Finite Group Databasebeta

← #82803

Subgroups — #82803

Order 384.

472 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
1C2×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×#889C211yesmaximal, characteristic, complemented
3#5230C211yesmaximal, characteristic, complemented
4#5230C211yesmaximal, characteristic, complemented
5#5210C211yesmaximal, characteristic
6#5210C211yesmaximal, characteristic
7#4985C211yesmaximal, characteristic, complemented
8#4983C211yesmaximal, characteristic, complemented
9#4980C211yesmaximal, characteristic, complemented
10#4975C211yesmaximal, characteristic, complemented
11#4972C211yesmaximal, characteristic, complemented
12#4949C211yesmaximal, characteristic
13#4948C211yesmaximal, characteristic
14#4945C211yesmaximal, characteristic
15#4944C211yesmaximal, characteristic
16#2720—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C3×C22⋊Q8—24no—
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42C2×C4×Dic3C2211yescharacteristic
43#889—24no—
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#125—31no—
55C2×C22⋊Q8D311yescharacteristic
56#400—31no—
57#399—31no—
58#398—31no—
59#396—31no—
60#395—31no—
61#395—31no—
62#389—31no—
63#388—31no—
64#388—31no—
65#387—31no—
66#385—31no—
67#385—31no—
68#384—31no—
69C24×C3C2311yescharacteristic
70C2×C3×Q8—22no—
71C2×C3×Q8—41no—
72C2×C3×Q8—41no—
73C22×C3×C4—22no—
74C22×C3×C4—41no—
75C22×C3×C4—41no—
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C3×C4⋊C4—22no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4D414yes—
93C3×C22⋊C4—22no—
94C3×C22⋊C4—22no—
95#217—22no—
96#217—41no—
97#217—41no—
98#217Q814yescomplemented
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101Dic3⋊C4—22no—
102Dic3⋊C4—22no—
103Dic3⋊C4—22no—
104Dic3⋊C4—22no—
105Dic3⋊C4—41no—
106Dic3⋊C4—41no—
107Dic3⋊C4—41no—
108Dic3⋊C4—41no—
109Dic3⋊C4—41no—
110Dic3⋊C4—41no—
111C4×Dic3—22no—
112C4×Dic3—22no—
113C22×Q8C2×D311yescharacteristic
114C23×C4C2×D311yescharacteristic
115#125—64no—
116C22⋊Q8—24no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4C2×D311yescharacteristic
130C2×C4⋊C4C2×D311yescharacteristic
131C2×C4⋊C4C2×D311yescharacteristic
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—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3C2×Q812yes—
157C23×C3C2411yescharacteristic
158C3×Q8—44no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
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—44no—
175C2×C3×C4C2×D412yes—
176C2×C3×C4C2×D412yes—
177C2×C3×C4C2×D412yes—
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
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×Dic3C2×Q812yes—
200C2×Dic3C2×Q812yes—
201C2×Dic3#4112yes—
202C24C22×D311yescharacteristic
203C2×Q8—22no—
204C2×Q8—41no—
205C2×Q8—41no—
206C22×C4—22no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4C22×D311yescharacteristic
218C22×C4C22×D311yescharacteristic
219C22×C4C22×D311yescharacteristic
220C22×C4C22×D311yescharacteristic
221C22×C4C22×D311yescharacteristic
222C22×C4C22×D311yescharacteristic
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—62no—
226C4⋊C4—62no—
227C4⋊C4—62no—
228C4⋊C4—62no—
229C4⋊C4—62no—
230C4⋊C4—62no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C4⋊C4#6714yes—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—34no—
241C22⋊C4—62no—
242C22⋊C4—121no—
243C22⋊C4—121no—
244C42—62no—
245C42—62no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3C22×D411yescharacteristic
254C22×C3C22×Q811yescharacteristic
255C22×C3C2×#4111yescharacteristic
256C22×C3H(5,𝔽2)11yescharacteristic
257C22×C3H(5,𝔽2)11yescharacteristic
258C22×C32−1+411yescharacteristic
259C22×C32−1+411yescharacteristic
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266Dic3—22no—
267Dic3—22no—
268Dic3—22no—
269Dic3—41no—
270Dic3—41no—
271Dic3—41no—
272Dic3—41no—
273Dic3—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23D3×Q812yes—
281C23C23×D311yescharacteristic
282Q8—44no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—44no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4C2×#6712yes—
324C2×C4C2×#6712yes—
325C2×C4C2×#6712yes—
326C2×C3—42no—
327C2×C3#53511yescharacteristic
328C2×C3#53911yescharacteristic
329C2×C3#54411yescharacteristic
330C2×C3#54711yescharacteristic
331C2×C3D4×Q811yescharacteristic
332C2×C3#55411yescharacteristic
333C2×C3#56311yescharacteristic
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22#100411yescharacteristic
342C22#100411yescharacteristic
343C22C2×D3×Q811yescharacteristic
344C22#100711yescharacteristic
345C22D3×#4111yescharacteristic
346C22#101011yescharacteristic
347C22C22×#6711yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—62no—
355C4—62no—
356C4—62no—
357C4—121no—
358C4—121no—
359C4—121no—
360C4—121no—
361C4—121no—
362C3#272011yescharacteristic, complemented, Sylow
363C2—42no—
364C2#562311yescharacteristic
365C2#563611yescharacteristic
366C2#563811yescharacteristic
367C2#564611yescharacteristic
368C2#580511yescharacteristic
369C2Q8×#6711yescharacteristic
370C2#582411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC24×C3C23
hypercenterC2×C22⋊Q8D3
hypocenterC3#2720
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#889C2semidirect
3. #5230C2semidirect
4. #5230C2semidirect
7. #4985C2semidirect
8. #4983C2semidirect
9. #4980C2semidirect
10. #4975C2semidirect
11. #4972C2semidirect
98. #217Q8semidirect
362. C3#2720semidirect

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×#8892
#52302
#52102
#49852
#49832
#49802
#49752
#49722
#49492
#49482
#49452
#49442
#27203