Finite Group Databasebeta

← #10692656

Subgroups — #10692656

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#73550C211yesmaximal, characteristic
2#73445C211yesmaximal, characteristic, complemented
3#69977C211yesmaximal, characteristic
4#69971C211yesmaximal, characteristic, complemented
5#69444C211yesmaximal, characteristic
6C3×#1297C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9414—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16#4958—21no—
17#4958—21no—
18#4943—21no—
19#4942—21no—
20#4940—22no—
21#4933—22no—
22#4672C412yescomplemented
23#4659C412yescomplemented
24#4652—22no—
25#4470—22no—
26#4469—22no—
27#4451—22no—
28#2591—31no—
29#2313—31no—
30#1507—31no—
31#1507—31no—
32#1463—31no—
33#1297—31no—
34#1297D311yescharacteristic, complemented
35C23×C3×C4C2311yescharacteristic
36C22×#67—21no—
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×D12C2×C411yescharacteristic
40C2×C3×M16—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42—21no—
45C2×#217—21no—
46#930—21no—
47#930—21no—
48#930—21no—
49#930—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—22no—
59C2×Dic3⋊C4C2×C412yes—
60C2×Dic3⋊C4D412yes—
61C2×C4×Dic3—21no—
62C2×#208—21no—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#95C2×C412yes—
66C3×#95D412yes—
67C3⋊#95—22no—
68C3⋊#95—24no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95D412yes—
72#830—21no—
73#830—21no—
74C2×#121—31no—
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⋊C4C2×D311yescharacteristic
89C4×C22⋊C4C2×D311yescharacteristic
90#327—62no—
91#327—62no—
92#324—62no—
93C24×C3C2×D411yescharacteristic
94C23×D3—21no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—81no—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×#67—44no—
105C2×#67—41no—
106C2×#67—41no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—22no—
113C22×Dic3—42no—
114C22×Dic3—81no—
115C22×Dic3C22⋊C412yes—
116C22×Dic3C22⋊C412yes—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—81no—
120C3×M16—81no—
121C2×C3×C8—41no—
122C3×C22⋊C4—42no—
123C3×C22⋊C4—42no—
124C3×C42—81no—
125C3×C42—81no—
126#217—42no—
127(C2×D3)⋊C4—44no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—82no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133Dic3⋊C4—22no—
134Dic3⋊C4—22no—
135Dic3⋊C4—22no—
136Dic3⋊C4—22no—
137Dic3⋊C4—82no—
138C4×Dic3—81no—
139#208—81no—
140C2×C3⋊C8—41no—
141C22×D4—31no—
142C22×D4—61no—
143C23×C4—31no—
144C23×C4—31no—
145C23×C4C22×D311yescharacteristic
146C2×M16—21no—
147C2×M16—61no—
148#121—61no—
149#121—61no—
150#121—61no—
151#121—61no—
152C2×C4⋊C4—32no—
153C2×C4⋊C4—32no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—62no—
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—21no—
168C2×C42—61no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173#95—32no—
174#95—62no—
175#95—62no—
176#95—62no—
177#95—64no—
178#95C4×D312yes—
179#95D1212yes—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—41no—
183C23×C3C2×C22⋊C411yescharacteristic
184C23×C3#12011yescharacteristic
185C23×C3#12311yescharacteristic
186C22×D3—42no—
187C22×D3—42no—
188C22×D3—41no—
189C22×D3—42no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206#67—82no—
207#67—82no—
208C2×Dic3—22no—
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—42no—
221C2×Dic3—81no—
222D12—81no—
223D12—81no—
224C3×C8—81no—
225C3⋊C8—81no—
226C24—61no—
227C24D3×D411yescharacteristic
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—124no—
231C2×D4—121no—
232C2×D4—121no—
233C2×D4—241no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—32no—
237C22×C4—32no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—62no—
244C22×C4—81no—
245C22×C4—122no—
246C22×C4—241no—
247C22×C4C2×C4×D311yescharacteristic
248C22×C4C2×D1211yescharacteristic
249C22×C4#23511yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4#23911yescharacteristic
253M16—81no—
254M16—241no—
255C2×C8—41no—
256C2×C8—121no—
257C4⋊C4—62no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—242no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—124no—
267C22⋊C4—122no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C22⋊C4—242no—
271C42—81no—
272C42—81no—
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#38611yescharacteristic
283C22×C3#42112yes—
284C22×C3#45412yes—
285C2×D3—84no—
286C2×D3—81no—
287C2×D3—81no—
288C2×D3—81no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—82no—
294C3×C4—82no—
295Dic3—42no—
296Dic3—42no—
297Dic3—82no—
298Dic3—81no—
299C23—22no—
300C23—22no—
301C23—41no—
302C23—122no—
303C23—122no—
304C23—121no—
305C23—122no—
306C23D3×C22⋊C411yescharacteristic
307C23#88211yescharacteristic
308C23#89411yescharacteristic
309D4—241no—
310D4—241no—
311D4—242no—
312D4—242no—
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—42no—
324C2×C4—42no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4—241no—
339C2×C4C4×D1211yescharacteristic
340C2×C4#88411yescharacteristic
341C2×C4#89311yescharacteristic
342C2×C4#89511yescharacteristic
343C8—81no—
344C8—241no—
345C2×C3—22no—
346C2×C3—22no—
347C2×C3—41no—
348C2×C3#188811yescharacteristic
349C2×C3#189711yescharacteristic
350C2×C3#191311yescharacteristic
351D3—161no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—44no—
360C22—244no—
361C22—241no—
362C22—241no—
363C22—241no—
364C22#466511yescharacteristic
365C22#482112yes—
366C22#482712yes—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—82no—
373C4—122no—
374C4—122no—
375C4—242no—
376C4—241no—
377C3#941411yescharacteristic, complemented, Sylow
378C2—22no—
379C2—22no—
380C2—41no—
381C2—481no—
382C2#6875711yescharacteristic
383C2#6876611yescharacteristic
384C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1297D3
hypocenterC3#9414
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
2. #73445C2semidirect
4. #69971C2semidirect
6. C3×#1297C2semidirect
7. #67618C2semidirect
22. #4672C4semidirect
23. #4659C4semidirect
34. #1297D3semidirect
377. C3#9414semidirect

Maximal subgroups

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

subgroupindex
#735502
#734452
#699772
#699712
#694442
C3×#12972
#676182
#94143