Finite Group Databasebeta

← #74387

Subgroups — #74387

Order 384.

528 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×#878C211yesmaximal, characteristic
2C3×#385C211yesmaximal, characteristic
3#5210C211yesmaximal, characteristic
4C2×C3⋊#95C211yesmaximal, characteristic
5#4978C211yesmaximal, characteristic, complemented
6#4976C211yesmaximal, characteristic, complemented
7#4973C211yesmaximal, characteristic, complemented
8#4972C211yesmaximal, characteristic, complemented
9#4942C211yesmaximal, characteristic
10Dic3×C22⋊C4C211yesmaximal, characteristic
11Dic3×C22⋊C4C211yesmaximal, characteristic
12#4653C211yesmaximal, characteristic, complemented
13#4652C211yesmaximal, characteristic, complemented
14#4652C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2329—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36C2×C4×Dic3C2211yescharacteristic
37C2×C4×Dic3C2211yescharacteristic
38#878C418yescomplemented
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×C22⋊Q8—31no—
54#391—31no—
55#391—31no—
56#389—31no—
57#389—31no—
58#387—31no—
59#387—31no—
60#387—31no—
61#387—31no—
62#385—31no—
63#385D311yescharacteristic
64#380—31no—
65C4×C22⋊C4—31no—
66C4×C22⋊C4—31no—
67C2×#95—31no—
68C24×C3C2311yescharacteristic
69C22×C3×C4—22no—
70C22×C3×C4—41no—
71C22×C3×C4—41no—
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×Dic3—41no—
79C22×Dic3—41no—
80C22×Dic3—41no—
81C22×Dic3—41no—
82C22×Dic3C2×C414yes—
83C22×Dic3Q814yescomplemented
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C2×Dic6—41no—
93C2×Dic6—41no—
94C2×Dic6C2×C414yescomplemented
95C3×C4⋊C4—41no—
96C3×C4⋊C4—41no—
97C3×C22⋊C4—22no—
98C3×C22⋊C4C2×C414yes—
99#217—22no—
100#217C2×C414yes—
101C4⋊Dic3C2×C414yescomplemented
102Dic3⋊C4—41no—
103Dic3⋊C4—41no—
104Dic3⋊C4—41no—
105Dic3⋊C4—41no—
106Dic3⋊C4C2×C414yescomplemented
107Dic3⋊C4C2×C414yescomplemented
108C4×Dic3—41no—
109C4×Dic3—41no—
110C4×Dic3—41no—
111C4×Dic3—41no—
112C22×Q8—31no—
113C23×C4—31no—
114C23×C4—31no—
115C23×C4C2×D311yescharacteristic
116C22⋊Q8—38no—
117C2×C4⋊C4—31no—
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⋊C4C2×D311yescharacteristic
125C2×C4⋊C4C2×D311yescharacteristic
126C2×C22⋊C4—31no—
127C2×C22⋊C4—31no—
128C2×C22⋊C4C2×D311yescharacteristic
129C2×C22⋊C4C2×D311yescharacteristic
130C2×C42—31no—
131C2×C42—31no—
132C2×C42—31no—
133C2×C42—31no—
134#95—31no—
135#95—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#95C2×D311yescharacteristic
147#95C2×D311yescharacteristic
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3C22×C412yes—
153C23×C3C2×Q812yes—
154C23×C3#4112yes—
155C23×C3C2411yescharacteristic
156C2×C3×C4—21no—
157C2×C3×C4—21no—
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—44no—
173C2×C3×C4C22×C412yes—
174C2×C3×C4C22×C412yes—
175C2×Dic3—21no—
176C2×Dic3—21no—
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—44no—
194C2×Dic3—44no—
195C2×Dic3C22×C412yes—
196C2×Dic3C22×C412yes—
197C2×Dic3C22×C412yes—
198C2×Dic3C22×C412yes—
199C2×Dic3C2×Q812yes—
200C2×Dic3C2×Q812yes—
201Dic6—44no—
202C24C22×D311yescharacteristic
203C2×Q8—34no—
204C2×Q8—121no—
205C2×Q8—121no—
206C22×C4—22no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—34no—
214C22×C4—34no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—121no—
220C22×C4—121no—
221C22×C4—121no—
222C22×C4—121no—
223C22×C4C22×D311yescharacteristic
224C22×C4C22×D311yescharacteristic
225C22×C4C22×D311yescharacteristic
226C22×C4C22×D311yescharacteristic
227C22×C4C22×D311yescharacteristic
228C22×C4C22×D311yescharacteristic
229C4⋊C4—34no—
230C4⋊C4—34no—
231C4⋊C4—34no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C22⋊C4—22no—
239C22⋊C4—34no—
240C22⋊C4—62no—
241C22⋊C4C4×D314yes—
242C42—121no—
243C42—121no—
244C42—121no—
245C42—121no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3C4×Q814yes—
253C22×C3C23×C411yescharacteristic
254C22×C3C22×Q811yescharacteristic
255C22×C3C2×#4111yescharacteristic
256C22×C3H(5,𝔽2)11yescharacteristic
257C22×C3H(5,𝔽2)11yescharacteristic
258C22×C3H(5,𝔽2)11yescharacteristic
259C22×C32−1+411yescharacteristic
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266Dic3—22no—
267Dic3—22no—
268Dic3—22no—
269Dic3—22no—
270Dic3—22no—
271Dic3—22no—
272Dic3—41no—
273Dic3—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D312yes—
279C23#23712yes—
280C23D3×Q812yes—
281C23C23×D311yescharacteristic
282Q8—124no—
283C2×C4—21no—
284C2×C4—21no—
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—32no—
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—124no—
325C2×C4—124no—
326C2×C4C2×C4×D312yes—
327C2×C4C2×C4×D312yes—
328C2×C3—24no—
329C2×C3C2×C4×Q811yescharacteristic
330C2×C3#51811yescharacteristic
331C2×C3#52011yescharacteristic
332C2×C3#53811yescharacteristic
333C2×C3#53911yescharacteristic
334C2×C3#54311yescharacteristic
335C2×C3#54411yescharacteristic
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22#88714yes—
343C22C22×C4×D311yescharacteristic
344C22C2×#23711yescharacteristic
345C22#100411yescharacteristic
346C22#100411yescharacteristic
347C22C2×D3×Q811yescharacteristic
348C22#100911yescharacteristic
349C22#101011yescharacteristic
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—62no—
357C4—62no—
358C4—62no—
359C4—62no—
360C4—62no—
361C4—62no—
362C4—121no—
363C4—121no—
364C3#232911yescharacteristic, complemented, Sylow
365C2—24no—
366C2C2×#88711yescharacteristic
367C2#554511yescharacteristic
368C2#554611yescharacteristic
369C2#559311yescharacteristic
370C2#562311yescharacteristic
371C2#563411yescharacteristic
372C2#566511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#385C2
socle, cosocleC23×C3C24
prime radicalC24×C3C23
hypercenter#385D3
hypocenterC3#2329
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
5. #4978C2semidirect
6. #4976C2semidirect
7. #4973C2semidirect
8. #4972C2semidirect
12. #4653C2semidirect
13. #4652C2semidirect
14. #4652C2semidirect
15. #4646C2semidirect
38. #878C4semidirect
83. C22×Dic3Q8semidirect
94. C2×Dic6C2×C4semidirect
101. C4⋊Dic3C2×C4semidirect
106. Dic3⋊C4C2×C4semidirect
107. Dic3⋊C4C2×C4semidirect
364. C3#2329semidirect

Maximal subgroups

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

subgroupindex
C2×#8782
C3×#3852
#52102
C2×C3⋊#952
#49782
#49762
#49732
#49722
#49422
Dic3×C22⋊C42
#46532
#46522
#46462
#23293