Finite Group Databasebeta

← #80609

Subgroups — #80609

Order 384.

364 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5675C211yesmaximal, characteristic
2#5583C211yesmaximal, characteristic, complemented
3C3×#464C211yesmaximal, characteristic, complemented
4#5186C211yesmaximal, characteristic, complemented
5#5183C211yesmaximal, characteristic, complemented
6#5173C211yesmaximal, characteristic
7#5167C211yesmaximal, characteristic, complemented
8#5054C211yesmaximal, characteristic, complemented
9#5031C211yesmaximal, characteristic, complemented
10#4849C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4809C211yesmaximal, characteristic, complemented
13#4797C211yesmaximal, characteristic, complemented
14#4783C211yesmaximal, characteristic
15#4768C211yesmaximal, characteristic, complemented
16#3223—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic
23#946C2211yescharacteristic
24#944C2211yescharacteristic
25#944C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#216C2211yescharacteristic
28C2×#216C2211yescharacteristic
29#936C2211yescharacteristic
30C2×#214C2211yescharacteristic
31C2×Dic12C2211yescharacteristic
32C2×Dic12C2211yescharacteristic
33#898—21no—
34#897C2211yescharacteristic
35#890C2211yescharacteristic
36#889—21no—
37#883—21no—
38#883—21no—
39#878—21no—
40#878—21no—
41#876—21no—
42#872C2211yescharacteristic
43#870—21no—
44#869C2211yescharacteristic
45C4×Dic6C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×#102C2211yescharacteristic
49#835C2211yescharacteristic
50#832C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53#816C2211yescharacteristic
54#816C2211yescharacteristic
55#815C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#810C2211yescharacteristic
58#810C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#541—31no—
61#536—31no—
62#497—31no—
63#494—31no—
64#491—31no—
65#488—31no—
66#464—31no—
67#464D311yescharacteristic, complemented
68#462—31no—
69#462—31no—
70#452—31no—
71#452—31no—
72#451—31no—
73#451—31no—
74#433—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C2×#67—21no—
79C22×Dic3—21no—
80D3×Q8—41no—
81D3×Q8—41no—
82#237—41no—
83#237—41no—
84C2×C4×D3—21no—
85C2×C4×D3C2311yescharacteristic
86C2×Dic6—21no—
87C2×Dic6C2311yescharacteristic
88C2×Dic6C2311yescharacteristic
89C2×Dic6C2311yescharacteristic
90C3×Q16—41no—
91C3×SD16—41no—
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97C3×C22⋊C4—21no—
98C3×C42—21no—
99C3×C42C2311yescharacteristic
100#217—21no—
101#217—21no—
102#216—41no—
103#216—41no—
104#216—41no—
105#214—41no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4D411yescharacteristic
109(C2×D3)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123Dic3⋊C4D411yescharacteristic
124C4×Dic3—21no—
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129Dic12—41no—
130C2×#41—31no—
131C22×Q8—31no—
132C2×Q16—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×Q16—31no—
136C2×Q16—31no—
137C2×Q16C2×D311yescharacteristic
138C2×SD16—31no—
139C2×SD16C2×D311yescharacteristic
140C4⋊Q8—31no—
141C4⋊Q8—31no—
142C4⋊Q8—31no—
143#126—61no—
144#126—61no—
145#125—31no—
146#125—61no—
147#125—61no—
148#124—31no—
149#124C2×D311yescharacteristic
150#123—61no—
151#123—61no—
152C22⋊Q8—31no—
153C22⋊Q8—31no—
154C22⋊Q8—61no—
155C22⋊Q8—61no—
156C4×Q8—31no—
157C4×Q8C2×D311yescharacteristic
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103—31no—
166#103C2×D311yescharacteristic
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C22×D3C2×D411yescharacteristic
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×D4—21no—
180C3×D4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—42no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—42no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C4×D3—21no—
197C4×D3—21no—
198C4×D3—42no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—21no—
203Dic6—21no—
204Dic6—21no—
205Dic6—42no—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210#41—121no—
211#41—121no—
212C2×Q8—31no—
213C2×Q8—31no—
214C2×Q8—31no—
215C2×Q8—61no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8C22×D311yescharacteristic
219C2×Q8C22×D311yescharacteristic
220C2×D4—61no—
221C2×D4C22×D311yescharacteristic
222C22×C4—31no—
223C22×C4—61no—
224C22×C4—61no—
225Q16—41no—
226Q16—121no—
227Q16—121no—
228Q16—121no—
229Q16—121no—
230Q16—121no—
231SD16—41no—
232SD16—121no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8C22×D311yescharacteristic
236C2×C8C22×D311yescharacteristic
237C4⋊C4—21no—
238C4⋊C4—31no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4C22×D311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—31no—
256C22⋊C4—31no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C42—21no—
262C42—31no—
263C42—61no—
264C42C22×D311yescharacteristic
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3C22×D411yescharacteristic
269C2×D3—21no—
270C2×D3—21no—
271C2×D3—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4H(5,𝔽2)11yescharacteristic
278C3×C4H(5,𝔽2)11yescharacteristic
279Dic3—21no—
280Dic3—21no—
281Dic3—21no—
282Dic3—41no—
283Dic3—41no—
284Dic3—41no—
285Dic3—41no—
286C23—21no—
287C23—31no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—61no—
293Q8—61no—
294Q8—61no—
295Q8—61no—
296Q8—61no—
297Q8—61no—
298Q8—122no—
299D4—21no—
300D4—21no—
301D4—122no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4D3×D411yescharacteristic
317C2×C4D3×D411yescharacteristic
318C2×C4C23×D311yescharacteristic
319C8—21no—
320C8—21no—
321C8—61no—
322C8—61no—
323C2×C3—81no—
324C2×C3#53511yescharacteristic
325C2×C3#57811yescharacteristic
326C2×C3#57811yescharacteristic
327D3—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—61no—
332C22—61no—
333C22—61no—
334C22C2×D3×D411yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—61no—
341C4—61no—
342C4—61no—
343C4—121no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4#100411yescharacteristic
348C4#100911yescharacteristic
349C3#322311yescharacteristic, complemented, Sylow
350C2—81no—
351C2—121no—
352C2#560711yescharacteristic
353C2#577111yescharacteristic
354C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#464C2
socleC22×C3C22×D4
prime radicalC2×#237C22
hypercenter#464D3
hypocenterC3#3223
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #5583C2semidirect
3. C3×#464C2semidirect
4. #5186C2semidirect
5. #5183C2semidirect
7. #5167C2semidirect
8. #5054C2semidirect
9. #5031C2semidirect
10. #4849C2semidirect
11. #4840C2semidirect
12. #4809C2semidirect
13. #4797C2semidirect
15. #4768C2semidirect
67. #464D3semidirect
349. C3#3223semidirect

Maximal subgroups

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

subgroupindex
#56752
#55832
C3×#4642
#51862
#51832
#51732
#51672
#50542
#50312
#48492
#48402
#48092
#47972
#47832
#47682
#32233