Finite Group Databasebeta

← #80879

Subgroups — #80879

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5614C211yesmaximal, characteristic, complemented
2#5611C211yesmaximal, characteristic, complemented
3C3×#469C211yesmaximal, characteristic, complemented
4#5157C211yesmaximal, characteristic, complemented
5#5152C211yesmaximal, characteristic, complemented
6#5127C211yesmaximal, characteristic, complemented
7#5121C211yesmaximal, characteristic, complemented
8#5035C211yesmaximal, characteristic, complemented
9#5034C211yesmaximal, characteristic, complemented
10D3×#106C211yesmaximal, characteristic, complemented
11#4856C211yesmaximal, characteristic, complemented
12#4777C211yesmaximal, characteristic, complemented
13#4774C211yesmaximal, characteristic, complemented
14#4766C211yesmaximal, characteristic, complemented
15#4761C211yesmaximal, characteristic, complemented
16#3345—31nomaximal, Sylow
17C2×#237—21no—
18C2×C3×D8C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic
20C3×#121C2211yescharacteristic
21C3×#121C2211yescharacteristic
22#938—21no—
23#938C2211yescharacteristic
24#938C2211yescharacteristic
25#935—21no—
26#935—21no—
27D4×Dic3C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29C4×#67C2211yescharacteristic
30#926C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×D3×C8C2211yescharacteristic
33D3×C4⋊C4C2211yescharacteristic
34D3×C4⋊C4C2211yescharacteristic
35#890C2211yescharacteristic
36#888C2211yescharacteristic
37#883—21no—
38#67⋊C4—21no—
39#879—21no—
40#878—21no—
41C3×#106C2211yescharacteristic
42C3×#102C2211yescharacteristic
43C3×#102C2211yescharacteristic
44#832C2211yescharacteristic
45#832C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#830C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50#818C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#817C2211yescharacteristic
54#815C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56#808C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#553—31no—
61#547—31no—
62#501—31no—
63#482—31no—
64#481—31no—
65#481—31no—
66#480—31no—
67#478—31no—
68#476—31no—
69#469D311yescharacteristic, complemented
70#466—31no—
71#442—31no—
72#435—31no—
73#428—31no—
74C2×#106—31no—
75C2×C3×D4—21no—
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C2×#67D411yescharacteristic
83C22×Dic3—21no—
84C22×Dic3—21no—
85C22×Dic3—21no—
86#237—44no—
87C2×C4×D3—21no—
88C2×C4×D3—21no—
89C2×C4×D3C2311yescharacteristic
90C2×Dic6—21no—
91C3×D8—22no—
92C3×M16—22no—
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C22⋊C4—21no—
98C3×C22⋊C4—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#217—21no—
104#217D411yescharacteristic
105(C2×D3)⋊C4D411yescharacteristic
106C4⋊Dic3—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117C4×Dic3—21no—
118C4×Dic3C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121D3×C8—22no—
122C2×#41—61no—
123C2×D8C2×D311yescharacteristic
124C2×M16C2×D311yescharacteristic
125C22×C8—31no—
126C4⋊Q8—31no—
127#126—61no—
128#125—31no—
129#123—61no—
130#123—61no—
131#123—61no—
132C22⋊Q8—61no—
133#121—31no—
134#121—31no—
135#121—61no—
136#121C2×D311yescharacteristic
137#121C2×D311yescharacteristic
138C4×D4—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—61no—
142#117—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146#107—31no—
147#107—31no—
148#107—31no—
149#106—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153#106—31no—
154#106C2×D311yescharacteristic
155C4⋊C8—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102C2×D311yescharacteristic
161#102C2×D311yescharacteristic
162C22⋊C8—31no—
163C22⋊C8—31no—
164#97—31no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3C2×D411yescharacteristic
168C22×D3C2×D411yescharacteristic
169C3×D4—21no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—42no—
174C3×D4—81no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2411yescharacteristic
181#67—22no—
182#67—42no—
183#67—42no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—81no—
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194C4×D3—21no—
195C4×D3—21no—
196C4×D3—42no—
197C4×D3—81no—
198Dic6—42no—
199C3×C8—21no—
200C3×C8#4112yescomplemented
201C3⋊C8—21no—
202C3⋊C8—21no—
203#41—124no—
204C2×Q8—61no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—31no—
208C2×D4—61no—
209C2×D4—61no—
210C2×D4C22×D311yescharacteristic
211C2×D4C22×D311yescharacteristic
212C22×C4—31no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4C22×D311yescharacteristic
219D8—22no—
220M16—22no—
221C2×C8—31no—
222C2×C8—31no—
223C2×C8—62no—
224C2×C8C22×D311yescharacteristic
225C2×C8C22×D311yescharacteristic
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4C22×D311yescharacteristic
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—31no—
242C22⋊C4—31no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C42—31no—
249C42—61no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3C22×D411yescharacteristic
260C2×D3—21no—
261C2×D3—21no—
262C2×D3AGL(1,ℤ/8)12yes—
263C3×C4—21no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4C2×#4111yescharacteristic
267C3×C42−1+411yescharacteristic
268Dic3—21no—
269Dic3—21no—
270Dic3—21no—
271Dic3—41no—
272Dic3—41no—
273Dic3—41no—
274Dic3—41no—
275C23—21no—
276C23—21no—
277C23—31no—
278C23D3×D411yescharacteristic
279Q8—122no—
280D4—21no—
281D4—21no—
282D4—21no—
283D4—21no—
284D4—42no—
285D4—62no—
286D4—81no—
287D4—122no—
288D4—122no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—241no—
306C2×C4—241no—
307C2×C4D3×D411yescharacteristic
308C2×C4C23×D311yescharacteristic
309C8—21no—
310C8—61no—
311C8—61no—
312C8#23712yescomplemented
313C2×C3—41no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3#54711yescharacteristic
317C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
318C2×C3#57711yescharacteristic
319D3—22no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—32no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—61no—
331C22—61no—
332C22C2×D3×D411yescharacteristic
333C4—21no—
334C4—41no—
335C4—41no—
336C4—61no—
337C4—61no—
338C4—61no—
339C4—121no—
340C4—121no—
341C4—121no—
342C4—121no—
343C4C2×#23711yescharacteristic
344C4#101011yescharacteristic
345C3#334511yescharacteristic, complemented, Sylow
346C2—41no—
347C2—62no—
348C2—81no—
349C2—81no—
350C2#561111yescharacteristic
351C2D3×AGL(1,ℤ/8)11yescharacteristic
352C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3345
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #5614C2semidirect
2. #5611C2semidirect
3. C3×#469C2semidirect
4. #5157C2semidirect
5. #5152C2semidirect
6. #5127C2semidirect
7. #5121C2semidirect
8. #5035C2semidirect
9. #5034C2semidirect
10. D3×#106C2semidirect
11. #4856C2semidirect
12. #4777C2semidirect
13. #4774C2semidirect
14. #4766C2semidirect
15. #4761C2semidirect
69. #469D3semidirect
200. C3×C8#41semidirect
312. C8#237semidirect
345. C3#3345semidirect

Maximal subgroups

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

subgroupindex
#56142
#56112
C3×#4692
#51572
#51522
#51272
#51212
#50352
#50342
D3×#1062
#48562
#47772
#47742
#47662
#47612
#33453