Finite Group Databasebeta

← #77174

Subgroups — #77174

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×#946C211yesmaximal, characteristic
2C2×#944C211yesmaximal, characteristic, complemented
3C2×#896C211yesmaximal, characteristic
4C2×D3×C4⋊C4C211yesmaximal, characteristic
5C3×#395C211yesmaximal, characteristic, complemented
6#4989C211yesmaximal, characteristic
7#4982C211yesmaximal, characteristic, complemented
8#4976C211yesmaximal, characteristic, complemented
9#4975C211yesmaximal, characteristic, complemented
10#4668C211yesmaximal, characteristic
11#4667C211yesmaximal, characteristic
12#4666C211yesmaximal, characteristic
13#4657C211yesmaximal, characteristic, complemented
14#4656C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2440—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25#946—24no—
26#944—24no—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
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#896—24no—
47D3×C4⋊C4—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#395D311yescharacteristic, complemented
64#389—31no—
65#389—31no—
66#385—31no—
67#385—31no—
68#384—31no—
69#380—31no—
70C4×C22⋊C4—31no—
71C23×D3C2311yescharacteristic
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C2×C3×Q8—41no—
75C22×C3×C4C2311yescharacteristic
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
89C2×C4×D3—22no—
90C2×C4×D3—22no—
91C2×C4×D3—22no—
92C2×C4×D3—22no—
93C2×C4×D3—22no—
94C2×C4×D3—22no—
95C2×C4×D3—41no—
96C2×C4×D3—41no—
97C2×C4×D3Q814yescomplemented
98C2×Dic6—22no—
99C2×Dic6—22no—
100C2×Dic6—41no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—41no—
104(C2×D3)⋊C4—22no—
105(C2×D3)⋊C4—22no—
106(C2×D3)⋊C4D414yes—
107(C2×D3)⋊C4Q814yescomplemented
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×Q8C2×D311yescharacteristic
125C23×C4—31no—
126C23×C4—31no—
127C23×C4—31no—
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—64no—
145C2×C4⋊C4C2×D311yescharacteristic
146C2×C4⋊C4C2×D311yescharacteristic
147C2×C4⋊C4C2×D311yescharacteristic
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C42—31no—
153C2×C42—31no—
154#95—31no—
155#95—31no—
156#95—31no—
157#95—31no—
158#95—31no—
159#95C2×D311yescharacteristic
160#95C2×D311yescharacteristic
161#95C2×D311yescharacteristic
162C23×C3C2411yescharacteristic
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—21no—
166C22×D3—21no—
167C22×D3C2×D412yes—
168C22×D3C2×Q812yes—
169C22×D3C2×Q812yes—
170C3×Q8—44no—
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—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4C2×D412yes—
189C2×C3×C4C2×D412yes—
190C2×C3×C4C2×Q812yes—
191C2×C3×C4C2×Q812yes—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
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×Dic3C2×D412yes—
208C2×Dic3C2×D412yes—
209C2×Dic3C2×D412yes—
210C2×Dic3C2×Q812yes—
211C2×Dic3C2×Q812yes—
212C2×Dic3#4112yes—
213C4×D3—28no—
214C4×D3—44no—
215C4×D3—44no—
216Dic6—44no—
217C24—31no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—62no—
222C2×Q8—62no—
223C2×Q8—121no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—34no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—62no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—121no—
239C22×C4—121no—
240C22×C4C22×D311yescharacteristic
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245C22×C4C22×D311yescharacteristic
246C22×C4C22×D311yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—34no—
250C4⋊C4—41no—
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—34no—
264C22⋊C4—34no—
265C22⋊C4—62no—
266C22⋊C4—62no—
267C42—62no—
268C42—62no—
269C22×C3C22×D411yescharacteristic
270C22×C3C22×D411yescharacteristic
271C22×C3C22×Q811yescharacteristic
272C22×C3C22×Q811yescharacteristic
273C22×C3C2×#4111yescharacteristic
274C22×C3H(5,𝔽2)11yescharacteristic
275C22×C32−1+411yescharacteristic
276C2×D3—22no—
277C2×D3—22no—
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—22no—
281C2×D3—22no—
282C2×D3C4⋊Q814yescomplemented
283C3×C4—22no—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290Dic3—22no—
291Dic3—22no—
292Dic3—22no—
293Dic3—41no—
294Dic3—41no—
295Dic3—41no—
296Dic3C22⋊Q814yescomplemented
297C23—32no—
298C23—32no—
299C23—32no—
300C23—61no—
301C23—61no—
302C23—61no—
303C23—61no—
304C23C23×D311yescharacteristic
305Q8—44no—
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—21no—
323C2×C4—21no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—32no—
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—124no—
347C2×C4—124no—
348C2×C4D3×D412yes—
349C2×C4D3×D412yes—
350C2×C4D3×Q812yes—
351C2×C4D3×Q812yes—
352C2×C3C2×C22⋊Q811yescharacteristic
353C2×C3C2×C4⋊Q811yescharacteristic
354C2×C3#54411yescharacteristic
355C2×C3D4211yescharacteristic
356C2×C3#54711yescharacteristic
357C2×C3D4×Q811yescharacteristic
358C2×C3#55411yescharacteristic
359D3—24no—
360C22—34no—
361C22—62no—
362C22—62no—
363C22—62no—
364C22—62no—
365C22—62no—
366C22—62no—
367C22C2×D3×D411yescharacteristic
368C22C2×D3×D411yescharacteristic
369C22#100411yescharacteristic
370C22C2×D3×Q811yescharacteristic
371C22C2×D3×Q811yescharacteristic
372C22#100711yescharacteristic
373C22D3×#4111yescharacteristic
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—34no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—62no—
383C4—62no—
384C4—62no—
385C4—121no—
386C4—121no—
387C4—121no—
388C3#244011yescharacteristic, complemented, Sylow
389C2—64no—
390C2#559011yescharacteristic
391C2D3×C22⋊Q811yescharacteristic
392C2#563111yescharacteristic
393C2#563611yescharacteristic
394C2D3×C4⋊Q811yescharacteristic
395C2#572711yescharacteristic
396C2#572911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC23×D3C23
hypercenter#395D3
hypocenterC3#2440
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#944C2semidirect
5. C3×#395C2semidirect
7. #4982C2semidirect
8. #4976C2semidirect
9. #4975C2semidirect
13. #4657C2semidirect
14. #4656C2semidirect
15. #4647C2semidirect
63. #395D3semidirect
97. C2×C4×D3Q8semidirect
107. (C2×D3)⋊C4Q8semidirect
282. C2×D3C4⋊Q8semidirect
296. Dic3C22⋊Q8semidirect
388. C3#2440semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×#9442
C2×#8962
C2×D3×C4⋊C42
C3×#3952
#49892
#49822
#49762
#49752
#46682
#46672
#46662
#46572
#46562
#46472
#24403