Finite Group Databasebeta

← #80857

Subgroups — #80857

Order 384.

358 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#5619C211yesmaximal, characteristic, complemented
2#5603C211yesmaximal, characteristic, complemented
3C3×#469C211yesmaximal, characteristic, complemented
4#5153C211yesmaximal, characteristic, complemented
5#5150C211yesmaximal, characteristic, complemented
6#5134C211yesmaximal, characteristic, complemented
7#5119C211yesmaximal, characteristic, complemented
8#5041C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4870C211yesmaximal, characteristic, complemented
11#4857C211yesmaximal, characteristic, complemented
12#4787C211yesmaximal, characteristic, complemented
13#4766C211yesmaximal, characteristic, complemented
14#4762C211yesmaximal, characteristic, complemented
15#4759C211yesmaximal, characteristic, complemented
16#3257—31nomaximal, Sylow
17C2×C3×D8C2211yescharacteristic
18C2×C3×M16C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C3×#121C2211yescharacteristic
21#940C2211yescharacteristic
22#939—21no—
23#938—21no—
24#936—21no—
25#936C2211yescharacteristic
26#935—21no—
27D4×Dic3C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C2×#213C2211yescharacteristic
31#930C2211yescharacteristic
32#926C2211yescharacteristic
33#924C2211yescharacteristic
34C22×C3⋊C8C2211yescharacteristic
35C2×#204C2211yescharacteristic
36#893C2211yescharacteristic
37#890C2211yescharacteristic
38#888C2211yescharacteristic
39#887C2211yescharacteristic
40#886—21no—
41#883—21no—
42#879—21no—
43#878—21no—
44C3×#106C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50#818C2211yescharacteristic
51#816C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56#810C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#542—31no—
61#540—31no—
62#479—31no—
63#478—31no—
64#476—31no—
65#474—31no—
66#469D311yescharacteristic, complemented
67#468—31no—
68#466—31no—
69#465—31no—
70#463—31no—
71#463—31no—
72#460—31no—
73#459—31no—
74#424—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×#67—21no—
83C22×Dic3—21no—
84C22×Dic3—21no—
85C2×D12C2311yescharacteristic
86C2×C4×D3—21no—
87C2×Dic6—21no—
88C2×Dic6C2311yescharacteristic
89C3×D8—41no—
90C3×M16—41no—
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96C3×C22⋊C4—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217D411yescharacteristic
103#217D411yescharacteristic
104#214—41no—
105#213—41no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3D411yescharacteristic
110C4⋊Dic3D411yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117C4×Dic3—21no—
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8—41no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123#204—41no—
124C2×SD16—31no—
125C2×SD16—31no—
126C2×D8—31no—
127C2×D8C2×D311yescharacteristic
128C2×M16C2×D311yescharacteristic
129C22×C8—31no—
130C4⋊Q8—31no—
131#127—31no—
132#126—61no—
133#125—31no—
134#124—31no—
135#124—61no—
136#123—61no—
137#123—61no—
138#123—61no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141#121—31no—
142#121—61no—
143#121—61no—
144#121C2×D311yescharacteristic
145#121C2×D311yescharacteristic
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—31no—
150#117—31no—
151#107—31no—
152#107—31no—
153#106—31no—
154#106C2×D311yescharacteristic
155C4⋊C8—31no—
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C23×C3—21no—
168C23×C3—21no—
169C23×C3C2×D411yescharacteristic
170C22×D3—21no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—21no—
174C3×D4—21no—
175C3×D4—81no—
176C3×D4—81no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2411yescharacteristic
183#67—81no—
184#67—81no—
185#67—81no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—81no—
190C2×Dic3—81no—
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195D12—21no—
196D12—21no—
197C4×D3—81no—
198Dic6—21no—
199Dic6—21no—
200Dic6—81no—
201C3×C8—21no—
202C3×C8—21no—
203C3⋊C8—21no—
204C3⋊C8—21no—
205C2×Q8—31no—
206C2×Q8—61no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—31no—
210C2×D4—61no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4C22×D311yescharacteristic
214C2×D4C22×D311yescharacteristic
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4C22×D311yescharacteristic
219SD16—121no—
220SD16—121no—
221D8—41no—
222D8—121no—
223M16—41no—
224C2×C8—31no—
225C2×C8—31no—
226C2×C8—121no—
227C2×C8C22×D311yescharacteristic
228C2×C8C22×D311yescharacteristic
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4C22×D311yescharacteristic
239C4⋊C4C22×D311yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—21no—
242C22⋊C4—31no—
243C22⋊C4—31no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C42—31no—
252C42—31no—
253C42—61no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3C22×D411yescharacteristic
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C3×C4—21no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4H(5,𝔽2)11yescharacteristic
271C3×C42−1+411yescharacteristic
272Dic3—21no—
273Dic3—21no—
274Dic3—21no—
275Dic3—21no—
276Dic3—41no—
277Dic3—41no—
278Dic3—41no—
279C23—21no—
280C23—21no—
281C23—61no—
282C23D3×D411yescharacteristic
283Q8—61no—
284Q8—61no—
285Q8—241no—
286D4—21no—
287D4—21no—
288D4—21no—
289D4—21no—
290D4—61no—
291D4—61no—
292D4—81no—
293D4—81no—
294D4—241no—
295D4—241no—
296D4—241no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—241no—
309C2×C4—241no—
310C2×C4—241no—
311C2×C4D3×D411yescharacteristic
312C2×C4C23×D311yescharacteristic
313C8—21no—
314C8—21no—
315C8—61no—
316C8—61no—
317C2×C3—41no—
318C2×C3—81no—
319C2×C3—81no—
320C2×C3#53711yescharacteristic
321C2×C3#57611yescharacteristic
322C2×C3#57711yescharacteristic
323D3—81no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—41no—
339C4—41no—
340C4—61no—
341C4—61no—
342C4—61no—
343C4—61no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4#100411yescharacteristic
348C4#101011yescharacteristic
349C3#325711yescharacteristic, complemented, Sylow
350C2—41no—
351C2—81no—
352C2—81no—
353C2—241no—
354C2#559711yescharacteristic
355C2#577411yescharacteristic
356C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3257
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. #5619C2semidirect
2. #5603C2semidirect
3. C3×#469C2semidirect
4. #5153C2semidirect
5. #5150C2semidirect
6. #5134C2semidirect
7. #5119C2semidirect
8. #5041C2semidirect
9. #5038C2semidirect
10. #4870C2semidirect
11. #4857C2semidirect
12. #4787C2semidirect
13. #4766C2semidirect
14. #4762C2semidirect
15. #4759C2semidirect
66. #469D3semidirect
349. C3#3257semidirect

Maximal subgroups

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

subgroupindex
#56192
#56032
C3×#4692
#51532
#51502
#51342
#51192
#50412
#50382
#48702
#48572
#47872
#47662
#47622
#47592
#32573