4—82no—
310D4—81no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4#93212yes—
321C2×C4C2×#21412yes—
322C2×C4#93711yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93811yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#93911yescharacteristic
327C2×C4#94011yescharacteristic
328C8—122no—
329C8—122no—
330C8—241no—
331C2×C3—22no—
332C2×C3—162no—
333C2×C3#200311yescharacteristic
334C2×C3#200411yescharacteristic
335C2×C3#201411yescharacteristic
336C22—21no—
337C22—21no—
338C22—88no—
339C22—82no—
340C22—82no—
341C22—82no—
342C22#521812yes—
343C22#522311yescharacteristic
344C4—21no—
345C4—21no—
346C4—44no—
347C4—82no—
348C4—241no—
349C4—241no—
350C4#506112yes—
351C4#506212yes—
352C4#507411yescharacteristic
353C4#507512yes—
354C4#507611yescharacteristic
355C3#1029411yescharacteristic, complemented, Sylow
356C2—22no—
357C2—162no—
358C2#7173011yescharacteristic
359C2#7173111yescharacteristic
360C2#7176811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2858C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenter#2858D3
hypocenterC3#10294
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2858

Complements

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

NH ≅ G/Ntype
2. #83223C2semidirect
4. #71773C2semidirect
6. #69396C2semidirect
7. #67611C2semidirect
130. C3×C42SD16semidirect
270. C42#214semidirect
355. C3#10294semidirect

Maximal subgroups

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

subgroupindex
C3×#28582
#832232
#831692
#717732
#717592
#693962
#676112
#102943