Finite Group Databasebeta

← #82247

Subgroups — #82247

Order 384.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#117C211yesmaximal, characteristic
2C2×#878C211yesmaximal, characteristic
3#5211C211yesmaximal, characteristic
4C2×C3⋊#95C211yesmaximal, characteristic
5#4986C211yesmaximal, characteristic, complemented
6#4982C211yesmaximal, characteristic, complemented
7#4982C211yesmaximal, characteristic, complemented
8#4979C211yesmaximal, characteristic, complemented
9#4950C211yesmaximal, characteristic
10#4946C211yesmaximal, characteristic
11#4942C211yesmaximal, characteristic
12#4933C211yesmaximal, characteristic, complemented
13#4933C211yesmaximal, characteristic, complemented
14#4929C211yesmaximal, characteristic, complemented
15#4928C211yesmaximal, characteristic, complemented
16#2438—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C3×#117—24no—
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C42C2211yescharacteristic
27C2×C3×C42C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39#878—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×C22⋊Q8—31no—
55C2×#117D311yescharacteristic
56#400—31no—
57#398—31no—
58#398—31no—
59#395—31no—
60#391—31no—
61#389—31no—
62#387—31no—
63#387—31no—
64#385—31no—
65#385—31no—
66#381—31no—
67#380—31no—
68C2×#95—31no—
69C24×C3C2311yescharacteristic
70C22×C3×C4—22no—
71C22×C3×C4—22no—
72C22×C3×C4Q814yescomplemented
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×Dic3—22no—
80C22×Dic3—22no—
81C22×Dic3—22no—
82C22×Dic3—22no—
83C22×Dic3—41no—
84C22×Dic3—41no—
85C22×Dic3C2311yescharacteristic
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×C22⋊C4—22no—
99C3×C22⋊C4D414yes—
100C3×C42—22no—
101C3×C42—22no—
102#217—22no—
103#217—22no—
104C4⋊Dic3—22no—
105C4⋊Dic3—22no—
106C4⋊Dic3—41no—
107Dic3⋊C4—22no—
108Dic3⋊C4—22no—
109Dic3⋊C4—22no—
110Dic3⋊C4—22no—
111Dic3⋊C4—41no—
112Dic3⋊C4—41no—
113C22×Q8—31no—
114C23×C4—31no—
115C23×C4—31no—
116C23×C4C2×D311yescharacteristic
117C22⋊Q8—64no—
118#117—24no—
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⋊C4C2×D311yescharacteristic
129C2×C4⋊C4C2×D311yescharacteristic
130C2×C22⋊C4—31no—
131C2×C22⋊C4—31no—
132C2×C22⋊C4C2×D311yescharacteristic
133C2×C22⋊C4C2×D311yescharacteristic
134C2×C42C2×D311yescharacteristic
135C2×C42C2×D311yescharacteristic
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—
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×C3C2×D412yes—
155C23×C3C2×Q812yes—
156C23×C3#4112yes—
157C23×C3C2411yescharacteristic
158C2×C3×C4—28no—
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×C4C2×D412yes—
174C2×C3×C4C2×D412yes—
175C2×C3×C4C2×Q812yes—
176C2×C3×C4C2×Q812yes—
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—21no—
198C2×Dic3—21no—
199C2×Dic3—44no—
200C2×Dic3—44no—
201C2×Dic3#4112yes—
202C2×Dic3#4112yes—
203Dic6—44no—
204C24C22×D311yescharacteristic
205C2×Q8—62no—
206C2×Q8—62no—
207C2×Q8—121no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4—62no—
221C22×C4—62no—
222C22×C4—121no—
223C22×C4—121no—
224C22×C4Dic614yescomplemented
225C22×C4C22×D311yescharacteristic
226C22×C4C22×D311yescharacteristic
227C22×C4C22×D311yescharacteristic
228C22×C4C22×D311yescharacteristic
229C22×C4C22×D311yescharacteristic
230C22×C4C22×D311yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—62no—
234C4⋊C4—62no—
235C4⋊C4—62no—
236C4⋊C4—62no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C22⋊C4—22no—
243C22⋊C4—62no—
244C22⋊C4—62no—
245C22⋊C4#6714yes—
246C42—22no—
247C42—22no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3C22⋊Q814yes—
255C22×C3C22×D411yescharacteristic
256C22×C3C22×Q811yescharacteristic
257C22×C3C2×#4111yescharacteristic
258C22×C3C2×#4111yescharacteristic
259C22×C3C2×#4111yescharacteristic
260C22×C3H(5,𝔽2)11yescharacteristic
261C22×C32−1+411yescharacteristic
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—41no—
267C3×C4—41no—
268Dic3—22no—
269Dic3—22no—
270Dic3—41no—
271Dic3—41no—
272Dic3—41no—
273Dic3—41no—
274Dic3—41no—
275Dic3—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23C2×Dic612yes—
281C23#23512yes—
282C23C2×#6712yes—
283C23C23×D311yescharacteristic
284Q8—124no—
285C2×C4—28no—
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—21no—
299C2×C4—21no—
300C2×C4—32no—
301C2×C4—32no—
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×C4—61no—
324C2×C4—124no—
325C2×C4—124no—
326C2×C4C2×Dic612yes—
327C2×C4C2×Dic612yes—
328C2×C4C2×#6712yes—
329C2×C4C2×#6712yes—
330C2×C3—24no—
331C2×C3C2×C22⋊Q811yescharacteristic
332C2×C3#53311yescharacteristic
333C2×C3#54411yescharacteristic
334C2×C3#54611yescharacteristic
335C2×C3#54711yescharacteristic
336C2×C3#55111yescharacteristic
337C2×C3#55211yescharacteristic
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22#92414yes—
345C22C22×Dic611yescharacteristic
346C22C2×#23511yescharacteristic
347C22D3×#4111yescharacteristic
348C22D3×#4111yescharacteristic
349C22#100911yescharacteristic
350C22#101011yescharacteristic
351C22C22×#6711yescharacteristic
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—62no—
359C4—62no—
360C4—121no—
361C4—121no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4—121no—
366C3#243811yescharacteristic, complemented, Sylow
367C2—24no—
368C2#551711yescharacteristic
369C2#551911yescharacteristic
370C2#552711yescharacteristic
371C2#553111yescharacteristic
372C2C2×#92411yescharacteristic
373C2#582411yescharacteristic
374C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#117C2
prime radicalC24×C3C23
hypercenterC2×#117D3
hypocenterC3#2438
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
5. #4986C2semidirect
6. #4982C2semidirect
7. #4982C2semidirect
8. #4979C2semidirect
12. #4933C2semidirect
13. #4933C2semidirect
14. #4929C2semidirect
15. #4928C2semidirect
72. C22×C3×C4Q8semidirect
224. C22×C4Dic6semidirect
366. C3#2438semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1172
C2×#8782
#52112
C2×C3⋊#952
#49862
#49822
#49792
#49502
#49462
#49422
#49332
#49292
#49282
#24383