Finite Group Databasebeta

← #80222

Subgroups — #80222

Order 384.

390 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5832C211yesmaximal, characteristic
2#5622C211yesmaximal, characteristic, complemented
3C3×#452C211yesmaximal, characteristic, complemented
4#5239C211yesmaximal, characteristic, complemented
5#5190C211yesmaximal, characteristic, complemented
6#5182C211yesmaximal, characteristic, complemented
7#5167C211yesmaximal, characteristic, complemented
8#5163C211yesmaximal, characteristic, complemented
9#5047C211yesmaximal, characteristic, complemented
10#4795C211yesmaximal, characteristic, complemented
11#4794C211yesmaximal, characteristic, complemented
12#4765C211yesmaximal, characteristic, complemented
13#4762C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4719C211yesmaximal, characteristic, complemented
16#3216—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22#947C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24#944C2211yescharacteristic
25#944C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#216C2211yescharacteristic
28C2×#215C2211yescharacteristic
29#940C2211yescharacteristic
30#939—21no—
31#939—21no—
32#936C2211yescharacteristic
33C2×#214C2211yescharacteristic
34C4×#67—21no—
35C4×#67—21no—
36#924C2211yescharacteristic
37C2×C4×Dic3C2211yescharacteristic
38C2×Dic12C2211yescharacteristic
39C2×#204C2211yescharacteristic
40#889—21no—
41#888C2211yescharacteristic
42#887—21no—
43#887C2211yescharacteristic
44#878—21no—
45#877—21no—
46C3×#103C2211yescharacteristic
47C3×#102C2211yescharacteristic
48C3×C22⋊C8C2211yescharacteristic
49#835C2211yescharacteristic
50#832C2211yescharacteristic
51#830C2211yescharacteristic
52#821C2211yescharacteristic
53#816C2211yescharacteristic
54#815C2211yescharacteristic
55#815C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58#810C2211yescharacteristic
59#809C2211yescharacteristic
60#533—31no—
61#532—31no—
62#497—31no—
63#497—31no—
64#489—31no—
65#489—31no—
66#462—31no—
67#462—31no—
68#460—31no—
69#460—31no—
70#452—31no—
71#452—31no—
72#452—31no—
73#452D311yescharacteristic, complemented
74#432—31no—
75C3×#41—41no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C22×Dic3D411yescharacteristic
87C22×Dic3D411yescharacteristic
88#235—41no—
89#235—41no—
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×Dic6—21no—
93C2×Dic6C2311yescharacteristic
94C2×Dic6C2311yescharacteristic
95C3×Q16—41no—
96C3×SD16—41no—
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102#217—21no—
103#217—21no—
104#217—21no—
105#216—41no—
106#216—41no—
107#215—41no—
108#214—41no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3D411yescharacteristic
123C4×Dic3D411yescharacteristic
124C4×Dic3C2311yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129#204—41no—
130C2×#41—31no—
131C2×#41C2×D311yescharacteristic
132C2×Q16—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×Q16C2×D311yescharacteristic
136C2×SD16—31no—
137C2×SD16—31no—
138C2×SD16—31no—
139C2×SD16C2×D311yescharacteristic
140C4⋊Q8—31no—
141C4⋊Q8—31no—
142C4⋊Q8—31no—
143#127—31no—
144#125—61no—
145#124—31no—
146#124—31no—
147C22⋊Q8—31no—
148C22⋊Q8—61no—
149C22⋊Q8C2×D311yescharacteristic
150#121—61no—
151#121—61no—
152C4×Q8—31no—
153C4×Q8—31no—
154C4×Q8—61no—
155C4×D4—61no—
156C4×D4—61no—
157#117—61no—
158C2×C42—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8C2×D311yescharacteristic
171#97—31no—
172#97—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—42no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—42no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—42no—
192#67—42no—
193#67—42no—
194C2×Dic3—22no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—41no—
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12—21no—
205C4×D3—42no—
206Dic6—21no—
207Dic6—21no—
208Dic6—21no—
209Dic6—21no—
210Dic6—42no—
211C3×C8—21no—
212C3×C8—21no—
213C3⋊C8—21no—
214C3⋊C8—21no—
215#41—41no—
216#41—41no—
217#41—121no—
218#41—121no—
219C2×Q8—31no—
220C2×Q8—31no—
221C2×Q8—61no—
222C2×Q8C22×D311yescharacteristic
223C2×Q8C22×D311yescharacteristic
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4C22×D311yescharacteristic
230C22×C4—21no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235Q16—41no—
236Q16—121no—
237Q16—121no—
238Q16—121no—
239SD16—41no—
240SD16—121no—
241SD16—121no—
242SD16—121no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8C22×D311yescharacteristic
246C2×C8C22×D311yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—31no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4C22×D311yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C42—31no—
265C42—31no—
266C42—31no—
267C42—31no—
268C42—61no—
269C42—61no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3C22×D411yescharacteristic
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4H(5,𝔽2)11yescharacteristic
285C3×C4H(5,𝔽2)11yescharacteristic
286Dic3—21no—
287Dic3—21no—
288Dic3—21no—
289Dic3—41no—
290Dic3—41no—
291Dic3—41no—
292Dic3#13712yes—
293C23—21no—
294C23—61no—
295C23D3×D411yescharacteristic
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—61no—
301Q8—61no—
302Q8—61no—
303Q8—61no—
304Q8—122no—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—61no—
309D4—61no—
310D4—122no—
311D4—122no—
312D4—122no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—42no—
323C2×C4—62no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—122no—
328C2×C4—121no—
329C2×C4D3×D411yescharacteristic
330C2×C4C23×D311yescharacteristic
331C8—21no—
332C8—21no—
333C8—61no—
334C8—61no—
335C2×C3—41no—
336C2×C3—81no—
337C2×C3#53511yescharacteristic
338C2×C3C2×#13711yescharacteristic
339C2×C3#57511yescharacteristic
340D3—81no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—121no—
348C22—121no—
349C22—121no—
350C22C2×D3×D411yescharacteristic
351C4—21no—
352C4—32no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—61no—
357C4—61no—
358C4—61no—
359C4—121no—
360C4—121no—
361C4—121no—
362C4#100411yescharacteristic
363C4#100411yescharacteristic
364C3#321611yescharacteristic, complemented, Sylow
365C2—41no—
366C2—81no—
367C2—241no—
368C2#559211yescharacteristic
369C2#576811yescharacteristic
370C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radical#5832C2
hypercenter#452D3
hypocenterC3#3216
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #5622C2semidirect
3. C3×#452C2semidirect
4. #5239C2semidirect
5. #5190C2semidirect
6. #5182C2semidirect
7. #5167C2semidirect
8. #5163C2semidirect
9. #5047C2semidirect
10. #4795C2semidirect
11. #4794C2semidirect
12. #4765C2semidirect
13. #4762C2semidirect
14. #4733C2semidirect
15. #4719C2semidirect
73. #452D3semidirect
364. C3#3216semidirect

Maximal subgroups

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

subgroupindex
#58322
#56222
C3×#4522
#52392
#51902
#51822
#51672
#51632
#50472
#47952
#47942
#47652
#47622
#47332
#47192
#32163