Finite Group Databasebeta

← #80197

Subgroups — #80197

Order 384.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5817C211yesmaximal, characteristic
2#5641C211yesmaximal, characteristic, complemented
3C3×#451C211yesmaximal, characteristic, complemented
4#5225C211yesmaximal, characteristic, complemented
5#5188C211yesmaximal, characteristic, complemented
6#5187C211yesmaximal, characteristic, complemented
7#5185C211yesmaximal, characteristic, complemented
8#5184C211yesmaximal, characteristic, complemented
9#5049C211yesmaximal, characteristic, complemented
10#4815C211yesmaximal, characteristic, complemented
11#4804C211yesmaximal, characteristic, complemented
12#4794C211yesmaximal, characteristic, complemented
13#4793C211yesmaximal, characteristic, complemented
14#4737C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic, complemented
16#3313—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22#947—21no—
23#947C2211yescharacteristic
24#946—21no—
25#946C2211yescharacteristic
26#946C2211yescharacteristic
27Q8×Dic3C2211yescharacteristic
28Q8×Dic3C2211yescharacteristic
29C2×#215C2211yescharacteristic
30#930—21no—
31#930C2211yescharacteristic
32C4×#67—21no—
33C4×#67C2211yescharacteristic
34#926C2211yescharacteristic
35C2×#208C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37#898—21no—
38#892C2211yescharacteristic
39#888C2211yescharacteristic
40#887—21no—
41#885—21no—
42#879—21no—
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#835C2211yescharacteristic
50#835C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#818C2211yescharacteristic
55#817C2211yescharacteristic
56#815C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58#809C2211yescharacteristic
59#807C2211yescharacteristic
60#556—31no—
61#548—31no—
62#489—31no—
63#485—31no—
64#482—31no—
65#477—31no—
66#468—31no—
67#467—31no—
68#464—31no—
69#452—31no—
70#451D311yescharacteristic, complemented
71#441—31no—
72#435—31no—
73#419—31no—
74#416—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×Q8C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67D411yescharacteristic
85#239—22no—
86#239—41no—
87C2×D12—21no—
88C2×D12C2311yescharacteristic
89C2×C4×D3—21no—
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C3×Q16—22no—
93C3×Q16—41no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4—21no—
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99#217—21no—
100#217D411yescharacteristic
101#215—41no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4D411yescharacteristic
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123#208—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D3×C8—41no—
127C2×#41—31no—
128C22×Q8C2×D311yescharacteristic
129C2×Q16C2×D311yescharacteristic
130C2×Q16C2×D311yescharacteristic
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#126—61no—
136#126—61no—
137#124—31no—
138#124—61no—
139#124—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8C2×D311yescharacteristic
144#121—31no—
145#121—61no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×Q8—61no—
149C4×D4—31no—
150C4×D4—61no—
151#117—31no—
152#117—31no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103C2×D311yescharacteristic
163#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169#97—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3—21no—
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—42no—
179C3×Q8#4112yes—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—42no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—41no—
190#67—42no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197D12—21no—
198D12—21no—
199D12—42no—
200C4×D3—21no—
201C4×D3—21no—
202C4×D3—42no—
203Dic6—42no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—62no—
209#41—121no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—61no—
214C2×Q8C22×D311yescharacteristic
215C2×Q8C22×D311yescharacteristic
216C2×Q8C22×D311yescharacteristic
217C2×D4—31no—
218C2×D4—31no—
219C2×D4—61no—
220C2×D4—61no—
221C22×C4—21no—
222C22×C4—31no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
225Q16—22no—
226Q16—41no—
227SD16—121no—
228M16—121no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—121no—
232C2×C8C22×D311yescharacteristic
233C2×C8C22×D311yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4C22×D311yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—31no—
247C22⋊C4—31no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C42—31no—
256C42—61no—
257C42—61no—
258C42—61no—
259C42—61no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3C22×D411yescharacteristic
264C2×D3—21no—
265C2×D3—21no—
266C2×D3—21no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C3×C4—22no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4C2×#4111yescharacteristic
276C3×C4H(5,𝔽2)11yescharacteristic
277Dic3—21no—
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283C23—31no—
284C23—61no—
285C23D3×D411yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—122no—
293Q8#23712yes—
294D4—61no—
295D4—61no—
296D4—122no—
297D4—121no—
298D4—122no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—42no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—122no—
315C2×C4D3×D411yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—61no—
320C8—61no—
321C2×C3—41no—
322C2×C3#54611yescharacteristic
323C2×C3#57511yescharacteristic
324C2×C3#57811yescharacteristic
325D3—41no—
326D3—81no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—61no—
331C22—61no—
332C22—61no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—61no—
343C4—61no—
344C4—61no—
345C4—121no—
346C4—121no—
347C4—121no—
348C4C2×#23711yescharacteristic
349C4#100411yescharacteristic
350C3#331311yescharacteristic, complemented, Sylow
351C2—41no—
352C2—121no—
353C2—241no—
354C2#559111yescharacteristic
355C2#576611yescharacteristic
356C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radical#5817C2
hypercenter#451D3
hypocenterC3#3313
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #5641C2semidirect
3. C3×#451C2semidirect
4. #5225C2semidirect
5. #5188C2semidirect
6. #5187C2semidirect
7. #5185C2semidirect
8. #5184C2semidirect
9. #5049C2semidirect
10. #4815C2semidirect
11. #4804C2semidirect
12. #4794C2semidirect
13. #4793C2semidirect
14. #4737C2semidirect
15. #4728C2semidirect
70. #451D3semidirect
350. C3#3313semidirect

Maximal subgroups

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

subgroupindex
#58172
#56412
C3×#4512
#52252
#51882
#51872
#51852
#51842
#50492
#48152
#48042
#47942
#47932
#47372
#47282
#33133