Finite Group Databasebeta

← #10692654

Subgroups — #10692654

Order 768.

552 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 93 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#82674C211yesmaximal, characteristic
2#73445C211yesmaximal, characteristic, complemented
3C3×#1507C211yesmaximal, characteristic
4#71673C211yesmaximal, characteristic
5#69971C211yesmaximal, characteristic, complemented
6#67683C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9414—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5222C412yescomplemented
14#5220—21no—
15#5220—21no—
16C4×#217C2211yescharacteristic
17C4×#217C2211yescharacteristic
18#5207C2211yescharacteristic
19#4985C412yescomplemented
20#4945—22no—
21#4943—21no—
22#4942—21no—
23#4933—22no—
24#4652—22no—
25C3×#327—22no—
26#4537—22no—
27#4469—22no—
28#2591—31no—
29#2313—31no—
30#1507—31no—
31#1507D311yescharacteristic
32#1463—31no—
33#1297—31no—
34#1297—31no—
35C22×C3×D4—21no—
36C22×C3×D4C2×C411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3D411yescharacteristic
39C23×Dic3D411yescharacteristic
40C2×C3×M16—21no—
41C3×#121—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×#121—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×#217—22no—
52C2×#217—22no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4C2×C412yes—
59C2×Dic3⋊C4D412yes—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62C2×#208—21no—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#95D412yes—
66C3⋊#95—22no—
67C3⋊#95—24no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95C2×C412yes—
71C3⋊#95D412yes—
72#830—21no—
73#830—21no—
74C2×#121C2×D311yescharacteristic
75C22×C4⋊C4—31no—
76#407—31no—
77#407C2×D311yescharacteristic
78#400—32no—
79#394—32no—
80#388—62no—
81#387—62no—
82#387—62no—
83#386—61no—
84#386—61no—
85#381—61no—
86#380—61no—
87C4×C22⋊C4—31no—
88C4×C22⋊C4—31no—
89C4×C22⋊C4C2×D311yescharacteristic
90#327—22no—
91#327—62no—
92#324—62no—
93C24×C3—21no—
94C24×C3C2×D411yescharacteristic
95C2×C3×D4—21no—
96C2×C3×D4—21no—
97C2×C3×D4—44no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—81no—
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4—81no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—22no—
115C22×Dic3—22no—
116C22×Dic3—42no—
117C22×Dic3—81no—
118C22×Dic3C22⋊C412yes—
119C22×Dic3C22⋊C412yes—
120C3×M16—81no—
121C2×C3×C8—41no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C22⋊C4—42no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C42—81no—
128#217—44no—
129#217—42no—
130#217—42no—
131#217—82no—
132Dic3⋊C4—22no—
133Dic3⋊C4—22no—
134Dic3⋊C4—22no—
135Dic3⋊C4—22no—
136Dic3⋊C4—82no—
137C4×Dic3—81no—
138C4×Dic3—81no—
139#208—81no—
140C2×C3⋊C8—41no—
141C22×D4—21no—
142C22×D4C4×D311yescharacteristic
143C23×C4—31no—
144C23×C4—31no—
145C23×C4C22×D311yescharacteristic
146C2×M16—21no—
147C2×M16—61no—
148#121—21no—
149#121—21no—
150#121—21no—
151#121—21no—
152C2×C4⋊C4—32no—
153C2×C4⋊C4—32no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4C4×D311yescharacteristic
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—62no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—62no—
166C2×C42—21no—
167C2×C42—61no—
168C2×C42—61no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173#95—32no—
174#95—32no—
175#95—62no—
176#95—62no—
177#95—62no—
178#95—64no—
179#95#6712yes—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—41no—
183C23×C3—42no—
184C23×C3—42no—
185C23×C3—41no—
186C23×C3—42no—
187C23×C3C2×C22⋊C411yescharacteristic
188C23×C3#12011yescharacteristic
189C23×C3#12311yescharacteristic
190C3×D4—81no—
191C3×D4—81no—
192C3×D4—82no—
193C3×D4—82no—
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—42no—
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—81no—
224C3×C8—81no—
225C3⋊C8—81no—
226C24—21no—
227C24D3×D411yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—44no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—81no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—32no—
237C22×C4—32no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—81no—
245C22×C4—122no—
246C22×C4—241no—
247C22×C4C2×C4×D311yescharacteristic
248C22×C4#23511yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4#23711yescharacteristic
252C22×C4C2×#6711yescharacteristic
253M16—81no—
254M16—241no—
255C2×C8—41no—
256C2×C8—121no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—62no—
263C4⋊C4—242no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—124no—
268C22⋊C4—122no—
269C22⋊C4—122no—
270C22⋊C4—242no—
271C42—81no—
272C42—241no—
273C42—241no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—44no—
282C22×C3—84no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3#38611yescharacteristic
287C22×C3#42112yes—
288C22×C3#45412yes—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—82no—
294C3×C4—81no—
295Dic3—42no—
296Dic3—42no—
297Dic3—82no—
298Dic3—82no—
299C23—22no—
300C23—22no—
301C23—41no—
302C23—42no—
303C23—42no—
304C23—41no—
305C23—42no—
306C23D3×C22⋊C411yescharacteristic
307C23#88311yescharacteristic
308C23#93711yescharacteristic
309D4—81no—
310D4—81no—
311D4—82no—
312D4—82no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—241no—
339C2×C4#67⋊C411yescharacteristic
340C2×C4#88411yescharacteristic
341C2×C4C4×#6711yescharacteristic
342C2×C4#93811yescharacteristic
343C8—81no—
344C8—241no—
345C2×C3—22no—
346C2×C3—22no—
347C2×C3—41no—
348C2×C3—161no—
349C2×C3#188811yescharacteristic
350C2×C3#189711yescharacteristic
351C2×C3#191311yescharacteristic
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—44no—
360C22—84no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#482112yes—
365C22#495611yescharacteristic
366C22#520112yes—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—81no—
373C4—122no—
374C4—122no—
375C4—242no—
376C4—242no—
377C3#941411yescharacteristic, complemented, Sylow
378C2—22no—
379C2—22no—
380C2—41no—
381C2—161no—
382C2#7003411yescharacteristic
383C2#7032511yescharacteristic
384C2#7038311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1507C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1507D3
hypocenterC3#9414
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #73445C2semidirect
5. #69971C2semidirect
6. #67683C2semidirect
7. #67618C2semidirect
13. #5222C4semidirect
19. #4985C4semidirect
377. C3#9414semidirect

Maximal subgroups

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

subgroupindex
#826742
#734452
C3×#15072
#716732
#699712
#676832
#676182
#94143