Finite Group Databasebeta

← #79843

Subgroups — #79843

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5579C211yesmaximal, characteristic, complemented
2#5576C211yesmaximal, characteristic
3C3×#441C211yesmaximal, characteristic, complemented
4#5187C211yesmaximal, characteristic, complemented
5#5186C211yesmaximal, characteristic, complemented
6#5028C211yesmaximal, characteristic, complemented
7#5020C211yesmaximal, characteristic, complemented
8#4863C211yesmaximal, characteristic, complemented
9#4862C211yesmaximal, characteristic, complemented
10#4816C211yesmaximal, characteristic, complemented
11#4807C211yesmaximal, characteristic, complemented
12#4803C211yesmaximal, characteristic, complemented
13D3×#103C211yesmaximal, characteristic, complemented
14#4714C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3316—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21C3×C4×Q8C2211yescharacteristic
22#946C2211yescharacteristic
23#946C2211yescharacteristic
24C2×#216C2211yescharacteristic
25C2×#215C2211yescharacteristic
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27C2×D3×C8C2211yescharacteristic
28#897—21no—
29#897C2211yescharacteristic
30#896—21no—
31#896—21no—
32#895—21no—
33#895C2211yescharacteristic
34D3×C4⋊C4C2211yescharacteristic
35D3×C4⋊C4C2211yescharacteristic
36#875—21no—
37#875C2211yescharacteristic
38C4×D12—21no—
39C4×D12—21no—
40C4×D12C2211yescharacteristic
41#870—21no—
42#869C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#97C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53#816C2211yescharacteristic
54#816C2211yescharacteristic
55#810C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#554—31no—
61#548—31no—
62#489—31no—
63#483—31no—
64#481—31no—
65#475—31no—
66#470—31no—
67#465—31no—
68#464—31no—
69#452—31no—
70#451—31no—
71#441D311yescharacteristic, complemented
72#435—31no—
73#417—31no—
74C2×#103—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77#239—22no—
78#239—41no—
79D3×Q8—22no—
80D3×Q8—41no—
81C2×D12—21no—
82C2×D12—21no—
83C2×D12D411yescharacteristic
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×C4×D3—21no—
89C2×C4×D3C2311yescharacteristic
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C2×Dic6—21no—
93C2×Dic6C2311yescharacteristic
94C3×Q16—22no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C42—21no—
102C3×C42—21no—
103C3×C42C2311yescharacteristic
104#216—41no—
105#215—41no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4D411yescharacteristic
113(C2×D3)⋊C4D411yescharacteristic
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3D411yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125ΓL(1,𝔽25)—41no—
126D3×C8—41no—
127C2×#41—31no—
128C22×Q8—31no—
129C2×Q16—31no—
130C2×Q16C2×D311yescharacteristic
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#125—61no—
136#124—31no—
137#124—61no—
138C22⋊Q8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144#121—31no—
145#121—61no—
146C4×Q8C2×D311yescharacteristic
147C4×Q8C2×D311yescharacteristic
148C4×D4—31no—
149C4×D4—61no—
150C4×D4—61no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153#106—31no—
154#106—31no—
155#106C2×D311yescharacteristic
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⋊C8—31no—
169#97C2×D311yescharacteristic
170C22×D3—21no—
171C22×D3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8#4112yes—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189D12—22no—
190D12—21no—
191D12—21no—
192D12—42no—
193D12—42no—
194C4×D3—21no—
195C4×D3—21no—
196C4×D3—21no—
197C4×D3—21no—
198C4×D3—42no—
199C4×D3—42no—
200C4×D3—81no—
201C4×D3—81no—
202Dic6—21no—
203Dic6—21no—
204Dic6—42no—
205Dic6—81no—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210#41—62no—
211#41—121no—
212C2×Q8—31no—
213C2×Q8—61no—
214C2×Q8—61no—
215C2×Q8—62no—
216C2×Q8—121no—
217C2×Q8C22×D311yescharacteristic
218C2×Q8C22×D311yescharacteristic
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—61no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229Q16—22no—
230Q16—121no—
231SD16—121no—
232M16—121no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—121no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4C22×D311yescharacteristic
250C4⋊C4C22×D311yescharacteristic
251C22⋊C4—31no—
252C22⋊C4—31no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C42—21no—
260C42—21no—
261C42C22×D311yescharacteristic
262C22×C3C22×D411yescharacteristic
263C2×D3—21no—
264C2×D3—21no—
265C2×D3—21no—
266C2×D3—21no—
267C2×D3—21no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C2×D3#13712yes—
272C3×C4—22no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4C2×#4111yescharacteristic
279C3×C4H(5,𝔽2)11yescharacteristic
280Dic3—21no—
281Dic3—21no—
282Dic3—41no—
283Dic3—41no—
284Dic3—41no—
285C23—31no—
286C23—31no—
287C23—61no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—61no—
292Q8—61no—
293Q8—122no—
294Q8—241no—
295Q8#23512yes—
296D4—62no—
297D4—61no—
298D4—61no—
299D4—122no—
300D4—122no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4D3×D411yescharacteristic
319C2×C4D3×D411yescharacteristic
320C2×C4C23×D311yescharacteristic
321C8—21no—
322C8—21no—
323C8—61no—
324C8—61no—
325C2×C3#54611yescharacteristic
326C2×C3C2×#13711yescharacteristic
327C2×C3#57711yescharacteristic
328D3—22no—
329D3—41no—
330D3—81no—
331C22—32no—
332C22—61no—
333C22—61no—
334C22—61no—
335C22—61no—
336C22—61no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22C2×D3×D411yescharacteristic
341C4—21no—
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—61no—
348C4—61no—
349C4—121no—
350C4—121no—
351C4—121no—
352C4C2×#23511yescharacteristic
353C4#100911yescharacteristic
354C3#331611yescharacteristic, complemented, Sylow
355C2—62no—
356C2—121no—
357C2—241no—
358C2#555011yescharacteristic
359C2D3×#13711yescharacteristic
360C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#441C2
socleC22×C3C22×D4
prime radical#5576C2
hypercenter#441D3
hypocenterC3#3316
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #5579C2semidirect
3. C3×#441C2semidirect
4. #5187C2semidirect
5. #5186C2semidirect
6. #5028C2semidirect
7. #5020C2semidirect
8. #4863C2semidirect
9. #4862C2semidirect
10. #4816C2semidirect
11. #4807C2semidirect
12. #4803C2semidirect
13. D3×#103C2semidirect
14. #4714C2semidirect
15. #4706C2semidirect
71. #441D3semidirect
354. C3#3316semidirect

Maximal subgroups

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

subgroupindex
#55792
#55762
C3×#4412
#51872
#51862
#50282
#50202
#48632
#48622
#48162
#48072
#48032
D3×#1032
#47142
#47062
#33163