Finite Group Databasebeta

← #82799

Subgroups — #82799

Order 384.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×#879C211yesmaximal, characteristic
3#5230C211yesmaximal, characteristic, complemented
4#5229C211yesmaximal, characteristic, complemented
5#5211C211yesmaximal, characteristic
6#5210C211yesmaximal, characteristic
7#4986C211yesmaximal, characteristic, complemented
8#4983C211yesmaximal, characteristic, complemented
9#4982C211yesmaximal, characteristic, complemented
10#4978C211yesmaximal, characteristic, complemented
11#4977C211yesmaximal, characteristic, complemented
12#4976C211yesmaximal, characteristic, complemented
13#4952C211yesmaximal, characteristic
14#4952C211yesmaximal, characteristic
15#4951C211yesmaximal, characteristic
16#2698—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×C4×Dic3C2211yescharacteristic
38C2×C4×Dic3C2211yescharacteristic
39#879—24no—
40C3⋊#95C2211yescharacteristic
41C3⋊#95C2211yescharacteristic
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×#126—31no—
55C2×C22⋊Q8D311yescharacteristic
56#400—31no—
57#398—31no—
58#398—31no—
59#397—31no—
60#397—31no—
61#395—31no—
62#391—31no—
63#389—31no—
64#388—31no—
65#387—31no—
66#385—31no—
67#382—31no—
68#381—31no—
69C24×C3C2311yescharacteristic
70C2×C3×Q8—22no—
71C2×C3×Q8—22no—
72C2×C3×Q8—41no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
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⋊C4—22no—
93C3×C22⋊C4—22no—
94C3×C22⋊C4D414yes—
95#217—22no—
96#217—22no—
97#217—22no—
98#217—41no—
99C4⋊Dic3—22no—
100C4⋊Dic3—41no—
101Dic3⋊C4—22no—
102Dic3⋊C4—22no—
103Dic3⋊C4—22no—
104Dic3⋊C4—22no—
105Dic3⋊C4—41no—
106Dic3⋊C4—41no—
107C4×Dic3—22no—
108C4×Dic3—22no—
109C22×Q8C2×D311yescharacteristic
110C23×C4C2×D311yescharacteristic
111#126—64no—
112C22⋊Q8—24no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4C2×D311yescharacteristic
122C2×C4⋊C4C2×D311yescharacteristic
123C2×C4⋊C4C2×D311yescharacteristic
124C2×C22⋊C4—31no—
125C2×C22⋊C4—31no—
126C2×C22⋊C4—31no—
127C2×C22⋊C4—31no—
128C2×C22⋊C4C2×D311yescharacteristic
129C2×C22⋊C4C2×D311yescharacteristic
130C2×C42—31no—
131C2×C42—31no—
132#95—31no—
133#95—31no—
134#95—31no—
135#95—31no—
136#95—31no—
137#95—31no—
138#95—31no—
139#95—31no—
140#95—31no—
141#95—31no—
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3C2×D412yes—
153C23×C3C2411yescharacteristic
154C3×Q8—44no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
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—44no—
171C2×C3×C4C2×D412yes—
172C2×C3×C4C2×D412yes—
173C2×C3×C4#4112yes—
174C2×Dic3—21no—
175C2×Dic3—21no—
176C2×Dic3—21no—
177C2×Dic3—21no—
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#4112yes—
198C24C22×D311yescharacteristic
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—41no—
213C22×C4C22×D311yescharacteristic
214C22×C4C22×D311yescharacteristic
215C22×C4C22×D311yescharacteristic
216C22×C4C22×D311yescharacteristic
217C22×C4C22×D311yescharacteristic
218C22×C4C22×D311yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—62no—
223C4⋊C4—62no—
224C4⋊C4—62no—
225C4⋊C4—62no—
226C4⋊C4—62no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C4⋊C4—121no—
230C22⋊C4—22no—
231C22⋊C4—62no—
232C22⋊C4—62no—
233C22⋊C4—62no—
234C22⋊C4—121no—
235C22⋊C4#6714yes—
236C42—62no—
237C42—62no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3C22×D411yescharacteristic
246C22×C3C2×#4111yescharacteristic
247C22×C3C2×#4111yescharacteristic
248C22×C3H(5,𝔽2)11yescharacteristic
249C22×C32−1+411yescharacteristic
250C22×C32−1+411yescharacteristic
251C22×C32−1+411yescharacteristic
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—41no—
258Dic3—22no—
259Dic3—41no—
260Dic3—41no—
261Dic3—41no—
262Dic3—41no—
263Dic3—41no—
264Dic3—41no—
265Dic3—41no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23C2×#6712yes—
273C23C23×D311yescharacteristic
274Q8—44no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—32no—
291C2×C4—44no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
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#23712yes—
316C2×C4C2×#6712yes—
317C2×C4C2×#6712yes—
318C2×C3—42no—
319C2×C3#53611yescharacteristic
320C2×C3#54111yescharacteristic
321C2×C3#54211yescharacteristic
322C2×C3#54611yescharacteristic
323C2×C3#54711yescharacteristic
324C2×C3#55211yescharacteristic
325C2×C3#56311yescharacteristic
326C22—22no—
327C22—22no—
328C22—22no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22C2×#23711yescharacteristic
334C22#100411yescharacteristic
335C22#100711yescharacteristic
336C22#100711yescharacteristic
337C22D3×#4111yescharacteristic
338C22#101011yescharacteristic
339C22C22×#6711yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—62no—
347C4—121no—
348C4—121no—
349C4—121no—
350C4—121no—
351C4—121no—
352C4—121no—
353C4—121no—
354C3#269811yescharacteristic, complemented, Sylow
355C2—42no—
356C2#562511yescharacteristic
357C2#563911yescharacteristic
358C2#564011yescharacteristic
359C2#564611yescharacteristic
360C2#580611yescharacteristic
361C2#581611yescharacteristic
362C2#582411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC24×C3C23
hypercenterC2×C22⋊Q8D3
hypocenterC3#2698
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #5230C2semidirect
4. #5229C2semidirect
7. #4986C2semidirect
8. #4983C2semidirect
9. #4982C2semidirect
10. #4978C2semidirect
11. #4977C2semidirect
12. #4976C2semidirect
354. C3#2698semidirect

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×#8792
#52302
#52292
#52112
#52102
#49862
#49832
#49822
#49782
#49772
#49762
#49522
#49512
#26983