Finite Group Databasebeta

← #74416

Subgroups — #74416

Order 384.

608 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#924C211yesmaximal, characteristic
2C22×Dic3⋊C4C211yesmaximal, characteristic
3C2×#888C211yesmaximal, characteristic, complemented
4C2×#878C211yesmaximal, characteristic
5C3×#385C211yesmaximal, characteristic
6#4982C211yesmaximal, characteristic, complemented
7#4979C211yesmaximal, characteristic, complemented
8#4975C211yesmaximal, characteristic, complemented
9#4950C211yesmaximal, characteristic
10#4942C211yesmaximal, characteristic
11Dic3×C22⋊C4C211yesmaximal, characteristic
12#4657C211yesmaximal, characteristic, complemented
13#4656C211yesmaximal, characteristic, complemented
14#4647C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2440—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32#924—24no—
33C2×Dic3⋊C4—24no—
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46#888—24no—
47#878—24no—
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×C4⋊Q8—31no—
57C2×C22⋊Q8—31no—
58C2×C22⋊Q8—31no—
59C22×C4⋊C4—31no—
60#398—31no—
61#398—31no—
62#395—31no—
63#395—31no—
64#389—31no—
65#389—31no—
66#385—31no—
67#385D311yescharacteristic
68#384—31no—
69#380—31no—
70C4×C22⋊C4—31no—
71C24×C3C2311yescharacteristic
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—41no—
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×Dic3—22no—
82C22×Dic3—22no—
83C22×Dic3—22no—
84C22×Dic3—22no—
85C22×Dic3—41no—
86C22×Dic3Q814yescomplemented
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C2×Dic6—22no—
96C2×Dic6—22no—
97C2×Dic6—22no—
98C2×Dic6—22no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—41no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4Q814yescomplemented
105#217—22no—
106#217D414yes—
107C4⋊Dic3—22no—
108C4⋊Dic3—22no—
109C4⋊Dic3—22no—
110C4⋊Dic3—41no—
111Dic3⋊C4—22no—
112Dic3⋊C4—22no—
113Dic3⋊C4—22no—
114Dic3⋊C4—22no—
115Dic3⋊C4—22no—
116Dic3⋊C4—22no—
117Dic3⋊C4—22no—
118Dic3⋊C4—22no—
119Dic3⋊C4—22no—
120Dic3⋊C4D414yes—
121C4×Dic3—22no—
122C4×Dic3—22no—
123C22×Q8—31no—
124C22×Q8—31no—
125C23×C4—31no—
126C23×C4—31no—
127C23×C4C2×D311yescharacteristic
128C4⋊Q8—64no—
129C22⋊Q8—64no—
130C22⋊Q8—64no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—64no—
146C2×C4⋊C4C2×D311yescharacteristic
147C2×C4⋊C4C2×D311yescharacteristic
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4C2×D311yescharacteristic
151C2×C22⋊C4C2×D311yescharacteristic
152C2×C42—31no—
153C2×C42—31no—
154#95—31no—
155#95—31no—
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
160#95C2×D311yescharacteristic
161#95C2×D311yescharacteristic
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3C2×D412yes—
167C23×C3C2×Q812yes—
168C23×C3C2×Q812yes—
169C23×C3C2411yescharacteristic
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—21no—
184C2×C3×C4—21no—
185C2×C3×C4—44no—
186C2×C3×C4C2×D412yes—
187C2×C3×C4C2×Q812yes—
188C2×C3×C4C2×Q812yes—
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—28no—
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—44no—
208C2×Dic3C2×D412yes—
209C2×Dic3C2×D412yes—
210C2×Dic3C2×D412yes—
211C2×Dic3C2×D412yes—
212C2×Dic3C2×Q812yes—
213C2×Dic3C2×Q812yes—
214C2×Dic3#4112yes—
215Dic6—44no—
216Dic6—44no—
217C24C22×D311yescharacteristic
218C2×Q8—62no—
219C2×Q8—62no—
220C2×Q8—62no—
221C2×Q8—62no—
222C2×Q8—121no—
223C2×Q8—121no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—31no—
227C22×C4—34no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—41no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—62no—
240C22×C4—121no—
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245C22×C4C22×D311yescharacteristic
246C22×C4C22×D311yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—34no—
249C4⋊C4—41no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—62no—
257C4⋊C4—62no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—121no—
263C22⋊C4—22no—
264C22⋊C4—34no—
265C22⋊C4—62no—
266C22⋊C4Dic614yescomplemented
267C42—62no—
268C42—62no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3C4⋊Q814yescomplemented
276C22×C3C22×D411yescharacteristic
277C22×C3C22×D411yescharacteristic
278C22×C3C22×Q811yescharacteristic
279C22×C3C22×Q811yescharacteristic
280C22×C3C2×#4111yescharacteristic
281C22×C3H(5,𝔽2)11yescharacteristic
282C22×C32−1+411yescharacteristic
283C3×C4—22no—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—41no—
289Dic3—22no—
290Dic3—22no—
291Dic3—22no—
292Dic3—22no—
293Dic3—41no—
294Dic3—41no—
295Dic3—41no—
296Dic3C22⋊Q814yescomplemented
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23C2×Dic612yes—
302C23D3×D412yes—
303C23D3×Q812yes—
304C23C23×D311yescharacteristic
305Q8—124no—
306Q8—124no—
307C2×C4—21no—
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—32no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—44no—
330C2×C4—68no—
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—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—124no—
349C2×C4C2×Dic612yes—
350C2×C4C2×Dic612yes—
351C2×C4D3×D412yes—
352C2×C3—24no—
353C2×C3C2×C22⋊Q811yescharacteristic
354C2×C3C2×C4⋊Q811yescharacteristic
355C2×C3#54411yescharacteristic
356C2×C3D4211yescharacteristic
357C2×C3#54711yescharacteristic
358C2×C3D4×Q811yescharacteristic
359C2×C3#55411yescharacteristic
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22#88814yescomplemented
367C22C22×Dic611yescharacteristic
368C22C2×D3×D411yescharacteristic
369C22C2×D3×D411yescharacteristic
370C22#100411yescharacteristic
371C22C2×D3×Q811yescharacteristic
372C22D3×#4111yescharacteristic
373C22#101011yescharacteristic
374C4—22no—
375C4—22no—
376C4—22no—
377C4—34no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—62no—
382C4—62no—
383C4—62no—
384C4—62no—
385C4—121no—
386C4—121no—
387C4—121no—
388C3#244011yescharacteristic, complemented, Sylow
389C2—24no—
390C2C2×#88811yescharacteristic
391C2D4×Dic611yescharacteristic
392C2#553911yescharacteristic
393C2#559011yescharacteristic
394C2#562311yescharacteristic
395C2D3×C22⋊Q811yescharacteristic
396C2#565511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC24×C3C23
hypercenter#385D3
hypocenterC3#2440
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C2×#888C2semidirect
6. #4982C2semidirect
7. #4979C2semidirect
8. #4975C2semidirect
12. #4657C2semidirect
13. #4656C2semidirect
14. #4647C2semidirect
15. #4647C2semidirect
86. C22×Dic3Q8semidirect
104. C3×C22⋊C4Q8semidirect
266. C22⋊C4Dic6semidirect
275. C22×C3C4⋊Q8semidirect
296. Dic3C22⋊Q8semidirect
366. C22#888semidirect
388. C3#2440semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C22×Dic3⋊C42
C2×#8882
C2×#8782
C3×#3852
#49822
#49792
#49752
#49502
#49422
Dic3×C22⋊C42
#46572
#46562
#46472
#24403