Finite Group Databasebeta

← #10693000

Subgroups — #10693000

Order 768.

610 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 97 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#74074C211yesmaximal, characteristic
2#73435C211yesmaximal, characteristic
3#69975C211yesmaximal, characteristic
4#69971C211yesmaximal, characteristic
5#69442C211yesmaximal, characteristic
6C3×#1284C211yesmaximal, characteristic
7#67628C211yesmaximal, characteristic
8#9478—31nomaximal, Sylow
9C2×#926C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#877—21no—
12C3×#407C2211yescharacteristic
13C3×#381C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17#4950—21no—
18#4943—21no—
19Dic3×C22⋊C4—21no—
20C4×Dic3⋊C4—22no—
21#4661C412yescomplemented
22#4659C412yescomplemented
23#4653—22no—
24#4649—22no—
25#4555—22no—
26#4523—22no—
27#4469—22no—
28#2484—31no—
29#2274—31no—
30#1826—31no—
31#1826—31no—
32#1463—31no—
33#1284—31no—
34#1284D311yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C23×Dic3D411yescharacteristic
37C23×Dic3D411yescharacteristic
38C2×C3×M16—21no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C42D412yes—
42C2×#217—21no—
43C2×#217—21no—
44#926—21no—
45#926—21no—
46#926—21no—
47#926—21no—
48C2×C4⋊Dic3—22no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×Dic3⋊C4—22no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4C2×C412yes—
57C2×Dic3⋊C4D412yes—
58C2×C4×Dic3—22no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3D412yes—
62C2×#208—21no—
63#877—28no—
64C3×C22⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×#95—21no—
67C3×#95—21no—
68C3×#95C2×C412yes—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74#830—21no—
75#830—21no—
76C2×#117—31no—
77C2×#117—61no—
78C22×C4⋊C4—31no—
79#407—31no—
80#407C2×D311yescharacteristic
81#400—32no—
82#398—32no—
83#387—62no—
84#385—61no—
85#381—61no—
86#381C2×D311yescharacteristic
87C4×C4⋊C4—62no—
88C4×C4⋊C4—62no—
89C4×C22⋊C4—31no—
90C4×C22⋊C4—61no—
91C4×C22⋊C4C2×D311yescharacteristic
92#339—62no—
93#339—62no—
94#324—62no—
95C24×C3C2×D411yescharacteristic
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—42no—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—24no—
108C22×Dic3—22no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—42no—
114C22×Dic3C22⋊C412yes—
115C22×Dic3C22⋊C412yes—
116C3×M16—42no—
117C2×C3×C8—41no—
118C3×C22⋊C4—24no—
119C3×C22⋊C4—42no—
120C3×C42—42no—
121#217—24no—
122#217—41no—
123#217—41no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—44no—
129Dic3⋊C4—28no—
130Dic3⋊C4—22no—
131Dic3⋊C4—22no—
132Dic3⋊C4—22no—
133Dic3⋊C4—22no—
134C4×Dic3—28no—
135C4×Dic3—41no—
136C4×Dic3—41no—
137C4×Dic3—42no—
138C4×Dic3—82no—
139#208—42no—
140C2×C3⋊C8—41no—
141C23×C4—31no—
142C23×C4—31no—
143C23×C4C22×D311yescharacteristic
144C2×M16—21no—
145C2×M16—61no—
146#117—68no—
147#117—61no—
148#117—61no—
149#117—61no—
150#117—61no—
151C2×C4⋊C4—32no—
152C2×C4⋊C4—32no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—62no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—62no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C42—32no—
166C2×C42—62no—
167C2×C42—61no—
168C2×C42—62no—
169C2×C42D1212yes—
170C22⋊C8—21no—
171C22⋊C8—21no—
172C22⋊C8—61no—
173C22⋊C8—61no—
174#95—21no—
175#95—21no—
176#95—62no—
177#95—62no—
178#95—62no—
179#95—62no—
180#95—61no—
181#95C4×D312yes—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3—41no—
185C23×C3C2×C22⋊C411yescharacteristic
186C23×C3#12011yescharacteristic
187C23×C3#12311yescharacteristic
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—24no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×Dic3—22no—
205C2×Dic3—22no—
206C2×Dic3—24no—
207C2×Dic3—22no—
208C2×Dic3—22no—
209C2×Dic3—24no—
210C2×Dic3—22no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—81no—
220C2×Dic3#10414yes—
221C3×C8—81no—
222C3⋊C8—81no—
223C24D3×D411yescharacteristic
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—32no—
227C22×C4—32no—
228C22×C4—42no—
229C22×C4—64no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—62no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—122no—
238C22×C4C2×C4×D311yescharacteristic
239C22×C4C2×D1211yescharacteristic
240C22×C4#23511yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243C22×C4#23911yescharacteristic
244M16—42no—
245M16—122no—
246C2×C8—41no—
247C2×C8—121no—
248C4⋊C4—68no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—124no—
258C22⋊C4—24no—
259C22⋊C4—42no—
260C22⋊C4—64no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C42—42no—
264C42—68no—
265C42—122no—
266C42—121no—
267C42—121no—
268C42—242no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—44no—
277C22×C3#38611yescharacteristic
278C22×C3C2×#10412yes—
279C22×C3#45812yes—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—44no—
285C3×C4—82no—
286Dic3—24no—
287Dic3—44no—
288Dic3—42no—
289Dic3—81no—
290Dic3—81no—
291C23—22no—
292C23—22no—
293C23—41no—
294C23D3×C22⋊C411yescharacteristic
295C23#88211yescharacteristic
296C23#89411yescharacteristic
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—34no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—64no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—64no—
316C2×C4—62no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—122no—
325C2×C4—241no—
326C2×C4C4×D1211yescharacteristic
327C2×C4#88411yescharacteristic
328C2×C4#89311yescharacteristic
329C2×C4#89511yescharacteristic
330C8—81no—
331C8—241no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—41no—
335C2×C3#189011yescharacteristic
336C2×C3#189611yescharacteristic
337C2×C3#191211yescharacteristic
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—44no—
346C22#466511yescharacteristic
347C22#474212yes—
348C22D3×#10412yes—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—44no—
354C4—64no—
355C4—82no—
356C4—124no—
357C4—122no—
358C4—241no—
359C4—241no—
360C3#947811yescharacteristic, complemented, Sylow
361C2—22no—
362C2—22no—
363C2—41no—
364C2#6870811yescharacteristic
365C2#6875611yescharacteristic
366C2#6896311yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9478
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
21. #4661C4semidirect
22. #4659C4semidirect
360. C3#9478semidirect

Maximal subgroups

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

subgroupindex
#740742
#734352
#699752
#699712
#694422
C3×#12842
#676282
#94783