Finite Group Databasebeta

← #10692624

Subgroups — #10692624

Order 768.

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

normal subgroups only

This group has 618 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#82785C211yesmaximal, characteristic
2#73435C211yesmaximal, characteristic
3C3×#1508C211yesmaximal, characteristic
4#71688C211yesmaximal, characteristic
5#69971C211yesmaximal, characteristic
6#67683C211yesmaximal, characteristic
7#67618C211yesmaximal, characteristic
8#9408—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#877—21no—
12C3×#407C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5230C412yescomplemented
15C4×#217C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18#4983C412yescomplemented
19#4976—22no—
20#4946—21no—
21#4946—21no—
22Dic3×C22⋊C4—21no—
23C4×Dic3⋊C4—22no—
24#4649—22no—
25C3×#328—22no—
26#4538—22no—
27#4469—22no—
28#2583—31no—
29#2274—31no—
30#1508—31no—
31#1508D311yescharacteristic
32#1463—31no—
33#1297—31no—
34#1297—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C2×C3×M16—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C42—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×Dic3⋊C4—22no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—22no—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4C2×C412yes—
59C2×Dic3⋊C4D412yes—
60C2×C4×Dic3—22no—
61C2×C4×Dic3—22no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65#877—28no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95D412yes—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95C2×C412yes—
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C2×C22⋊Q8C2×D311yescharacteristic
77C2×#117—61no—
78C22×C4⋊C4—31no—
79#407—31no—
80#407C2×D311yescharacteristic
81#398—32no—
82#395—32no—
83#389—62no—
84#385—61no—
85#385—61no—
86C4×C4⋊C4—62no—
87C4×C4⋊C4—62no—
88C4×C22⋊C4—31no—
89C4×C22⋊C4—31no—
90C4×C22⋊C4—61no—
91C4×C22⋊C4C2×D311yescharacteristic
92#328—22no—
93#328—62no—
94#324—62no—
95C24×C3C2×D411yescharacteristic
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—81no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—81no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—24no—
112C22×Dic3—22no—
113C22×Dic3—21no—
114C22×Dic3—22no—
115C22×Dic3—21no—
116C22×Dic3—81no—
117C22×Dic3C22⋊C412yes—
118C22×Dic3C22⋊C412yes—
119C3×M16—81no—
120C2×C3×C8—41no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—24no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C42—81no—
129#217—24no—
130#217—24no—
131Dic3⋊C4—28no—
132Dic3⋊C4—22no—
133Dic3⋊C4—22no—
134Dic3⋊C4—22no—
135Dic3⋊C4—22no—
136Dic3⋊C4—44no—
137Dic3⋊C4—82no—
138C4×Dic3—28no—
139C4×Dic3—81no—
140C4×Dic3—82no—
141C4×Dic3—81no—
142#208—81no—
143C2×C3⋊C8—41no—
144C22×Q8C4×D311yescharacteristic
145C23×C4—31no—
146C23×C4—31no—
147C23×C4C22×D311yescharacteristic
148C2×M16—21no—
149C2×M16—61no—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152C22⋊Q8—21no—
153C22⋊Q8—21no—
154#117—68no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—32no—
157C2×C4⋊C4—32no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—62no—
164C2×C4⋊C4—62no—
165C2×C4⋊C4C4×D311yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C42—21no—
171C2×C42—62no—
172C2×C42—62no—
173C2×C42—61no—
174C2×C42—61no—
175C22⋊C8—21no—
176C22⋊C8—21no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179#95—32no—
180#95—32no—
181#95—62no—
182#95—62no—
183#95—62no—
184#95#6712yes—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—41no—
188C23×C3C2×C22⋊C411yescharacteristic
189C23×C3#12011yescharacteristic
190C23×C3#12311yescharacteristic
191C3×Q8—81no—
192C3×Q8—81no—
193C2×C3×C4—24no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—24no—
213C2×Dic3—22no—
214C2×Dic3—22no—
215C2×Dic3—24no—
216C2×Dic3—24no—
217C2×Dic3—22no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—81no—
225C2×Dic3#10414yes—
226C3×C8—81no—
227C3⋊C8—81no—
228C24D3×D411yescharacteristic
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—81no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—32no—
236C22×C4—32no—
237C22×C4—64no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—81no—
245C22×C4—241no—
246C22×C4C2×C4×D311yescharacteristic
247C22×C4#23511yescharacteristic
248C22×C4D3×D411yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4#23711yescharacteristic
251C22×C4C2×#6711yescharacteristic
252M16—81no—
253M16—241no—
254C2×C8—41no—
255C2×C8—121no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—68no—
261C4⋊C4—62no—
262C4⋊C4—62no—
263C4⋊C4—62no—
264C4⋊C4—62no—
265C4⋊C4—124no—
266C4⋊C4—242no—
267C22⋊C4—24no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—64no—
271C22⋊C4—64no—
272C42—68no—
273C42—81no—
274C42—241no—
275C42—241no—
276C42—242no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—44no—
285C22×C3#38611yescharacteristic
286C22×C3C2×#10412yes—
287C22×C3#45612yes—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—44no—
293C3×C4—81no—
294C3×C4—81no—
295Dic3—24no—
296Dic3—44no—
297Dic3—44no—
298Dic3—42no—
299C23—22no—
300C23—22no—
301C23—41no—
302C23D3×C22⋊C411yescharacteristic
303C23#88311yescharacteristic
304C23#93711yescharacteristic
305Q8—81no—
306Q8—81no—
307C2×C4—24no—
308C2×C4—34no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—64no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—64no—
327C2×C4—64no—
328C2×C4—62no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—241no—
336C2×C4#67⋊C411yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4C4×#6711yescharacteristic
339C2×C4#93811yescharacteristic
340C8—81no—
341C8—241no—
342C2×C3—22no—
343C2×C3—22no—
344C2×C3—41no—
345C2×C3#188611yescharacteristic
346C2×C3#189611yescharacteristic
347C2×C3#191311yescharacteristic
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22D3×#10412yes—
357C22#495611yescharacteristic
358C22#520512yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—44no—
364C4—64no—
365C4—81no—
366C4—81no—
367C4—124no—
368C4—124no—
369C4—122no—
370C3#940811yescharacteristic, complemented, Sylow
371C2—22no—
372C2—22no—
373C2—41no—
374C2#7003411yescharacteristic
375C2#7034511yescharacteristic
376C2#7038211yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1508C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1508D3
hypocenterC3#9408
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
14. #5230C4semidirect
18. #4983C4semidirect
370. C3#9408semidirect

Maximal subgroups

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

subgroupindex
#827852
#734352
C3×#15082
#716882
#699712
#676832
#676182
#94083