Finite Group Databasebeta

← #80075

Subgroups — #80075

Order 384.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5827C211yesmaximal, characteristic, complemented
2#5603C211yesmaximal, characteristic, complemented
3C3×#449C211yesmaximal, characteristic
4#5239C211yesmaximal, characteristic, complemented
5#5166C211yesmaximal, characteristic, complemented
6Dic3×SD16C211yesmaximal, characteristic, complemented
7#5153C211yesmaximal, characteristic, complemented
8#5152C211yesmaximal, characteristic, complemented
9#5034C211yesmaximal, characteristic, complemented
10#4791C211yesmaximal, characteristic, complemented
11#4789C211yesmaximal, characteristic, complemented
12#4761C211yesmaximal, characteristic, complemented
13#4758C211yesmaximal, characteristic, complemented
14#4721C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3125—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×SD16C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C3×#121C2211yescharacteristic
21Q8×Dic3C2211yescharacteristic
22Q8×Dic3C2211yescharacteristic
23C2×#216C2211yescharacteristic
24#936—21no—
25#936C2211yescharacteristic
26#935—21no—
27D4×Dic3—21no—
28D4×Dic3—21no—
29D4×Dic3C2211yescharacteristic
30D4×Dic3C2211yescharacteristic
31D4×Dic3C2211yescharacteristic
32C2×#214C2211yescharacteristic
33#926—21no—
34#926C2211yescharacteristic
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3C2211yescharacteristic
37#924C2211yescharacteristic
38#888C2211yescharacteristic
39#887C2211yescharacteristic
40#879—21no—
41#878—21no—
42C3×#103C2211yescharacteristic
43C3×#102C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45#835C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#830C2211yescharacteristic
50#817C2211yescharacteristic
51#817C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54#815C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#808C2211yescharacteristic
59#808C2211yescharacteristic
60#542—31no—
61#520—31no—
62#483—31no—
63#481—31no—
64#476—31no—
65#475—31no—
66#464—31no—
67#463—31no—
68#452—31no—
69#449D311yescharacteristic
70#442—31no—
71#441—31no—
72C4×SD16—31no—
73C4×SD16—31no—
74#433—31no—
75C3×#41—22no—
76C3×#41—22no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C22×Dic3—21no—
85C22×Dic3—21no—
86C22×Dic3—21no—
87C22×Dic3—21no—
88C2×Dic6—21no—
89C2×Dic6C2311yescharacteristic
90C3×SD16—22no—
91C3×D8—22no—
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96#217—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217D411yescharacteristic
103#217D411yescharacteristic
104#216—22no—
105#214—22no—
106C4⋊Dic3—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3D411yescharacteristic
110C4⋊Dic3D411yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121C4×Dic3C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×#41C2×D311yescharacteristic
125C2×Q16—31no—
126C2×SD16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×D8C2×D311yescharacteristic
129C4⋊Q8—31no—
130#126—61no—
131#124—31no—
132#124—61no—
133#123—61no—
134C22⋊Q8—31no—
135C22⋊Q8—61no—
136#121C2×D311yescharacteristic
137C4×Q8—31no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—61no—
144C4×D4—61no—
145#117—31no—
146#117—61no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149#106—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103C2×D311yescharacteristic
159#102—31no—
160#102—31no—
161#102—31no—
162#102C2×D311yescharacteristic
163C22⋊C8—31no—
164C22⋊C8C2×D311yescharacteristic
165#97—31no—
166C4×C8—31no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3C2×D411yescharacteristic
170C3×Q8—21no—
171C3×Q8#4112yes—
172C3×D4—24no—
173C3×D4—21no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—81no—
177C3×D4#4112yes—
178C2×C3×C4—24no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—81no—
190C2×Dic3—81no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197Dic6—21no—
198Dic6—21no—
199Dic6—81no—
200C3×C8—21no—
201C3×C8—21no—
202C3⋊C8—21no—
203C3⋊C8—21no—
204#41—22no—
205#41—22no—
206C2×Q8—31no—
207C2×Q8—61no—
208C2×Q8C22×D311yescharacteristic
209C2×D4—21no—
210C2×D4—21no—
211C2×D4C22×D311yescharacteristic
212C2×D4C22×D311yescharacteristic
213C22×C4—21no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4C22×D311yescharacteristic
219Q16—62no—
220SD16—22no—
221SD16—62no—
222D8—22no—
223C2×C8—31no—
224C2×C8—31no—
225C2×C8C22×D311yescharacteristic
226C2×C8C22×D311yescharacteristic
227C4⋊C4—31no—
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—31no—
241C22⋊C4—31no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C42—31no—
249C42—31no—
250C42—61no—
251C42—61no—
252C42—61no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3C22×D411yescharacteristic
263C3×C4—21no—
264C3×C4—22no—
265C3×C4—41no—
266C3×C4C2×#4111yescharacteristic
267C3×C4C2×#4111yescharacteristic
268Dic3—21no—
269Dic3—21no—
270Dic3—21no—
271Dic3—21no—
272Dic3—41no—
273Dic3—41no—
274Dic3—41no—
275Dic3—41no—
276C23—21no—
277C23—21no—
278C23D3×D411yescharacteristic
279Q8—21no—
280Q8—61no—
281Q8—61no—
282Q8—241no—
283Q8#23712yes—
284D4—24no—
285D4—21no—
286D4—21no—
287D4—21no—
288D4—81no—
289D4#23712yes—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—241no—
304C2×C4—241no—
305C2×C4—241no—
306C2×C4—241no—
307C2×C4D3×D411yescharacteristic
308C2×C4C23×D311yescharacteristic
309C8—21no—
310C8—21no—
311C8—61no—
312C8—61no—
313C2×C3—41no—
314C2×C3—42no—
315C2×C3—81no—
316C2×C3#52511yescharacteristic
317C2×C3#57711yescharacteristic
318C2×C3#57711yescharacteristic
319C22—21no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22C2×D3×D411yescharacteristic
329C4—21no—
330C4—22no—
331C4—41no—
332C4—61no—
333C4—61no—
334C4—61no—
335C4—61no—
336C4—121no—
337C4—121no—
338C4—121no—
339C4—121no—
340C4C2×#23711yescharacteristic
341C4C2×#23711yescharacteristic
342C3#312511yescharacteristic, complemented, Sylow
343C2—41no—
344C2—42no—
345C2—81no—
346C2#558611yescharacteristic
347C2#577011yescharacteristic
348C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#449C2
socleC22×C3C22×D4
hypercenter#449D3
hypocenterC3#3125
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #5827C2semidirect
2. #5603C2semidirect
4. #5239C2semidirect
5. #5166C2semidirect
6. Dic3×SD16C2semidirect
7. #5153C2semidirect
8. #5152C2semidirect
9. #5034C2semidirect
10. #4791C2semidirect
11. #4789C2semidirect
12. #4761C2semidirect
13. #4758C2semidirect
14. #4721C2semidirect
15. #4720C2semidirect
342. C3#3125semidirect

Maximal subgroups

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

subgroupindex
#58272
#56032
C3×#4492
#52392
#51662
Dic3×SD162
#51532
#51522
#50342
#47912
#47892
#47612
#47582
#47212
#47202
#31253