Finite Group Databasebeta

← #10687366

Subgroups — #10687366

Order 768.

718 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C3×#3493C211yesmaximal, characteristic
2#71710C212yesmaximal, complemented
3#71710C212yesmaximal, complemented
4#71594C211yesmaximal, characteristic, complemented
5#71584C211yesmaximal, characteristic, complemented
6#8463—31nomaximal, Sylow
7C3×#556—22no—
8C3×#555—22no—
9C3×#553—22no—
10C3×#552—22no—
11C3×#544C411yescharacteristic, complemented
12C3×#538—22no—
13C3×#537C411yescharacteristic, complemented
14C3×#529—22no—
15C3×#525C2212yes—
16C3×#520C412yescomplemented
17C2×C3×#117C2211yescharacteristic
18#5211C2212yes—
19C2×#830C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#4538—22no—
22#4538—22no—
23#4537—22no—
24#4537—22no—
25#4526—22no—
26#4524—22no—
27#3493D311yescharacteristic
28#1820—31no—
29#1818—31no—
30#1516—32no—
31#1516—32no—
32C2×C3×#41C2×C412yes—
33C23×C3×C4C2311yescharacteristic
34C3×C4⋊Q8—22no—
35C3×#126—24no—
36C3×#126—24no—
37C3×#125—22no—
38C3×#125—22no—
39C3×#124—22no—
40C3×#124—24no—
41C3×#123—24no—
42C3×#123—22no—
43C3×C22⋊Q8—22no—
44C3×C22⋊Q8—22no—
45C3×C22⋊Q8D412yes—
46C3×C22⋊Q8D412yes—
47C3×#121—24no—
48C3×#121—22no—
49C3×#121—22no—
50C3×#121D412yes—
51C3×#121D412yes—
52C3×#120—22no—
53C3×C4×Q8—22no—
54C3×C4×Q8—22no—
55C3×C4×D4—22no—
56C3×C4×D4—22no—
57C3×C4×D4—24no—
58C3×C4×D4—22no—
59C3×C4×D4—22no—
60C3×#117—24no—
61C3×#117—22no—
62C3×#117—22no—
63C3×#117—22no—
64C3×#117C2×C412yes—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4C2×C411yescharacteristic
67C2×C3×C4⋊C4C2×C411yescharacteristic
68C2×C3×C22⋊C4D412yes—
69C2×C3×C42D412yes—
70C2×#217—22no—
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75#830—22no—
76#830—22no—
77#556—22no—
78#555—22no—
79#553—22no—
80#552—22no—
81#544Dic311yescharacteristic, complemented
82#538—22no—
83#537Dic311yescharacteristic, complemented
84#529—22no—
85#525C2×D312yes—
86#520Dic312yescomplemented
87C2×#117C2×D311yescharacteristic
88C2×C22⋊C8—31no—
89C2×C22⋊C8—31no—
90#381—32no—
91#342—62no—
92#336—62no—
93#328—62no—
94#328—62no—
95#327—62no—
96#327—62no—
97C24×C3C2×D411yescharacteristic
98C3×#41—44no—
99C2×C3×Q8—22no—
100C2×C3×Q8C22⋊C412yes—
101C2×C3×D4—22no—
102C2×C3×D4—22no—
103C2×C3×D4—22no—
104C2×C3×D4—22no—
105C2×C3×D4C22⋊C412yes—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—42no—
109C22×C3×C4C22⋊C412yes—
110C22×C3×C4C22⋊C412yes—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D412yes—
115C22×Dic3—42no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4—24no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4C22⋊C412yes—
125C3×C4⋊C4C22⋊C412yes—
126C3×C22⋊C4—24no—
127C3×C22⋊C4—24no—
128C3×C22⋊C4—22no—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—22no—
133C3×C22⋊C4—22no—
134C3×C42—24no—
135C3×C42—22no—
136C3×C42—22no—
137C3×C42—22no—
138#217—44no—
139C2×C3⋊C8—42no—
140C2×C3⋊C8—42no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C2×#41C2×Dic312yes—
144C23×C4C22×D311yescharacteristic
145C22×C8—61no—
146C22×C8—61no—
147C4⋊Q8—22no—
148#126—24no—
149#126—24no—
150#125—22no—
151#125—22no—
152#124—22no—
153#124—24no—
154#123—24no—
155#123—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8#6712yes—
159C22⋊Q8#6712yes—
160#121—24no—
161#121—22no—
162#121—22no—
163#121#6712yes—
164#121#6712yes—
165#120—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—24no—
171C4×D4—22no—
172C4×D4—22no—
173#117—24no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117C2×Dic312yes—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C2×Dic311yescharacteristic
180C2×C4⋊C4C2×Dic311yescharacteristic
181C2×C22⋊C4—62no—
182C2×C22⋊C4#6712yes—
183C2×C42#6712yes—
184C22⋊C8—62no—
185C22⋊C8—62no—
186#95—62no—
187#95—62no—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3C2×C22⋊C411yescharacteristic
193C23×C3#12012yes—
194C3×Q8—42no—
195C3×Q8—82no—
196C3×D4—42no—
197C3×D4—44no—
198C3×D4—82no—
199C3×D4—82no—
200C3×D4—82no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—44no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—84no—
214C2×C3×C4C2×C22⋊C412yes—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×Dic3—82no—
218C2×Dic3—82no—
219C2×Dic3—82no—
220C3⋊C8—81no—
221C3⋊C8—81no—
222C24C2×#6711yescharacteristic
223#41—44no—
224C2×Q8—22no—
225C2×Q8#21712yes—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4#21712yes—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—122no—
235C22×C4#21712yes—
236C22×C4#21712yes—
237C22×C4C22×Dic311yescharacteristic
238C22×C4C2×#6711yescharacteristic
239C22×C4C2×#6712yes—
240C22×C4C2×#6712yes—
241C2×C8—122no—
242C2×C8—122no—
243C2×C8—121no—
244C2×C8—121no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4#21712yes—
254C4⋊C4#21712yes—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—22no—
258C22⋊C4—24no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—124no—
264C42—24no—
265C42—22no—
266C42—22no—
267C42—22no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—42no—
276C22×C3—82no—
277C22×C3#37911yescharacteristic
278C22×C3#41012yes—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—22no—
282C3×C4—42no—
283C3×C4—42no—
284C3×C4—42no—
285C3×C4—42no—
286C3×C4—42no—
287Dic3—162no—
288C23—22no—
289C23—22no—
290C23—41no—
291C23—41no—
292C23C2×#21711yescharacteristic
293C23#95312yes—
294Q8—42no—
295Q8—82no—
296D4—42no—
297D4—44no—
298D4—82no—
299D4—82no—
300D4—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—84no—
314C2×C4—242no—
315C2×C4—242no—
316C2×C4—242no—
317C2×C4C2×#21712yes—
318C2×C4#95311yescharacteristic
319C2×C4#95311yescharacteristic
320C8—241no—
321C8—241no—
322C2×C3—22no—
323C2×C3—42no—
324C2×C3—82no—
325C2×C3#178211yescharacteristic
326C2×C3#179411yescharacteristic
327C2×C3#179411yescharacteristic
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—82no—
337C22#524012yes—
338C22#524711yescharacteristic
339C4—21no—
340C4—21no—
341C4—22no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—482no—
348C3#846311yescharacteristic, complemented, Sylow
349C2—22no—
350C2—42no—
351C2—82no—
352C2#7213911yescharacteristic
353C2#7214611yescharacteristic
354C2#7214611yescharacteristic
subgroupG/H
centerC22#5247
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3493C2
socleC22×C3#379
cosocleC23×C3×C4C23
hypercenter#3493D3
hypocenterC3#8463
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3493

Complements

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

NH ≅ G/Ntype
2. #71710C2semidirect
3. #71710C2semidirect
4. #71594C2semidirect
5. #71584C2semidirect
11. C3×#544C4semidirect
13. C3×#537C4semidirect
16. C3×#520C4semidirect
81. #544Dic3semidirect
83. #537Dic3semidirect
86. #520Dic3semidirect
348. C3#8463semidirect

Maximal subgroups

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

subgroupindex
C3×#34932
#717102
#715942
#715842
#84633