Finite Group Databasebeta

← #82800

Subgroups — #82800

Order 384.

506 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#885C211yesmaximal, characteristic
3#5231C211yesmaximal, characteristic, complemented
4#5229C211yesmaximal, characteristic, complemented
5#5213C211yesmaximal, characteristic
6#5211C211yesmaximal, characteristic, complemented
7#4994C211yesmaximal, characteristic, complemented
8#4992C211yesmaximal, characteristic, complemented
9#4990C211yesmaximal, characteristic, complemented
10#4979C211yesmaximal, characteristic, complemented
11#4978C211yesmaximal, characteristic, complemented
12#4973C211yesmaximal, characteristic, complemented
13#4959C211yesmaximal, characteristic
14#4952C211yesmaximal, characteristic, complemented
15#4951C211yesmaximal, characteristic, complemented
16#2700—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C3×C22⋊Q8—24no—
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41#885—24no—
42C3⋊#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#124—31no—
55C2×C22⋊Q8D311yescharacteristic, complemented
56#397—31no—
57#397—31no—
58#396—31no—
59#395—31no—
60#394—31no—
61#393—31no—
62#391—31no—
63#391—31no—
64#388—31no—
65#388—31no—
66#387—31no—
67#386—31no—
68#381—31no—
69C24×C3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C3×Q8—41no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—41no—
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C2×#67—22no—
84C2×#67—22no—
85C2×#67—41no—
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C2×Dic6—22no—
94C2×Dic6—22no—
95C2×Dic6—41no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—22no—
99C3×C22⋊C4—22no—
100C3×C22⋊C4D414yes—
101#217—22no—
102#217—41no—
103#217—41no—
104(C2×D3)⋊C4—22no—
105(C2×D3)⋊C4—22no—
106(C2×D3)⋊C4—22no—
107(C2×D3)⋊C4—41no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110C4⋊Dic3—41no—
111Dic3⋊C4—22no—
112Dic3⋊C4—41no—
113C4×Dic3—22no—
114C4×Dic3—41no—
115C22×Q8—31no—
116C22×Q8C2×D311yescharacteristic
117C22×D4—31no—
118C23×C4C2×D311yescharacteristic
119#124—64no—
120C22⋊Q8—24no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4C2×D311yescharacteristic
125C2×C4⋊C4C2×D311yescharacteristic
126C2×C4⋊C4C2×D311yescharacteristic
127C2×C22⋊C4—31no—
128C2×C22⋊C4—31no—
129C2×C22⋊C4—31no—
130C2×C22⋊C4—31no—
131C2×C22⋊C4—31no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—31no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4C2×D311yescharacteristic
137C2×C22⋊C4C2×D311yescharacteristic
138C2×C42—31no—
139C2×C42—31no—
140#95—31no—
141#95—31no—
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3C2×D412yes—
159C23×C3C2411yescharacteristic
160C22×D3—21no—
161C22×D3—21no—
162C22×D3—21no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—21no—
166C22×D3—21no—
167C3×Q8—44no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—44no—
184C2×C3×C4C2×D412yes—
185C2×C3×C4C2×D412yes—
186C2×C3×C4#4112yes—
187#67—44no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3#4112yes—
209Dic6—44no—
210C24—31no—
211C24C22×D311yescharacteristic
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—41no—
215C2×Q8—62no—
216C2×Q8—62no—
217C2×Q8—121no—
218C2×D4—62no—
219C2×D4—62no—
220C2×D4—121no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—41no—
231C22×C4C22×D311yescharacteristic
232C22×C4C22×D311yescharacteristic
233C22×C4C22×D311yescharacteristic
234C22×C4C22×D311yescharacteristic
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—62no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—22no—
244C22⋊C4—62no—
245C22⋊C4—62no—
246C22⋊C4—62no—
247C22⋊C4—62no—
248C22⋊C4—121no—
249C22⋊C4—121no—
250C22⋊C4—121no—
251C22⋊C4—121no—
252C22⋊C4—121no—
253C22⋊C4#6714yes—
254C42—62no—
255C42—121no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3C22×D411yescharacteristic
264C22×C3C2×#4111yescharacteristic
265C22×C3C2×#4111yescharacteristic
266C22×C3H(5,𝔽2)11yescharacteristic
267C22×C3H(5,𝔽2)11yescharacteristic
268C22×C32−1+411yescharacteristic
269C22×C32−1+411yescharacteristic
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—41no—
283Dic3—22no—
284Dic3—41no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288Dic3—41no—
289Dic3—41no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—61no—
297C23—61no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—61no—
302C23—61no—
303C23C2×#6712yes—
304C23C23×D311yescharacteristic
305Q8—44no—
306Q8—124no—
307D4—124no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—32no—
324C2×C4—44no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4#23712yes—
346C2×C4C2×#6712yes—
347C2×C4C2×#6712yes—
348C2×C3—42no—
349C2×C3#53611yescharacteristic
350C2×C3#54211yescharacteristic
351C2×C3#54211yescharacteristic
352C2×C3#54611yescharacteristic
353C2×C3#54611yescharacteristic
354C2×C3#55611yescharacteristic
355C2×C3#56211yescharacteristic
356D3—81no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22C2×#23711yescharacteristic
372C22#100411yescharacteristic
373C22#100711yescharacteristic
374C22#100711yescharacteristic
375C22D3×#4111yescharacteristic
376C22#100911yescharacteristic
377C22C22×#6711yescharacteristic
378C4—22no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—62no—
385C4—121no—
386C4—121no—
387C4—121no—
388C4—121no—
389C4—121no—
390C4—121no—
391C3#270011yescharacteristic, complemented, Sylow
392C2—42no—
393C2—241no—
394C2#564011yescharacteristic
395C2#564111yescharacteristic
396C2#564311yescharacteristic
397C2#564711yescharacteristic
398C2#580611yescharacteristic
399C2#581611yescharacteristic
400C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC22×#67C22
hypercenterC2×C22⋊Q8D3
hypocenterC3#2700
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
3. #5231C2semidirect
4. #5229C2semidirect
6. #5211C2semidirect
7. #4994C2semidirect
8. #4992C2semidirect
9. #4990C2semidirect
10. #4979C2semidirect
11. #4978C2semidirect
12. #4973C2semidirect
14. #4952C2semidirect
15. #4951C2semidirect
55. C2×C22⋊Q8D3semidirect
391. C3#2700semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C2×#8852
#52312
#52292
#52132
#52112
#49942
#49922
#49902
#49792
#49782
#49732
#49592
#49522
#49512
#27003