Finite Group Databasebeta

← #10708216

Subgroups — #10708216

Order 768.

452 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 57 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#76823C211yesmaximal, characteristic
2#69701C211yesmaximal, characteristic
3#68958C211yesmaximal, characteristic
4#68117C211yesmaximal, characteristic
5C3×#1351C211yesmaximal, characteristic
6#67492C211yesmaximal, characteristic
7#67447C211yesmaximal, characteristic
8#12252—31nomaximal, Sylow
9C2×#944—21no—
10#5483C2211yescharacteristic
11C2×#878—21no—
12C3×#393C2211yescharacteristic
13#5228—21no—
14#4950—21no—
15#4949C2211yescharacteristic
16#4946C2211yescharacteristic
17#4945—21no—
18#4722—21no—
19#4657—21no—
20#4656—21no—
21#4652—21no—
22C3×#342C2211yescharacteristic
23C3×#323C2211yescharacteristic
24#4527—21no—
25#4525C2211yescharacteristic
26#4467—21no—
27#4452—21no—
28#2677—31no—
29#2062—31no—
30#1614—31no—
31#1602—31no—
32#1352—31no—
33#1351D311yescharacteristic
34#1326—31no—
35C22×C3×Q8D411yescharacteristic
36C23×Dic3D411yescharacteristic
37C22×Dic6—21no—
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4C2311yescharacteristic
41C2×#217—21no—
42C2×#217—21no—
43C2×#217D411yescharacteristic
44#944—44no—
45C2×#216—41no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—41no—
48C2×C4⋊Dic3D411yescharacteristic
49#924—21no—
50#924—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×C4×Dic3—21no—
57#878—44no—
58#878—41no—
59#878—41no—
60C3×C22⋊C8—21no—
61C3×#95—21no—
62C3×#95D411yescharacteristic
63#835—41no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—41no—
68C3⋊#95D411yescharacteristic
69#830—21no—
70#818—41no—
71#816—41no—
72#543—31no—
73C2×C4⋊Q8—61no—
74C2×C22⋊Q8—61no—
75#484—61no—
76#451—61no—
77#399—31no—
78#398—61no—
79#395—61no—
80#393C2×D311yescharacteristic
81#388—61no—
82#387—61no—
83#385—31no—
84#385—61no—
85#342—61no—
86#342C2×D311yescharacteristic
87#328—61no—
88#328—61no—
89#323—31no—
90#323C2×D311yescharacteristic
91C24×C3C2×D411yescharacteristic
92C2×C3×Q8—42no—
93C2×C3×Q8—41no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—41no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—42no—
106C22×Dic3—41no—
107C2×Dic6—21no—
108C2×Dic6—21no—
109C2×Dic6—42no—
110C2×Dic6—42no—
111C2×Dic6—81no—
112C2×C3×C8—41no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—41no—
117#217—21no—
118#217—42no—
119#217—41no—
120#217—41no—
121#217Q1612yescomplemented
122#216—82no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—42no—
126C4⋊Dic3—41no—
127C4⋊Dic3—82no—
128Dic3⋊C4—24no—
129Dic3⋊C4—42no—
130Dic3⋊C4—42no—
131Dic3⋊C4—42no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—42no—
135C4×Dic3—42no—
136C2×C3⋊C8—41no—
137C22×Q8—61no—
138C22×Q8D1211yescharacteristic
139C23×C4—31no—
140C2×Q16—121no—
141C4⋊Q8—124no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144C22⋊Q8—121no—
145C22⋊Q8—121no—
146C22⋊Q8—124no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—121no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4C22×D311yescharacteristic
161C2×C42—61no—
162#107—121no—
163#103—121no—
164#103—121no—
165C22⋊C8—21no—
166C22⋊C8—61no—
167#95—21no—
168#95—31no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—121no—
173#95D1211yescharacteristic
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3#12011yescharacteristic
179C23×C3C2×Q1611yescharacteristic
180C23×C3#13711yescharacteristic
181C3×Q8—42no—
182C3×Q8—81no—
183C2×C3×C4—22no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×Dic3—22no—
197C2×Dic3—22no—
198C2×Dic3—22no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214Dic6—84no—
215Dic6—81no—
216Dic6—81no—
217C3×C8—81no—
218C3⋊C8—81no—
219C24D3×D411yescharacteristic
220C2×Q8—42no—
221C2×Q8—41no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—122no—
225C2×Q8—122no—
226C2×Q8—241no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—122no—
237C22×C4—121no—
238C22×C4C2×D1211yescharacteristic
239C22×C4D3×D411yescharacteristic
240Q16—242no—
241C2×C8—41no—
242C2×C8—121no—
243C4⋊C4—64no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—121no—
247C4⋊C4—122no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—122no—
251C4⋊C4—122no—
252C4⋊C4—122no—
253C4⋊C4—122no—
254C4⋊C4—242no—
255C22⋊C4—22no—
256C22⋊C4—32no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—61no—
261C22⋊C4—122no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C42—122no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—81no—
274C22×C3#45111yescharacteristic
275C22×C3#45611yescharacteristic
276C22×C3B(4,𝔽2)11yescharacteristic
277C3×C4—41no—
278C3×C4—42no—
279C3×C4—41no—
280C3×C4—81no—
281C3×C4—82no—
282Dic3—42no—
283Dic3—42no—
284Dic3—42no—
285Dic3—81no—
286Dic3—81no—
287Dic3—81no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23#88211yescharacteristic
293C23#91511yescharacteristic
294C23D3×Q1611yescharacteristic
295Q8—42no—
296Q8—81no—
297Q8—244no—
298Q8—241no—
299Q8—241no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—122no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4—241no—
331C8—81no—
332C8—241no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3#160511yescharacteristic
338C2×C3#219611yescharacteristic
339C2×C3#220111yescharacteristic
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22#474111yescharacteristic
350C22#480911yescharacteristic
351C22#482611yescharacteristic
352C4—41no—
353C4—42no—
354C4—41no—
355C4—81no—
356C4—82no—
357C4—122no—
358C4—122no—
359C4—122no—
360C4—241no—
361C4—241no—
362C4—241no—
363C3#1225211yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2#6832711yescharacteristic
369C2#6913011yescharacteristic
370C2#6915511yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1351C2
socleC22×C3B(4,𝔽2)
prime radicalC24×C3C2×D4
hypercenter#1351D3
hypocenterC3#12252
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
121. #217Q16semidirect
363. C3#12252semidirect

Maximal subgroups

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

subgroupindex
#768232
#697012
#689582
#681172
C3×#13512
#674922
#674472
#122523