Finite Group Databasebeta

← #74983

Subgroups — #74983

Order 384.

508 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#888C211yesmaximal, characteristic, complemented
2C2×#872C211yesmaximal, characteristic
3C3×#387C211yesmaximal, characteristic, complemented
4#4994C211yesmaximal, characteristic
5#4994C211yesmaximal, characteristic
6#4976C211yesmaximal, characteristic, complemented
7#4929C211yesmaximal, characteristic, complemented
8#4673C211yesmaximal, characteristic
9#4673C211yesmaximal, characteristic
10#4667C211yesmaximal, characteristic
11#4666C211yesmaximal, characteristic
12#4655C211yesmaximal, characteristic, complemented
13#4650C211yesmaximal, characteristic, complemented
14#4647C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2439—31nomaximal, Sylow
17C22×C4×D3C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C42C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×C4×Dic3C2211yescharacteristic
38C2×C4×Dic3C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40#888—24no—
41#872—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×C4⋊Q8—31no—
55C2×#117—31no—
56#396—31no—
57#396—31no—
58#393—31no—
59#393—31no—
60#391—31no—
61#391—31no—
62#389—31no—
63#389—31no—
64#387—31no—
65#387D311yescharacteristic, complemented
66#382—31no—
67#380—31no—
68C4×C22⋊C4—31no—
69C23×D3C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C2×C4×D3—22no—
85C2×C4×D3—22no—
86C2×C4×D3Q814yescomplemented
87C2×Dic6—22no—
88C2×Dic6—22no—
89C2×Dic6—22no—
90C2×Dic6—22no—
91C2×Dic6—41no—
92C2×Dic6—41no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C42—22no—
96(C2×D3)⋊C4—22no—
97(C2×D3)⋊C4—22no—
98(C2×D3)⋊C4—22no—
99(C2×D3)⋊C4—22no—
100(C2×D3)⋊C4—41no—
101(C2×D3)⋊C4—41no—
102C4⋊Dic3—22no—
103C4⋊Dic3—41no—
104Dic3⋊C4—22no—
105Dic3⋊C4—22no—
106Dic3⋊C4—22no—
107Dic3⋊C4—22no—
108Dic3⋊C4—41no—
109Dic3⋊C4D414yes—
110C4×Dic3—22no—
111C4×Dic3—22no—
112C4×Dic3—22no—
113C22×Q8—31no—
114C22×Q8—31no—
115C23×C4—31no—
116C4⋊Q8—64no—
117#117—64no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4C2×D311yescharacteristic
127C2×C4⋊C4C2×D311yescharacteristic
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×C42—31no—
135C2×C42—31no—
136C2×C42—31no—
137C2×C42C2×D311yescharacteristic
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#95C2×D311yescharacteristic
147#95C2×D311yescharacteristic
148#95C2×D311yescharacteristic
149#95C2×D311yescharacteristic
150C23×C3C2411yescharacteristic
151C22×D3—21no—
152C22×D3—21no—
153C22×D3—21no—
154C22×D3—21no—
155C22×D3—21no—
156C22×D3—21no—
157C22×D3C2×Q812yes—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
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×C4C2×D412yes—
177C2×C3×C4C2×Q812yes—
178C2×C3×C4#4112yes—
179C2×Dic3—21no—
180C2×Dic3—21no—
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×Dic3C2×D412yes—
196C2×Dic3C2×D412yes—
197C2×Dic3C2×Q812yes—
198C2×Dic3#4112yes—
199C2×Dic3#4112yes—
200C4×D3—28no—
201Dic6—44no—
202Dic6—44no—
203C24—31no—
204C2×Q8—62no—
205C2×Q8—62no—
206C2×Q8—62no—
207C2×Q8—62no—
208C2×Q8—121no—
209C2×Q8—121no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—34no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4C22×D311yescharacteristic
221C22×C4C22×D311yescharacteristic
222C22×C4C22×D311yescharacteristic
223C22×C4C22×D311yescharacteristic
224C22×C4C22×D311yescharacteristic
225C22×C4C22×D311yescharacteristic
226C22×C4C22×D311yescharacteristic
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—34no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—62no—
233C4⋊C4—62no—
234C4⋊C4—62no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C22⋊C4—62no—
238C22⋊C4—62no—
239C22⋊C4—62no—
240C22⋊C4—62no—
241C22⋊C4—121no—
242C22⋊C4—121no—
243C42—22no—
244C42—62no—
245C42—62no—
246C42—62no—
247C22×C3C22×D411yescharacteristic
248C22×C3C22×Q811yescharacteristic
249C22×C3C2×#4111yescharacteristic
250C22×C3C2×#4111yescharacteristic
251C22×C3C2×#4111yescharacteristic
252C22×C3H(5,𝔽2)11yescharacteristic
253C22×C32−1+411yescharacteristic
254C2×D3—22no—
255C2×D3—22no—
256C2×D3—22no—
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—41no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—41no—
268Dic3—22no—
269Dic3—22no—
270Dic3—41no—
271Dic3—41no—
272Dic3—41no—
273Dic3—41no—
274Dic3C22⋊Q814yes—
275C23—32no—
276C23—61no—
277C23—61no—
278C23—61no—
279C23—61no—
280C23—61no—
281C23—61no—
282C23C23×D311yescharacteristic
283Q8—124no—
284Q8—124no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
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—32no—
304C2×C4—32no—
305C2×C4—32no—
306C2×C4—32no—
307C2×C4—32no—
308C2×C4—68no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
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#23512yes—
326C2×C4D3×D412yes—
327C2×C4D3×Q812yes—
328C2×C3C2×C22⋊Q811yescharacteristic
329C2×C3#53311yescharacteristic
330C2×C3#54411yescharacteristic
331C2×C3#54611yescharacteristic
332C2×C3#54811yescharacteristic
333C2×C3#55511yescharacteristic
334C2×C3#55611yescharacteristic
335D3—42no—
336C22—62no—
337C22—62no—
338C22—62no—
339C22—121no—
340C22—121no—
341C22—121no—
342C22—121no—
343C22C2×#23511yescharacteristic
344C22C2×D3×D411yescharacteristic
345C22C2×D3×Q811yescharacteristic
346C22D3×#4111yescharacteristic
347C22D3×#4111yescharacteristic
348C22#100911yescharacteristic
349C22#101011yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—34no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—62no—
359C4—62no—
360C4—121no—
361C4—121no—
362C4—121no—
363C4—121no—
364C3#243911yescharacteristic, complemented, Sylow
365C2—122no—
366C2#553211yescharacteristic
367C2#555211yescharacteristic
368C2#557811yescharacteristic
369C2D3×C22⋊Q811yescharacteristic
370C2#565411yescharacteristic
371C2#568911yescharacteristic
372C2#570211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#387C2
prime radicalC23×D3C23
hypercenter#387D3
hypocenterC3#2439
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#888C2semidirect
3. C3×#387C2semidirect
6. #4976C2semidirect
7. #4929C2semidirect
12. #4655C2semidirect
13. #4650C2semidirect
14. #4647C2semidirect
15. #4646C2semidirect
65. #387D3semidirect
86. C2×C4×D3Q8semidirect
364. C3#2439semidirect

Maximal subgroups

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

subgroupindex
C2×#8882
C2×#8722
C3×#3872
#49942
#49762
#49292
#46732
#46672
#46662
#46552
#46502
#46472
#46462
#24393