Finite Group Databasebeta

← #80880

Subgroups — #80880

Order 384.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5618C211yesmaximal, characteristic, complemented
2#5616C211yesmaximal, 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#5033C211yesmaximal, characteristic, complemented
10#4860C211yesmaximal, characteristic, complemented
11#4856C211yesmaximal, characteristic, complemented
12#4784C211yesmaximal, characteristic, complemented
13#4779C211yesmaximal, characteristic, complemented
14#4763C211yesmaximal, characteristic, complemented
15#4759C211yesmaximal, characteristic, complemented
16#3347—31nomaximal, Sylow
17C2×C3×D8C2211yescharacteristic
18C2×C3×M16C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C3×#121C2211yescharacteristic
21#939—21no—
22#939—21no—
23#938C2211yescharacteristic
24#938C2211yescharacteristic
25#936—21no—
26D4×Dic3—21no—
27D4×Dic3C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29C4×#67C2211yescharacteristic
30#926C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×D3×C8C2211yescharacteristic
33#892C2211yescharacteristic
34#892C2211yescharacteristic
35#890C2211yescharacteristic
36#888C2211yescharacteristic
37#886—21no—
38#885—21no—
39#879—21no—
40#877—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#807C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#555—31no—
61#553—31no—
62#501—31no—
63#482—31no—
64#482—31no—
65#482—31no—
66#480—31no—
67#478—31no—
68#474—31no—
69#469D311yescharacteristic, complemented
70#466—31no—
71#442—31no—
72#435—31no—
73#428—31no—
74#427—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—
86C2×C4×D3C2311yescharacteristic
87C2×Dic6—21no—
88C3×D8—22no—
89C3×M16—22no—
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C3×C4⋊C4C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C22⋊C4—21no—
95C3×C22⋊C4—21no—
96#217—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#217D411yescharacteristic
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4D411yescharacteristic
105C4⋊Dic3—21no—
106C4⋊Dic3—21no—
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4D411yescharacteristic
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120D3×C8—22no—
121C2×D8C2×D311yescharacteristic
122C2×M16C2×D311yescharacteristic
123C22×C8—31no—
124C4⋊Q8—31no—
125#126—61no—
126#125—31no—
127#124—61no—
128#124—61no—
129#123—61no—
130#121—31no—
131#121—31no—
132#121—61no—
133#121—61no—
134#121C2×D311yescharacteristic
135#121C2×D311yescharacteristic
136C4×D4—31no—
137C4×D4—31no—
138C4×D4—31no—
139C4×D4—61no—
140#117—31no—
141#117—31no—
142#117—31no—
143#117—61no—
144C2×C4⋊C4—31no—
145#107—31no—
146#107—31no—
147#107—31no—
148#107—31no—
149#107—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153#106C2×D311yescharacteristic
154C4⋊C8—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102C2×D311yescharacteristic
160#102C2×D311yescharacteristic
161C22⋊C8—31no—
162C22⋊C8—31no—
163#97—31no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3C2×D411yescharacteristic
167C22×D3C2×D411yescharacteristic
168C3×D4—21no—
169C3×D4—21no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4—42no—
173C3×D4—81no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2411yescharacteristic
180#67—41no—
181#67—81no—
182#67—81no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—42no—
188C2×Dic3—81no—
189C2×Dic3—81no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C4×D3—21no—
194C4×D3—21no—
195Dic6—81no—
196C3×C8—21no—
197C3×C8#4112yes—
198C3⋊C8—21no—
199C3⋊C8—21no—
200C2×Q8—61no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—31no—
204C2×D4—61no—
205C2×D4—61no—
206C2×D4C22×D311yescharacteristic
207C2×D4C22×D311yescharacteristic
208C22×C4—31no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4C22×D311yescharacteristic
213D8—22no—
214M16—22no—
215C2×C8—31no—
216C2×C8—31no—
217C2×C8—62no—
218C2×C8C22×D311yescharacteristic
219C2×C8C22×D311yescharacteristic
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—31no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4C22×D311yescharacteristic
230C4⋊C4C22×D311yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—21no—
233C22⋊C4—31no—
234C22⋊C4—31no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C42—31no—
243C42—61no—
244C42—61no—
245C42—61no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3C22×D411yescharacteristic
256C2×D3—21no—
257C2×D3—21no—
258C2×D3—21no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4C2×#4111yescharacteristic
263C3×C42−1+411yescharacteristic
264Dic3—21no—
265Dic3—21no—
266Dic3—21no—
267Dic3—41no—
268Dic3—41no—
269Dic3—41no—
270Dic3—41no—
271C23—21no—
272C23—21no—
273C23—31no—
274C23D3×D411yescharacteristic
275Q8—241no—
276D4—21no—
277D4—21no—
278D4—21no—
279D4—21no—
280D4—42no—
281D4—81no—
282D4—121no—
283D4—241no—
284D4—241no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—122no—
299C2×C4—241no—
300C2×C4—241no—
301C2×C4D3×D411yescharacteristic
302C2×C4C23×D311yescharacteristic
303C8—21no—
304C8—61no—
305C8—61no—
306C8#23712yes—
307C2×C3—41no—
308C2×C3—81no—
309C2×C3—81no—
310C2×C3#54711yescharacteristic
311C2×C3#57511yescharacteristic
312C2×C3#57611yescharacteristic
313D3—41no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—61no—
324C22—61no—
325C22—61no—
326C22C2×D3×D411yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—61no—
331C4—61no—
332C4—61no—
333C4—121no—
334C4—121no—
335C4—121no—
336C4—121no—
337C4C2×#23711yescharacteristic
338C4#101011yescharacteristic
339C3#334711yescharacteristic, complemented, Sylow
340C2—41no—
341C2—81no—
342C2—81no—
343C2—121no—
344C2#561111yescharacteristic
345C2#577311yescharacteristic
346C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3347
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. #5618C2semidirect
2. #5616C2semidirect
3. C3×#469C2semidirect
4. #5157C2semidirect
5. #5152C2semidirect
6. #5127C2semidirect
7. #5121C2semidirect
8. #5035C2semidirect
9. #5033C2semidirect
10. #4860C2semidirect
11. #4856C2semidirect
12. #4784C2semidirect
13. #4779C2semidirect
14. #4763C2semidirect
15. #4759C2semidirect
69. #469D3semidirect
339. C3#3347semidirect

Maximal subgroups

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

subgroupindex
#56182
#56162
C3×#4692
#51572
#51522
#51272
#51212
#50352
#50332
#48602
#48562
#47842
#47792
#47632
#47592
#33473