Finite Group Databasebeta

← #76791

Subgroups — #76791

Order 384.

540 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#946C211yesmaximal, characteristic, complemented
2C2×#877C211yesmaximal, characteristic, complemented
3C3×#393C211yesmaximal, characteristic, complemented
4#5229C211yesmaximal, characteristic, complemented
5#4961C211yesmaximal, characteristic
6#4958C211yesmaximal, characteristic
7#4956C211yesmaximal, characteristic
8#4950C211yesmaximal, characteristic, complemented
9#4944C211yesmaximal, characteristic, complemented
10#4675C211yesmaximal, characteristic, complemented
11#4670C211yesmaximal, characteristic, complemented
12#4666C211yesmaximal, characteristic, complemented
13#4660C211yesmaximal, characteristic, complemented
14#4655C211yesmaximal, characteristic, complemented
15#4654C211yesmaximal, characteristic, complemented
16#2477—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26#946—24no—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×C4×Dic3C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41#877—24no—
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×#117—31no—
56#400—31no—
57#397—31no—
58#394—31no—
59#393D311yescharacteristic, complemented
60#391—31no—
61#388—31no—
62#388—31no—
63#387—31no—
64#387—31no—
65#386—31no—
66#386—31no—
67#385—31no—
68C4×C22⋊C4—31no—
69C24×C3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C3×Q8—41no—
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—22no—
81C2×#67—22no—
82C2×#67—41no—
83C22×Dic3—22no—
84C22×Dic3—22no—
85C22×Dic3—22no—
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C2×C4×D3—22no—
94C2×C4×D3—22no—
95C2×C4×D3—41no—
96C3×C22⋊C4—22no—
97C3×C22⋊C4—22no—
98C3×C22⋊C4—41no—
99#217—22no—
100#217—22no—
101(C2×D3)⋊C4—22no—
102(C2×D3)⋊C4—22no—
103(C2×D3)⋊C4—22no—
104(C2×D3)⋊C4—22no—
105(C2×D3)⋊C4—22no—
106C4⋊Dic3—22no—
107C4⋊Dic3—22no—
108C4⋊Dic3—41no—
109Dic3⋊C4—22no—
110Dic3⋊C4—22no—
111Dic3⋊C4D414yescomplemented
112C4×Dic3—22no—
113C4×Dic3—22no—
114C4×Dic3—41no—
115C22×Q8C2×D311yescharacteristic
116C22×D4—31no—
117C23×C4—31no—
118C23×C4—31no—
119C22⋊Q8—64no—
120#117—64no—
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×C22⋊C4—31no—
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×C22⋊C4C2×D311yescharacteristic
135C2×C22⋊C4C2×D311yescharacteristic
136C2×C22⋊C4C2×D311yescharacteristic
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#95—31no—
148#95—31no—
149#95C2×D311yescharacteristic
150#95C2×D311yescharacteristic
151#95C2×D311yescharacteristic
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3#4112yes—
159C23×C3C2411yescharacteristic
160C22×D3—21no—
161C22×D3—21no—
162C22×D3—21no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—21no—
166C22×D3#4112yes—
167C3×Q8—44no—
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×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×C4C2×D412yes—
185C2×C3×C4#4112yes—
186#67—44no—
187C2×Dic3—28no—
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—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3C2×D412yes—
207C2×Dic3C2×D412yes—
208C2×Dic3#4112yes—
209C4×D3—44no—
210C24—31no—
211C24C22×D311yescharacteristic
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—41no—
215C2×D4—62no—
216C2×D4—62no—
217C2×D4—121no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—62no—
230C22×C4—121no—
231C22×C4C22×D311yescharacteristic
232C22×C4C22×D311yescharacteristic
233C22×C4C22×D311yescharacteristic
234C22×C4C22×D311yescharacteristic
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237C4⋊C4—34no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—62no—
241C4⋊C4—62no—
242C4⋊C4—121no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—41no—
246C22⋊C4—62no—
247C22⋊C4—62no—
248C22⋊C4—62no—
249C22⋊C4—62no—
250C22⋊C4—62no—
251C22⋊C4—62no—
252C22⋊C4—62no—
253C42—62no—
254C42—62no—
255C42—121no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3C22×D411yescharacteristic
264C22×C3C2×#4111yescharacteristic
265C22×C3C2×#4111yescharacteristic
266C22×C3C2×#4111yescharacteristic
267C22×C3C2×#4111yescharacteristic
268C22×C3H(5,𝔽2)11yescharacteristic
269C22×C32−1+411yescharacteristic
270C2×D3—22no—
271C2×D3—22no—
272C2×D3—22no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—41no—
283Dic3—22no—
284Dic3—22no—
285Dic3—22no—
286Dic3—41no—
287Dic3—41no—
288Dic3—41no—
289Dic3—41no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—32no—
297C23—61no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—61no—
302C23—61no—
303C23#23912yes—
304C23C23×D311yescharacteristic
305Q8—44no—
306D4—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—32no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—68no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
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—124no—
346C2×C4D3×D412yes—
347C2×C4#23712yes—
348C2×C3—42no—
349C2×C3#52511yescharacteristic
350C2×C3#52911yescharacteristic
351C2×C3#54211yescharacteristic
352C2×C3#54611yescharacteristic
353C2×C3#54811yescharacteristic
354C2×C3#55111yescharacteristic
355C2×C3#55211yescharacteristic
356D3—42no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—62no—
365C22—62no—
366C22—62no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22C2×D3×D411yescharacteristic
372C22C2×#23711yescharacteristic
373C22#100411yescharacteristic
374C22C2×#23911yescharacteristic
375C22D3×#4111yescharacteristic
376C22D3×#4111yescharacteristic
377C22#101011yescharacteristic
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—62no—
385C4—62no—
386C4—62no—
387C4—121no—
388C4—121no—
389C4—121no—
390C4—121no—
391C3#247711yescharacteristic, complemented, Sylow
392C2—42no—
393C2—122no—
394C2#559111yescharacteristic
395C2#562711yescharacteristic
396C2#563711yescharacteristic
397C2#564211yescharacteristic
398C2#566911yescharacteristic
399C2#567711yescharacteristic
400C2#568011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC22×#67C22
hypercenter#393D3
hypocenterC3#2477
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#946C2semidirect
2. C2×#877C2semidirect
3. C3×#393C2semidirect
4. #5229C2semidirect
8. #4950C2semidirect
9. #4944C2semidirect
10. #4675C2semidirect
11. #4670C2semidirect
12. #4666C2semidirect
13. #4660C2semidirect
14. #4655C2semidirect
15. #4654C2semidirect
59. #393D3semidirect
111. Dic3⋊C4D4semidirect
391. C3#2477semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×#8772
C3×#3932
#52292
#49612
#49582
#49562
#49502
#49442
#46752
#46702
#46662
#46602
#46552
#46542
#24773