Finite Group Databasebeta

← #10694713

Subgroups — #10694713

Order 768.

510 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#5116C211yesmaximal, characteristic
2C3×#1752C211yesmaximal, characteristic
3#71970C211yesmaximal, characteristic
4#69946C212yesmaximal
5#69457C212yesmaximal
6#9784—31nomaximal, Sylow
7C23×C3×C8C2211yescharacteristic
8C2×#924C2212yes—
9C22×Dic12C411yescharacteristic, complemented
10C2×C3×C22⋊C8C2211yescharacteristic
11C2×C3×#97—21no—
12C3×C4×C22⋊C4C2211yescharacteristic
13#5212C2212yes—
14#5116—21no—
15#5116—21no—
16#5116—21no—
17#5116—21no—
18C2×#818C411yescharacteristic, complemented
19C2×#817—21no—
20C2×#816C412yescomplemented
21C3×#336—21no—
22#4551—21no—
23#4527—24no—
24#4467—24no—
25C2×#467—31no—
26#1996—32no—
27#1807—31no—
28#1752D311yescharacteristic
29#1520—32no—
30C23×C3×C4C2311yescharacteristic
31C22×Dic6C2×C412yes—
32C22×C3×C8—21no—
33C22×C3×C8—42no—
34C22×C3×C8C2×C411yescharacteristic
35C22×C3×C8C2×C411yescharacteristic
36C22×C3×C8D411yescharacteristic
37C22×C3×C8D411yescharacteristic
38C22×C3×C8D411yescharacteristic
39C22×C3×C8D411yescharacteristic
40C2×C3×C22⋊C4D412yes—
41C2×C3×C42—21no—
42C2×#217—22no—
43C2×#217—22no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3C2×C412yes—
46#924—22no—
47#924—22no—
48#924—22no—
49#924—22no—
50C2×Dic3⋊C4—22no—
51C2×Dic12—21no—
52C2×Dic12—21no—
53C2×Dic12—42no—
54C3×C22⋊C8—21no—
55C3×C22⋊C8D412yes—
56C3×#97—42no—
57C3×#95—21no—
58C3⋊#95—22no—
59#818—21no—
60#818—21no—
61#817—42no—
62#816—22no—
63#816—22no—
64C22×Q16—31no—
65C23×C8C2×D311yescharacteristic
66C2×C22⋊Q8—32no—
67#467—61no—
68#467—61no—
69#467—61no—
70#467—61no—
71C2×#107—31no—
72C2×#106—61no—
73C2×#103—32no—
74C2×C22⋊C8C2×D311yescharacteristic
75C2×#97—21no—
76#380—32no—
77C4×C22⋊C4C2×D311yescharacteristic
78#342—64no—
79#340—61no—
80#336—21no—
81#328—64no—
82C24×C3C2×D411yescharacteristic
83C22×C3×C4—22no—
84C22×C3×C4—41no—
85C22×C3×C4—41no—
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4111yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×Dic3—22no—
93C22×Dic3—22no—
94C22×Dic3—42no—
95C2×Dic6—22no—
96C2×Dic6—22no—
97C2×Dic6—42no—
98C2×Dic6—42no—
99C2×C3×C8—22no—
100C2×C3×C8—22no—
101C2×C3×C8—22no—
102C2×C3×C8—22no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C2×C3×C8C22⋊C411yescharacteristic
109C2×C3×C8C22⋊C411yescharacteristic
110C2×C3×C8C22⋊C411yescharacteristic
111C2×C3×C8C22⋊C411yescharacteristic
112C3×C22⋊C4—24no—
113C3×C42—41no—
114C3×C42—41no—
115#217—42no—
116#217—42no—
117#217—44no—
118C4⋊Dic3—22no—
119C4⋊Dic3—22no—
120C4⋊Dic3—42no—
121C4⋊Dic3—42no—
122Dic3⋊C4—42no—
123Dic3⋊C4—42no—
124Dic12—42no—
125Dic12—42no—
126C22×Q8—32no—
127C23×C4C22×D311yescharacteristic
128C2×Q16—61no—
129C2×Q16—61no—
130C2×Q16—122no—
131C22×C8—21no—
132C22×C8—42no—
133C22×C8C4×D311yescharacteristic
134C22×C8C4×D311yescharacteristic
135C22×C8D1211yescharacteristic
136C22×C8D1211yescharacteristic
137C22×C8#6711yescharacteristic
138C22×C8#6711yescharacteristic
139C22⋊Q8—62no—
140C22⋊Q8—62no—
141C22⋊Q8—62no—
142C22⋊Q8—62no—
143C2×C4⋊C4—32no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4—62no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4D1212yes—
149C2×C42—21no—
150#107—61no—
151#107—61no—
152#106—122no—
153#103—62no—
154#103—62no—
155C22⋊C8—21no—
156C22⋊C8D1212yes—
157#97—42no—
158#95—21no—
159#95—62no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3C4×D411yescharacteristic
163C23×C3#12111yescharacteristic
164C23×C3#12711yescharacteristic
165C23×C3#13714yes—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4C2×C22⋊C411yescharacteristic
179C2×C3×C4C4×D411yescharacteristic
180C2×C3×C4#12111yescharacteristic
181C2×C3×C4#12411yescharacteristic
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—82no—
185C2×Dic3—82no—
186C2×Dic3—82no—
187C2×Dic3—82no—
188C2×Dic3—82no—
189Dic6—42no—
190Dic6—42no—
191Dic6—82no—
192C3×C8—21no—
193C3×C8—21no—
194C3×C8—21no—
195C3×C8—42no—
196C3×C8#9912yes—
197C24C2×D1211yescharacteristic
198C2×Q8—62no—
199C2×Q8—62no—
200C2×Q8—122no—
201C2×Q8—122no—
202C22×C4—22no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—122no—
208C22×C4C2×C4×D311yescharacteristic
209C22×C4C2×D1211yescharacteristic
210C22×C4C2×D1211yescharacteristic
211C22×C4#23511yescharacteristic
212C22×C4#23511yescharacteristic
213C22×C4C2×#6711yescharacteristic
214Q16—122no—
215Q16—122no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8(C2×D3)⋊C411yescharacteristic
226C2×C8(C2×D3)⋊C411yescharacteristic
227C2×C8(C2×D3)⋊C411yescharacteristic
228C2×C8(C2×D3)⋊C411yescharacteristic
229C4⋊C4—62no—
230C4⋊C4—62no—
231C4⋊C4—122no—
232C4⋊C4—122no—
233C4⋊C4—122no—
234C4⋊C4—122no—
235C22⋊C4—24no—
236C22⋊C4—122no—
237C22⋊C4—122no—
238C22⋊C4—124no—
239C42—41no—
240C42—41no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—44no—
248C22×C3#39011yescharacteristic
249C22×C3#44112yes—
250C22×C3#47012yes—
251C22×C3#49712yes—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—82no—
256C3×C4C2×#9911yescharacteristic
257C3×C4#41011yescharacteristic
258Dic3—82no—
259Dic3—82no—
260Dic3—162no—
261C23—41no—
262C23—41no—
263C23C4×D1211yescharacteristic
264C23#87411yescharacteristic
265C23#90914yes—
266C23#93011yescharacteristic
267Q8—122no—
268Q8—122no—
269Q8—242no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—242no—
285C2×C4—242no—
286C2×C4—242no—
287C2×C4—242no—
288C2×C4—242no—
289C2×C4C4×D1211yescharacteristic
290C2×C4#87511yescharacteristic
291C2×C4C2×(C2×D3)⋊C411yescharacteristic
292C2×C4#93011yescharacteristic
293C8—21no—
294C8—21no—
295C8—21no—
296C8—42no—
297C8#80612yes—
298C2×C3—22no—
299C2×C3—22no—
300C2×C3—22no—
301C2×C3#196411yescharacteristic
302C2×C3#197111yescharacteristic
303C2×C3#197611yescharacteristic
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—44no—
311C22#471512yes—
312C22#471612yes—
313C22#493911yescharacteristic
314C22#513712yes—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—82no—
319C4—242no—
320C4—242no—
321C4—482no—
322C4C2×#80611yescharacteristic
323C4#500611yescharacteristic
324C3#978411yescharacteristic, complemented, Sylow
325C2—22no—
326C2—22no—
327C2—22no—
328C2#6945611yescharacteristic
329C2#6982411yescharacteristic
330C2#6989711yescharacteristic
subgroupG/H
centerC22#4939
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1752C2
socleC22×C3#390
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1752D3
hypocenterC3#9784
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
9. C22×Dic12C4semidirect
18. C2×#818C4semidirect
20. C2×#816C4semidirect
324. C3#9784semidirect

Maximal subgroups

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

subgroupindex
C2×#51162
C3×#17522
#719702
#699462
#694572
#97843