Finite Group Databasebeta

← #75837

Subgroups — #75837

Order 384.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#883C211yesmaximal, characteristic
2C2×#67⋊C4C211yesmaximal, characteristic
3C2×#872C211yesmaximal, characteristic, complemented
4C3×#388C211yesmaximal, characteristic, complemented
5#4989C211yesmaximal, characteristic, complemented
6#4972C211yesmaximal, characteristic, complemented
7#4956C211yesmaximal, characteristic
8#4946C211yesmaximal, characteristic, complemented
9#4945C211yesmaximal, characteristic, complemented
10Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
11C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
12#4666C211yesmaximal, characteristic, complemented
13D3×#95C211yesmaximal, characteristic, complemented
14#4653C211yesmaximal, characteristic, complemented
15#4652C211yesmaximal, characteristic, complemented
16#2359—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C42C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×C4×Dic3C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42#883C418yescomplemented
43#67⋊C4—24no—
44#872—24no—
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C2×#123—31no—
56C2×C4×D4—31no—
57C2×#117—31no—
58#389—31no—
59#388—31no—
60#388D311yescharacteristic, complemented
61#387—31no—
62#387—31no—
63#386—31no—
64#385—31no—
65#385—31no—
66C4×C4⋊C4—31no—
67C4×C22⋊C4—31no—
68C4×C22⋊C4—31no—
69C2×#95—31no—
70C24×C3C2311yescharacteristic
71C23×D3C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—22no—
79C2×#67—22no—
80C2×#67C2×C414yes—
81C22×Dic3—22no—
82C22×Dic3—22no—
83C22×Dic3—41no—
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C2×C4×D3—22no—
92C2×C4×D3—22no—
93C2×C4×D3—22no—
94C2×C4×D3—22no—
95C2×C4×D3—22no—
96C2×C4×D3—22no—
97C2×C4×D3—41no—
98C2×C4×D3—41no—
99C2×C4×D3C2×C414yescomplemented
100C3×C4⋊C4—41no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—41no—
103C3×C22⋊C4C2×C414yescomplemented
104C3×C42—22no—
105#217—22no—
106#217C2×C414yescomplemented
107(C2×D3)⋊C4—22no—
108(C2×D3)⋊C4C2×C414yescomplemented
109(C2×D3)⋊C4D414yes—
110C4⋊Dic3C2×C414yescomplemented
111Dic3⋊C4—22no—
112Dic3⋊C4—22no—
113Dic3⋊C4—22no—
114Dic3⋊C4—22no—
115Dic3⋊C4C2×C414yescomplemented
116C4×Dic3—22no—
117C4×Dic3—22no—
118C4×Dic3—22no—
119C4×Dic3—41no—
120C22×D4—31no—
121C23×C4—31no—
122C23×C4—31no—
123C23×C4—31no—
124C23×C4—31no—
125#123—38no—
126C4×D4—64no—
127#117—64no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4C2×D311yescharacteristic
135C2×C22⋊C4—31no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4C2×D311yescharacteristic
141C2×C22⋊C4C2×D311yescharacteristic
142C2×C22⋊C4C2×D311yescharacteristic
143C2×C42—31no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—31no—
147C2×C42C2×D311yescharacteristic
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95—31no—
156#95C2×D311yescharacteristic
157#95C2×D311yescharacteristic
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3C22×C412yes—
165C23×C3C2411yescharacteristic
166C22×D3—21no—
167C22×D3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3C22×C412yes—
171C22×D3C2×D412yes—
172C22×D3#4112yes—
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×C4C22×C412yes—
189C2×C3×C4C22×C412yes—
190C2×C3×C4C2×D412yes—
191#67—28no—
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×Dic3—44no—
208C2×Dic3C22×C412yes—
209C2×Dic3C22×C412yes—
210C2×Dic3C22×C412yes—
211C2×Dic3C2×D412yes—
212C2×Dic3#4112yes—
213C2×Dic3#4112yes—
214C4×D3—28no—
215C4×D3—44no—
216C4×D3—44no—
217C24—31no—
218C24C22×D311yescharacteristic
219C2×D4—34no—
220C2×D4—62no—
221C2×D4—62no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—34no—
229C22×C4—31no—
230C22×C4—62no—
231C22×C4—62no—
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×C4—121no—
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245C22×C4C22×D311yescharacteristic
246C22×C4C22×D311yescharacteristic
247C4⋊C4—34no—
248C4⋊C4—34no—
249C4⋊C4—41no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C22⋊C4—22no—
255C22⋊C4—34no—
256C22⋊C4—34no—
257C22⋊C4—34no—
258C22⋊C4—41no—
259C22⋊C4—62no—
260C22⋊C4—62no—
261C22⋊C4C4×D314yescomplemented
262C42—22no—
263C42—62no—
264C42—62no—
265C42—62no—
266C42—121no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3C23×C411yescharacteristic
275C22×C3C22×D411yescharacteristic
276C22×C3C2×#4111yescharacteristic
277C22×C3C2×#4111yescharacteristic
278C22×C3C2×#4111yescharacteristic
279C22×C3H(5,𝔽2)11yescharacteristic
280C22×C32−1+411yescharacteristic
281C2×D3—22no—
282C2×D3—22no—
283C2×D3—22no—
284C2×D3—22no—
285C2×D3—22no—
286C2×D3—22no—
287C2×D3C4×D414yes—
288C3×C4—22no—
289C3×C4—22no—
290C3×C4—22no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294Dic3—22no—
295Dic3—22no—
296Dic3—22no—
297Dic3—22no—
298Dic3—22no—
299Dic3—22no—
300Dic3—41no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—32no—
308C23—32no—
309C23—32no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23—61no—
314C23C2×C4×D312yes—
315C23C23×D311yescharacteristic
316D4—68no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—68no—
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—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—124no—
355C2×C4—124no—
356C2×C4—124no—
357C2×C4C2×C4×D312yes—
358C2×C4C2×C4×D312yes—
359C2×C4D3×D412yes—
360C2×C3—42no—
361C2×C3C2×C4×D411yescharacteristic
362C2×C3C4×#4111yescharacteristic
363C2×C3#52011yescharacteristic
364C2×C3#54611yescharacteristic
365C2×C3#54711yescharacteristic
366C2×C3#55111yescharacteristic
367C2×C3#55211yescharacteristic
368D3—24no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—34no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—62no—
378C22—62no—
379C22—62no—
380C22—62no—
381C22—62no—
382C22—62no—
383C22C22×C4×D311yescharacteristic
384C22C2×D3×D411yescharacteristic
385C22#100411yescharacteristic
386C22D3×#4111yescharacteristic
387C22D3×#4111yescharacteristic
388C22D3×#4111yescharacteristic
389C22#101011yescharacteristic
390C4—22no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—62no—
397C4—62no—
398C4—62no—
399C4—62no—
400C4—62no—
401C4—62no—
402C4—121no—
403C3#235911yescharacteristic, complemented, Sylow
404C2—42no—
405C2—64no—
406C2#552211yescharacteristic
407C2C4×D3×D411yescharacteristic
408C2#554611yescharacteristic
409C2#561211yescharacteristic
410C2#565511yescharacteristic
411C2#567911yescharacteristic
412C2#570811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#388C2
socle, cosocleC23×C3C24
prime radicalC22×#67C22
hypercenter#388D3
hypocenterC3#2359
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. C2×#872C2semidirect
4. C3×#388C2semidirect
5. #4989C2semidirect
6. #4972C2semidirect
8. #4946C2semidirect
9. #4945C2semidirect
10. Dic3×C22⋊C4C2semidirect
11. C4×Dic3⋊C4C2semidirect
12. #4666C2semidirect
13. D3×#95C2semidirect
14. #4653C2semidirect
15. #4652C2semidirect
42. #883C4semidirect
60. #388D3semidirect
99. C2×C4×D3C2×C4semidirect
103. C3×C22⋊C4C2×C4semidirect
106. #217C2×C4semidirect
108. (C2×D3)⋊C4C2×C4semidirect
110. C4⋊Dic3C2×C4semidirect
115. Dic3⋊C4C2×C4semidirect
261. C22⋊C4C4×D3semidirect
403. C3#2359semidirect

Maximal subgroups

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

subgroupindex
C2×#8832
C2×#67⋊C42
C2×#8722
C3×#3882
#49892
#49722
#49562
#49462
#49452
Dic3×C22⋊C42
C4×Dic3⋊C42
#46662
D3×#952
#46532
#46522
#23593