Finite Group Databasebeta

← #77739

Subgroups — #77739

Order 384.

528 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×#892C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic, complemented
3C3×#398C211yesmaximal, characteristic, complemented
4#4994C211yesmaximal, characteristic
5#4994C211yesmaximal, characteristic
6#4993C211yesmaximal, characteristic
7#4982C211yesmaximal, characteristic, complemented
8Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
9Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10#4673C211yesmaximal, characteristic
11#4666C211yesmaximal, characteristic
12#4666C211yesmaximal, characteristic
13#4660C211yesmaximal, characteristic, complemented
14#4656C211yesmaximal, characteristic, complemented
15#4649C211yesmaximal, characteristic, complemented
16#2396—31nomaximal, Sylow
17C22×C4×D3C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×C4×Dic3C2211yescharacteristic
37C2×C4×Dic3C2211yescharacteristic
38C2×C4×Dic3C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42#892—24no—
43#887—24no—
44C3×#95C2211yescharacteristic
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×C4×Q8—31no—
55C2×#117—31no—
56#400—31no—
57#399—31no—
58#398—31no—
59#398D311yescharacteristic, complemented
60#396—31no—
61#396—31no—
62#395—31no—
63#393—31no—
64C4×C4⋊C4—31no—
65C4×C4⋊C4—31no—
66C4×C4⋊C4—31no—
67C4×C22⋊C4—31no—
68C4×C22⋊C4—31no—
69C23×D3C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C2×C4×D3—22no—
85C2×C4×D3—22no—
86C2×C4×D3Q814yescomplemented
87C2×Dic6—22no—
88C2×Dic6—22no—
89C2×Dic6—22no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94(C2×D3)⋊C4—22no—
95(C2×D3)⋊C4—22no—
96(C2×D3)⋊C4—22no—
97(C2×D3)⋊C4—22no—
98(C2×D3)⋊C4—22no—
99(C2×D3)⋊C4—22no—
100C4⋊Dic3—22no—
101C4⋊Dic3—22no—
102C4⋊Dic3—22no—
103Dic3⋊C4—22no—
104Dic3⋊C4—22no—
105Dic3⋊C4—22no—
106Dic3⋊C4—22no—
107C4×Dic3—22no—
108C4×Dic3—22no—
109C4×Dic3—22no—
110C4×Dic3—22no—
111C4×Dic3—22no—
112C4×Dic3D414yes—
113C22×Q8—31no—
114C23×C4—31no—
115C4×Q8—64no—
116#117—64no—
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⋊C4C2×D311yescharacteristic
125C2×C4⋊C4C2×D311yescharacteristic
126C2×C4⋊C4C2×D311yescharacteristic
127C2×C4⋊C4C2×D311yescharacteristic
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×C42—31no—
135C2×C42—31no—
136C2×C42—31no—
137C2×C42—31no—
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#95C2×D311yescharacteristic
148#95C2×D311yescharacteristic
149#95C2×D311yescharacteristic
150C23×C3C2411yescharacteristic
151C22×D3—21no—
152C22×D3—21no—
153C22×D3—21no—
154C22×D3—21no—
155C22×D3—21no—
156C22×D3—21no—
157C22×D3C2×Q812yes—
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—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4C2×D412yes—
176C2×C3×C4C2×Q812yes—
177C2×C3×C4#4112yes—
178C2×C3×C4#4112yes—
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×Dic3C2×D412yes—
195C2×Dic3C2×D412yes—
196C2×Dic3C2×Q812yes—
197C2×Dic3#4112yes—
198C2×Dic3#4112yes—
199C2×Dic3#4112yes—
200C4×D3—28no—
201Dic6—28no—
202C24—31no—
203C2×Q8—62no—
204C2×Q8—62no—
205C2×Q8—62no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—34no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—62no—
215C22×C4—62no—
216C22×C4C22×D311yescharacteristic
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—22no—
226C4⋊C4—22no—
227C4⋊C4—62no—
228C4⋊C4—62no—
229C4⋊C4—62no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—62no—
233C4⋊C4—62no—
234C22⋊C4—62no—
235C22⋊C4—62no—
236C22⋊C4—62no—
237C22⋊C4—62no—
238C22⋊C4—62no—
239C22⋊C4—62no—
240C42—34no—
241C42—62no—
242C42—62no—
243C42—62no—
244C42—62no—
245C42—62no—
246C22×C3C22×D411yescharacteristic
247C22×C3C22×Q811yescharacteristic
248C22×C3C2×#4111yescharacteristic
249C22×C3C2×#4111yescharacteristic
250C22×C3C2×#4111yescharacteristic
251C22×C3C2×#4111yescharacteristic
252C22×C3C2×#4111yescharacteristic
253C2×D3—22no—
254C2×D3—22no—
255C2×D3—22no—
256C2×D3—41no—
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267Dic3—22no—
268Dic3—22no—
269Dic3—22no—
270Dic3—41no—
271Dic3—41no—
272Dic3—41no—
273Dic3C22⋊Q814yes—
274C23—32no—
275C23—61no—
276C23—61no—
277C23—61no—
278C23—61no—
279C23—61no—
280C23—61no—
281C23C23×D311yescharacteristic
282Q8—68no—
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—21no—
299C2×C4—21no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—32no—
306C2×C4—68no—
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×C4D3×D412yes—
323C2×C4#23712yes—
324C2×C4D3×Q812yes—
325C2×C4#23912yes—
326C2×C3C2×C22⋊Q811yescharacteristic
327C2×C3#52511yescharacteristic
328C2×C3#52911yescharacteristic
329C2×C3#52911yescharacteristic
330C2×C3#53211yescharacteristic
331C2×C3#53311yescharacteristic
332C2×C3#53311yescharacteristic
333D3—42no—
334C22—62no—
335C22—62no—
336C22—62no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22—121no—
341C22C2×D3×D411yescharacteristic
342C22C2×#23711yescharacteristic
343C22C2×D3×Q811yescharacteristic
344C22C2×#23911yescharacteristic
345C22D3×#4111yescharacteristic
346C22D3×#4111yescharacteristic
347C22D3×#4111yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—34no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—62no—
357C4—62no—
358C4—62no—
359C4—121no—
360C4—121no—
361C4—121no—
362C3#239611yescharacteristic, complemented, Sylow
363C2—122no—
364C2#559611yescharacteristic
365C2D3×C22⋊Q811yescharacteristic
366C2#562711yescharacteristic
367C2#564911yescharacteristic
368C2#568811yescharacteristic
369C2#569111yescharacteristic
370C2#572811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC23×D3C23
hypercenter#398D3
hypocenterC3#2396
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#887C2semidirect
3. C3×#398C2semidirect
7. #4982C2semidirect
8. Dic3×C4⋊C4C2semidirect
9. Dic3×C4⋊C4C2semidirect
13. #4660C2semidirect
14. #4656C2semidirect
15. #4649C2semidirect
59. #398D3semidirect
86. C2×C4×D3Q8semidirect
362. C3#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#8922
C2×#8872
C3×#3982
#49942
#49932
#49822
Dic3×C4⋊C42
#46732
#46662
#46602
#46562
#46492
#23963