Finite Group Databasebeta

← #74433

Subgroups — #74433

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×#924C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic, complemented
3C3×#385C211yesmaximal, characteristic
4C4×#217C211yesmaximal, characteristic
5#4980C211yesmaximal, characteristic, complemented
6#4979C211yesmaximal, characteristic, complemented
7#4976C211yesmaximal, characteristic, complemented
8#4948C211yesmaximal, characteristic
9#4947C211yesmaximal, characteristic
10#4945C211yesmaximal, characteristic
11#4945C211yesmaximal, characteristic
12#4656C211yesmaximal, characteristic, complemented
13#4653C211yesmaximal, characteristic, complemented
14#4652C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2465—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30#924—24no—
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
40C2×C4×Dic3C2211yescharacteristic
41#887—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×C4×Q8—31no—
56#396—31no—
57#395—31no—
58#395—31no—
59#393—31no—
60#391—31no—
61#389—31no—
62#388—31no—
63#388—31no—
64#387—31no—
65#387—31no—
66#387—31no—
67#385D311yescharacteristic
68C4×C22⋊C4—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×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C2×Dic6—22no—
88C2×Dic6—22no—
89C2×Dic6—22no—
90C2×Dic6—22no—
91C2×Dic6—22no—
92C2×Dic6—41no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C22⋊C4—22no—
96C3×C22⋊C4—41no—
97#217—22no—
98#217—22no—
99#217—22no—
100#217Q814yescomplemented
101C4⋊Dic3—22no—
102C4⋊Dic3—22no—
103Dic3⋊C4—22no—
104Dic3⋊C4—22no—
105Dic3⋊C4—22no—
106Dic3⋊C4—41no—
107Dic3⋊C4—41no—
108Dic3⋊C4D414yes—
109C4×Dic3—22no—
110C4×Dic3—22no—
111C4×Dic3—22no—
112C4×Dic3—41no—
113C22×Q8—31no—
114C22×Q8—31no—
115C23×C4C2×D311yescharacteristic
116C22⋊Q8—64no—
117C4×Q8—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⋊C4C2×D311yescharacteristic
133C2×C22⋊C4C2×D311yescharacteristic
134C2×C42—31no—
135C2×C42—31no—
136C2×C42—31no—
137C2×C42—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×C3C2×Q812yes—
157C23×C3C2411yescharacteristic
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—44no—
174C2×C3×C4C2×D412yes—
175C2×C3×C4#4112yes—
176C2×C3×C4#4112yes—
177C2×Dic3—21no—
178C2×Dic3—21no—
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×Dic3—21no—
196C2×Dic3C2×D412yes—
197C2×Dic3C2×D412yes—
198C2×Dic3C2×Q812yes—
199C2×Dic3C2×Q812yes—
200C2×Dic3#4112yes—
201Dic6—28no—
202Dic6—44no—
203C24C22×D311yescharacteristic
204C2×Q8—62no—
205C2×Q8—62no—
206C2×Q8—62no—
207C2×Q8—62no—
208C2×Q8—62no—
209C2×Q8—121no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—41no—
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—22no—
238C22⋊C4—34no—
239C22⋊C4—41no—
240C22⋊C4—62no—
241C22⋊C4—62no—
242C22⋊C4—62no—
243C42—62no—
244C42—62no—
245C42—62no—
246C42—121no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3C22×D411yescharacteristic
255C22×C3C22×Q811yescharacteristic
256C22×C3C2×#4111yescharacteristic
257C22×C3C2×#4111yescharacteristic
258C22×C3C2×#4111yescharacteristic
259C22×C3H(5,𝔽2)11yescharacteristic
260C22×C32−1+411yescharacteristic
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267Dic3—22no—
268Dic3—22no—
269Dic3—22no—
270Dic3—22no—
271Dic3—22no—
272Dic3—41no—
273Dic3—41no—
274Dic3—41no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23D3×Q812yes—
282C23C23×D311yescharacteristic
283Q8—68no—
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—32no—
301C2×C4—32no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—44no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
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×C4#23712yes—
328C2×C3—42no—
329C2×C3#52511yescharacteristic
330C2×C3#53311yescharacteristic
331C2×C3#54211yescharacteristic
332C2×C3#54611yescharacteristic
333C2×C3D4×Q811yescharacteristic
334C2×C3#55411yescharacteristic
335C2×C3#55611yescharacteristic
336C22—22no—
337C22—22no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22C2×#23511yescharacteristic
344C22C2×D3×D411yescharacteristic
345C22C2×#23711yescharacteristic
346C22#100411yescharacteristic
347C22C2×D3×Q811yescharacteristic
348C22D3×#4111yescharacteristic
349C22#101011yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—62no—
357C4—62no—
358C4—62no—
359C4—62no—
360C4—62no—
361C4—121no—
362C4—121no—
363C4—121no—
364C3#246511yescharacteristic, complemented, Sylow
365C2—42no—
366C2#551111yescharacteristic
367C2#554211yescharacteristic
368C2#554711yescharacteristic
369C2#559111yescharacteristic
370C2#562211yescharacteristic
371C2#563211yescharacteristic
372C2#566211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC24×C3C23
hypercenter#385D3
hypocenterC3#2465
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#887C2semidirect
5. #4980C2semidirect
6. #4979C2semidirect
7. #4976C2semidirect
12. #4656C2semidirect
13. #4653C2semidirect
14. #4652C2semidirect
15. #4646C2semidirect
100. #217Q8semidirect
364. C3#2465semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C2×#8872
C3×#3852
C4×#2172
#49802
#49792
#49762
#49482
#49472
#49452
#46562
#46532
#46522
#46462
#24653