Finite Group Databasebeta

← #80059

Subgroups — #80059

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5816C211yesmaximal, characteristic, complemented
2#5603C211yesmaximal, characteristic, complemented
3C3×#448C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5168C211yesmaximal, characteristic, complemented
6#5167C211yesmaximal, characteristic, complemented
7#5165C211yesmaximal, characteristic, complemented
8#5162C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic
10#4798C211yesmaximal, characteristic, complemented
11#4795C211yesmaximal, characteristic, complemented
12#4794C211yesmaximal, characteristic, complemented
13#4792C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3225—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×#121C2211yescharacteristic
22#947C2211yescharacteristic
23#947C2211yescharacteristic
24#946—21no—
25#946—21no—
26#944C2211yescharacteristic
27#944C2211yescharacteristic
28C2×#216C2211yescharacteristic
29C2×#216C2211yescharacteristic
30#936—21no—
31#936C2211yescharacteristic
32#935—21no—
33D4×Dic3C2211yescharacteristic
34C2×#214C2211yescharacteristic
35C2×#213C2211yescharacteristic
36#929—21no—
37#929—21no—
38#926C2211yescharacteristic
39#924C2211yescharacteristic
40C2×Dic12C2211yescharacteristic
41C2×#204C2211yescharacteristic
42#888C2211yescharacteristic
43#887C2211yescharacteristic
44#879—21no—
45#878—21no—
46C3×#103C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×C22⋊C8C2211yescharacteristic
49#835C2211yescharacteristic
50#835C2211yescharacteristic
51#830C2211yescharacteristic
52#821C2211yescharacteristic
53#816C2211yescharacteristic
54#815C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#810C2211yescharacteristic
59#809C2211yescharacteristic
60#542—31no—
61#536—31no—
62#497—31no—
63#495—31no—
64#489—31no—
65#487—31no—
66#464—31no—
67#463—31no—
68#462—31no—
69#461—31no—
70#452—31no—
71#451—31no—
72#449—31no—
73#448D311yescharacteristic, complemented
74#433—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—41no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C22×Dic3—21no—
86#235—41no—
87#235—41no—
88C2×D12C2311yescharacteristic
89C2×C4×D3—21no—
90C2×Dic6—21no—
91C2×Dic6C2311yescharacteristic
92C2×Dic6C2311yescharacteristic
93C3×SD16—41no—
94C3×SD16—41no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#217D411yescharacteristic
104#217D411yescharacteristic
105#216—41no—
106#216—41no—
107#214—41no—
108#213—41no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113C4⋊Dic3D411yescharacteristic
114C4⋊Dic3D411yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123C4×Dic3—21no—
124C4×Dic3C2311yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129#204—41no—
130C2×#41—31no—
131C22×Q8C2×D311yescharacteristic
132C2×Q16—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16C2×D311yescharacteristic
138C2×SD16C2×D311yescharacteristic
139C2×D8—31no—
140C4⋊Q8—31no—
141C4⋊Q8—31no—
142C4⋊Q8—31no—
143#126—61no—
144#124—31no—
145#124—31no—
146#124—31no—
147#124—61no—
148#123—61no—
149#123—61no—
150#123—61no—
151C22⋊Q8—31no—
152C22⋊Q8—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155#121C2×D311yescharacteristic
156C4×Q8—31no—
157C4×D4—31no—
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103C2×D311yescharacteristic
166#103C2×D311yescharacteristic
167#102—31no—
168#102—31no—
169C22⋊C8—31no—
170C22⋊C8C2×D311yescharacteristic
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×Q8—42no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—81no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—42no—
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—42no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—81no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201D12—21no—
202D12—21no—
203C4×D3—42no—
204Dic6—21no—
205Dic6—21no—
206Dic6—21no—
207Dic6—21no—
208Dic6—81no—
209C3×C8—21no—
210C3×C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—121no—
214#41—121no—
215C2×Q8—21no—
216C2×Q8—31no—
217C2×Q8—31no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—61no—
221C2×Q8C22×D311yescharacteristic
222C2×Q8C22×D311yescharacteristic
223C2×D4—21no—
224C2×D4—31no—
225C2×D4—61no—
226C2×D4C22×D311yescharacteristic
227C22×C4—21no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4C22×D311yescharacteristic
231Q16—121no—
232Q16—121no—
233Q16—121no—
234SD16—41no—
235SD16—41no—
236SD16—121no—
237SD16—121no—
238D8—121no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8C22×D311yescharacteristic
242C2×C8C22×D311yescharacteristic
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4C22×D311yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—31no—
256C22⋊C4—31no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C42—31no—
266C42—31no—
267C42—61no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3C22×D411yescharacteristic
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C3×C4—21no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4H(5,𝔽2)11yescharacteristic
283C3×C4H(5,𝔽2)11yescharacteristic
284Dic3—21no—
285Dic3—21no—
286Dic3—21no—
287Dic3—21no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291C23—21no—
292C23—61no—
293C23D3×D411yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—61no—
300Q8—61no—
301Q8—61no—
302Q8—61no—
303Q8—241no—
304D4—21no—
305D4—21no—
306D4—61no—
307D4—61no—
308D4—81no—
309D4—122no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—42no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—122no—
324C2×C4—241no—
325C2×C4D3×D411yescharacteristic
326C2×C4C23×D311yescharacteristic
327C8—21no—
328C8—21no—
329C8—61no—
330C8—61no—
331C2×C3—41no—
332C2×C3—81no—
333C2×C3#53511yescharacteristic
334C2×C3#57711yescharacteristic
335C2×C3#57811yescharacteristic
336D3—81no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22C2×D3×D411yescharacteristic
347C4—21no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—61no—
352C4—61no—
353C4—61no—
354C4—61no—
355C4—121no—
356C4—121no—
357C4—121no—
358C4#100411yescharacteristic
359C4#100411yescharacteristic
360C3#322511yescharacteristic, complemented, Sylow
361C2—41no—
362C2—81no—
363C2—241no—
364C2#559211yescharacteristic
365C2#576211yescharacteristic
366C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5037C2
hypercenter#448D3
hypocenterC3#3225
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #5816C2semidirect
2. #5603C2semidirect
3. C3×#448C2semidirect
4. #5228C2semidirect
5. #5168C2semidirect
6. #5167C2semidirect
7. #5165C2semidirect
8. #5162C2semidirect
10. #4798C2semidirect
11. #4795C2semidirect
12. #4794C2semidirect
13. #4792C2semidirect
14. #4733C2semidirect
15. #4720C2semidirect
73. #448D3semidirect
360. C3#3225semidirect

Maximal subgroups

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

subgroupindex
#58162
#56032
C3×#4482
#52282
#51682
#51672
#51652
#51622
#50372
#47982
#47952
#47942
#47922
#47332
#47202
#32253