Finite Group Databasebeta

← #74472

Subgroups — #74472

Order 384.

544 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#924C211yesmaximal, characteristic
2C2×#878C211yesmaximal, characteristic
3C3×#385C211yesmaximal, characteristic
4C2×C3⋊#95C211yesmaximal, characteristic
5#4984C211yesmaximal, characteristic, complemented
6#4982C211yesmaximal, characteristic, complemented
7#4979C211yesmaximal, characteristic, complemented
8#4972C211yesmaximal, characteristic, complemented
9#4950C211yesmaximal, characteristic
10#4942C211yesmaximal, characteristic
11Dic3×C22⋊C4C211yesmaximal, characteristic
12#4657C211yesmaximal, characteristic, complemented
13#4656C211yesmaximal, characteristic, complemented
14#4646C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2549—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⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41#878—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×C22⋊Q8—31no—
56#398—31no—
57#398—31no—
58#395—31no—
59#395—31no—
60#391—31no—
61#391—31no—
62#389—31no—
63#389—31no—
64#385—31no—
65#385D311yescharacteristic
66#380—31no—
67C4×C22⋊C4—31no—
68C2×#95—31no—
69C24×C3C2311yescharacteristic
70C22×C3×C4—22no—
71C22×C3×C4—22no—
72C22×C3×C4—41no—
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×Dic3—22no—
80C22×Dic3—22no—
81C22×Dic3—41no—
82C22×Dic3—41no—
83C22×Dic3—41no—
84C22×Dic3Q814yescomplemented
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C2×Dic6—22no—
94C2×Dic6—22no—
95C2×Dic6—22no—
96C2×Dic6—22no—
97C2×Dic6—41no—
98C2×Dic6—41no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—41no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4Q814yescomplemented
103#217—22no—
104#217D414yes—
105C4⋊Dic3—22no—
106C4⋊Dic3—22no—
107C4⋊Dic3—22no—
108C4⋊Dic3—41no—
109Dic3⋊C4—22no—
110Dic3⋊C4—22no—
111Dic3⋊C4—22no—
112Dic3⋊C4—22no—
113Dic3⋊C4—41no—
114Dic3⋊C4—41no—
115C4×Dic3—41no—
116C4×Dic3—41no—
117C22×Q8—31no—
118C22×Q8—31no—
119C23×C4—31no—
120C23×C4—31no—
121C23×C4C2×D311yescharacteristic
122C22⋊Q8—64no—
123C22⋊Q8—64no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—31no—
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×C4⋊C4C2×D311yescharacteristic
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C22⋊C4C2×D311yescharacteristic
139C2×C22⋊C4C2×D311yescharacteristic
140C2×C42—31no—
141C2×C42—31no—
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95C2×D311yescharacteristic
153#95C2×D311yescharacteristic
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3C2×D412yes—
159C23×C3C2×Q812yes—
160C23×C3C2×Q812yes—
161C23×C3C2411yescharacteristic
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
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—44no—
178C2×C3×C4C2×Q812yes—
179C2×C3×C4C2×Q812yes—
180C2×C3×C4#4112yes—
181C2×Dic3—21no—
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—44no—
201C2×Dic3—44no—
202C2×Dic3C2×D412yes—
203C2×Dic3C2×D412yes—
204C2×Dic3C2×Q812yes—
205C2×Dic3C2×Q812yes—
206C2×Dic3#4112yes—
207Dic6—44no—
208Dic6—44no—
209C24C22×D311yescharacteristic
210C2×Q8—62no—
211C2×Q8—62no—
212C2×Q8—62no—
213C2×Q8—62no—
214C2×Q8—121no—
215C2×Q8—121no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—34no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—41no—
228C22×C4—62no—
229C22×C4—62no—
230C22×C4—121no—
231C22×C4—121no—
232C22×C4—121no—
233C22×C4C22×D311yescharacteristic
234C22×C4C22×D311yescharacteristic
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237C22×C4C22×D311yescharacteristic
238C22×C4C22×D311yescharacteristic
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—62no—
242C4⋊C4—62no—
243C4⋊C4—62no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C22⋊C4—22no—
252C22⋊C4—34no—
253C22⋊C4—62no—
254C22⋊C4Dic614yescomplemented
255C42—121no—
256C42—121no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3C4⋊Q814yes—
264C22×C3C22×D411yescharacteristic
265C22×C3C22×Q811yescharacteristic
266C22×C3C22×Q811yescharacteristic
267C22×C3C2×#4111yescharacteristic
268C22×C3C2×#4111yescharacteristic
269C22×C3H(5,𝔽2)11yescharacteristic
270C22×C3H(5,𝔽2)11yescharacteristic
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277Dic3—22no—
278Dic3—22no—
279Dic3—22no—
280Dic3—22no—
281Dic3—22no—
282Dic3—41no—
283Dic3—41no—
284Dic3—41no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23C2×Dic612yes—
290C23D3×D412yes—
291C23D3×Q812yes—
292C23C23×D311yescharacteristic
293Q8—124no—
294Q8—124no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—32no—
311C2×C4—32no—
312C2×C4—32no—
313C2×C4—32no—
314C2×C4—32no—
315C2×C4—44no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
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—124no—
336C2×C4—124no—
337C2×C4C2×Dic612yes—
338C2×C4C2×Dic612yes—
339C2×C4#23712yes—
340C2×C3—24no—
341C2×C3C2×C4⋊Q811yescharacteristic
342C2×C3#53311yescharacteristic
343C2×C3#54311yescharacteristic
344C2×C3#54611yescharacteristic
345C2×C3#54611yescharacteristic
346C2×C3#55411yescharacteristic
347C2×C3#55411yescharacteristic
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22#88814yes—
355C22C22×Dic611yescharacteristic
356C22C2×D3×D411yescharacteristic
357C22C2×#23711yescharacteristic
358C22#100411yescharacteristic
359C22C2×D3×Q811yescharacteristic
360C22D3×#4111yescharacteristic
361C22#100911yescharacteristic
362C4—22no—
363C4—22no—
364C4—22no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—62no—
369C4—62no—
370C4—62no—
371C4—62no—
372C4—62no—
373C4—121no—
374C4—121no—
375C4—121no—
376C3#254911yescharacteristic, complemented, Sylow
377C2—24no—
378C2C2×#88811yescharacteristic
379C2#553911yescharacteristic
380C2#554311yescharacteristic
381C2#559111yescharacteristic
382C2#562211yescharacteristic
383C2#563411yescharacteristic
384C2#565411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC24×C3C23
hypercenter#385D3
hypocenterC3#2549
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
5. #4984C2semidirect
6. #4982C2semidirect
7. #4979C2semidirect
8. #4972C2semidirect
12. #4657C2semidirect
13. #4656C2semidirect
14. #4646C2semidirect
15. #4646C2semidirect
84. C22×Dic3Q8semidirect
102. C3×C22⋊C4Q8semidirect
254. C22⋊C4Dic6semidirect
376. C3#2549semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C2×#8782
C3×#3852
C2×C3⋊#952
#49842
#49822
#49792
#49722
#49502
#49422
Dic3×C22⋊C42
#46572
#46562
#46462
#25493