Finite Group Databasebeta

← #74663

Subgroups — #74663

Order 384.

568 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×D4×Dic3C211yesmaximal, characteristic
2C3×#386C211yesmaximal, characteristic
3#5220C211yesmaximal, characteristic
4#5210C211yesmaximal, characteristic, complemented
5C2×C3⋊#95C211yesmaximal, characteristic, complemented
6#4946C211yesmaximal, characteristic, complemented
7#4945C211yesmaximal, characteristic, complemented
8#4944C211yesmaximal, characteristic, complemented
9#4943C211yesmaximal, characteristic, complemented
10#4942C211yesmaximal, characteristic, complemented
11Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
12#4654C211yesmaximal, characteristic, complemented
13#4653C211yesmaximal, characteristic, complemented
14#4651C211yesmaximal, characteristic, complemented
15#4650C211yesmaximal, characteristic, complemented
16#2318—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C23×Dic3C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30D4×Dic3C418yescomplemented
31C2×C4⋊Dic3C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×C4×Dic3C2211yescharacteristic
37C2×C4×Dic3C2211yescharacteristic
38C2×C4×Dic3C2211yescharacteristic
39C3×#95C2211yescharacteristic
40C3×#95C2211yescharacteristic
41C3⋊#95C2211yescharacteristic
42C3⋊#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C2×C4×D4—31no—
54#388—31no—
55#388—31no—
56#387—31no—
57#387—31no—
58#386—31no—
59#386D311yescharacteristic
60#385—31no—
61#385—31no—
62#383—31no—
63#382—31no—
64#381—31no—
65#380—31no—
66C4×C22⋊C4—31no—
67C2×#95—31no—
68C24×C3C2311yescharacteristic
69C24×C3C2311yescharacteristic
70C2×C3×D4—22no—
71C2×C3×D4—22no—
72C2×C3×D4C2×C414yes—
73C22×C3×C4—22no—
74C22×C3×C4—41no—
75C22×C3×C4—41no—
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×Dic3—22no—
82C22×Dic3—41no—
83C22×Dic3—41no—
84C22×Dic3—41no—
85C22×Dic3—41no—
86C22×Dic3—41no—
87C22×Dic3—41no—
88C22×Dic3C2×C414yes—
89C22×Dic3C2×C414yescomplemented
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C3×C22⋊C4—22no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101#217—22no—
102#217—41no—
103#217—41no—
104#217C2×C414yes—
105#217C2×C414yescomplemented
106C4⋊Dic3C2×C414yescomplemented
107Dic3⋊C4—41no—
108Dic3⋊C4—41no—
109Dic3⋊C4—41no—
110Dic3⋊C4—41no—
111C4×Dic3—41no—
112C4×Dic3—41no—
113C4×Dic3C2×C414yescomplemented
114C22×D4C2×D311yescharacteristic
115C23×C4—31no—
116C23×C4—31no—
117C23×C4—31no—
118C23×C4C2×D311yescharacteristic
119C4×D4—38no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C22⋊C4—31no—
126C2×C22⋊C4—31no—
127C2×C22⋊C4—31no—
128C2×C22⋊C4—31no—
129C2×C22⋊C4—31no—
130C2×C22⋊C4C2×D311yescharacteristic
131C2×C22⋊C4C2×D311yescharacteristic
132C2×C22⋊C4C2×D311yescharacteristic
133C2×C42—31no—
134C2×C42—31no—
135C2×C42—31no—
136#95—31no—
137#95—31no—
138#95—31no—
139#95—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#95C2×D311yescharacteristic
149#95C2×D311yescharacteristic
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3C22×C412yes—
161C23×C3C22×C412yes—
162C23×C3#4112yes—
163C23×C3#4112yes—
164C23×C3C2411yescharacteristic
165C3×D4—28no—
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×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—44no—
181C2×C3×C4C22×C412yes—
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×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—44no—
203C2×Dic3—44no—
204C2×Dic3—44no—
205C2×Dic3C22×C412yes—
206C2×Dic3C22×C412yes—
207C2×Dic3C22×C412yes—
208C2×Dic3C22×C412yes—
209C24C22×D311yescharacteristic
210C24C22×D311yescharacteristic
211C2×D4—22no—
212C2×D4—22no—
213C2×D4C4×D314yes—
214C22×C4—22no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—34no—
220C22×C4—34no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—62no—
228C22×C4—121no—
229C22×C4—121no—
230C22×C4—121no—
231C22×C4—121no—
232C22×C4—121no—
233C22×C4—121no—
234C22×C4C22×D311yescharacteristic
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237C22×C4C22×D311yescharacteristic
238C22×C4C22×D311yescharacteristic
239C4⋊C4—34no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C22⋊C4—22no—
245C22⋊C4—34no—
246C22⋊C4—34no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—62no—
250C22⋊C4—121no—
251C22⋊C4—121no—
252C42—34no—
253C42—121no—
254C42—121no—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3#11714yes—
269C22×C3C23×C411yescharacteristic
270C22×C3C2×#4111yescharacteristic
271C22×C3C2×#4111yescharacteristic
272C22×C3H(5,𝔽2)11yescharacteristic
273C22×C3H(5,𝔽2)11yescharacteristic
274C22×C3H(5,𝔽2)11yescharacteristic
275C22×C32−1+411yescharacteristic
276C3×C4—22no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281Dic3—22no—
282Dic3—22no—
283Dic3—22no—
284Dic3—22no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288Dic3—41no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23C2×C4×D312yes—
300C23C2×C4×D312yes—
301C23#23712yes—
302C23#23712yes—
303C23C23×D311yescharacteristic
304D4—28no—
305C2×C4—21no—
306C2×C4—21no—
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—32no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—32no—
323C2×C4—44no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
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—124no—
345C2×C4—124no—
346C2×C4—124no—
347C2×C4C2×C4×D312yes—
348C2×C3—24no—
349C2×C3—42no—
350C2×C3C2×#11711yescharacteristic
351C2×C3#51811yescharacteristic
352C2×C3#52011yescharacteristic
353C2×C3#53811yescharacteristic
354C2×C3#53811yescharacteristic
355C2×C3#53911yescharacteristic
356C2×C3#53911yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22#87714yes—
371C22C22×C4×D311yescharacteristic
372C22C2×#23711yescharacteristic
373C22C2×#23711yescharacteristic
374C22#100411yescharacteristic
375C22#100411yescharacteristic
376C22#100911yescharacteristic
377C22#101011yescharacteristic
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—62no—
384C4—62no—
385C4—62no—
386C4—62no—
387C4—121no—
388C4—121no—
389C4—121no—
390C4—121no—
391C3#231811yescharacteristic, complemented, Sylow
392C2—24no—
393C2—42no—
394C2C2×#87711yescharacteristic
395C2#554511yescharacteristic
396C2#554611yescharacteristic
397C2#559311yescharacteristic
398C2#560211yescharacteristic
399C2#561711yescharacteristic
400C2#566511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#386C2
socle, cosocleC23×C3C24
prime radicalC22×C3×D4C22
hypercenter#386D3
hypocenterC3#2318
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #5210C2semidirect
5. C2×C3⋊#95C2semidirect
6. #4946C2semidirect
7. #4945C2semidirect
8. #4944C2semidirect
9. #4943C2semidirect
10. #4942C2semidirect
11. Dic3×C22⋊C4C2semidirect
12. #4654C2semidirect
13. #4653C2semidirect
14. #4651C2semidirect
15. #4650C2semidirect
30. D4×Dic3C4semidirect
89. C22×Dic3C2×C4semidirect
105. #217C2×C4semidirect
106. C4⋊Dic3C2×C4semidirect
113. C4×Dic3C2×C4semidirect
391. C3#2318semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C3×#3862
#52202
#52102
C2×C3⋊#952
#49462
#49452
#49442
#49432
#49422
Dic3×C22⋊C42
#46542
#46532
#46512
#46502
#23183