Finite Group Databasebeta

← #74653

Subgroups — #74653

Order 384.

414 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
1C3×#385C211yesmaximal, characteristic, complemented
2#5213C211yesmaximal, characteristic
3#5210C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic, complemented
6#4982C211yesmaximal, characteristic, complemented
7#4981C211yesmaximal, characteristic, complemented
8#4961C211yesmaximal, characteristic
9#4959C211yesmaximal, characteristic
10#4952C211yesmaximal, characteristic, complemented
11#4948C211yesmaximal, characteristic, complemented
12#4676C211yesmaximal, characteristic, complemented
13#4674C211yesmaximal, characteristic, complemented
14#4660C211yesmaximal, characteristic, complemented
15#4658C211yesmaximal, characteristic, complemented
16#2827—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C3×#95C2211yescharacteristic
41C3×#95C2211yescharacteristic
42C3⋊#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52#400—31no—
53#400—31no—
54#399—31no—
55#398—31no—
56#398—31no—
57#397—31no—
58#397—31no—
59#397—31no—
60#396—31no—
61#396—31no—
62#394—31no—
63#394—31no—
64#386—31no—
65#385—31no—
66#385D311yescharacteristic, complemented
67C24×C3C2311yescharacteristic
68C23×D3C2311yescharacteristic
69C22×C3×C4—41no—
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
78C2×#67—22no—
79C2×#67—41no—
80C2×#67—41no—
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C3×C4⋊C4—41no—
89C3×C4⋊C4—41no—
90C3×C22⋊C4—22no—
91C3×C22⋊C4—41no—
92#217—22no—
93#217—41no—
94#217—41no—
95(C2×D3)⋊C4—41no—
96(C2×D3)⋊C4—41no—
97(C2×D3)⋊C4—41no—
98(C2×D3)⋊C4—41no—
99(C2×D3)⋊C4—41no—
100(C2×D3)⋊C4—41no—
101C4⋊Dic3—41no—
102C4⋊Dic3—41no—
103Dic3⋊C4—41no—
104Dic3⋊C4—41no—
105Dic3⋊C4—41no—
106Dic3⋊C4—41no—
107Dic3⋊C4—41no—
108Dic3⋊C4—41no—
109C22×D4—31no—
110C23×C4C2×D311yescharacteristic
111C2×C4⋊C4—31no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4C2×D311yescharacteristic
120C2×C4⋊C4C2×D311yescharacteristic
121C2×C22⋊C4—31no—
122C2×C22⋊C4—31no—
123C2×C22⋊C4—31no—
124C2×C22⋊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
132#95—31no—
133#95—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#95C2×D311yescharacteristic
143#95C2×D311yescharacteristic
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3#4112yes—
151C23×C3C2411yescharacteristic
152C22×D3—21no—
153C22×D3—21no—
154C22×D3—21no—
155C22×D3—21no—
156C22×D3—21no—
157C22×D3—21no—
158C22×D3—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×C4—21no—
177C2×C3×C4—44no—
178#67—44no—
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×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C24—31no—
201C24C22×D311yescharacteristic
202C2×D4—62no—
203C2×D4—121no—
204C2×D4—121no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4C22×D311yescharacteristic
216C22×C4C22×D311yescharacteristic
217C22×C4C22×D311yescharacteristic
218C22×C4C22×D311yescharacteristic
219C22×C4C22×D311yescharacteristic
220C22×C4C22×D311yescharacteristic
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—121no—
224C4⋊C4—121no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C4⋊C4—121no—
230C4⋊C4—121no—
231C22⋊C4—22no—
232C22⋊C4—41no—
233C22⋊C4—62no—
234C22⋊C4—121no—
235C22⋊C4—121no—
236C22⋊C4—121no—
237C22⋊C4—121no—
238C22⋊C4—121no—
239C22⋊C4—121no—
240C22⋊C4—121no—
241C22⋊C4—121no—
242C22×C3—22no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3C2×#4111yescharacteristic
250C22×C3H(5,𝔽2)11yescharacteristic
251C22×C3H(5,𝔽2)11yescharacteristic
252C22×C3H(5,𝔽2)11yescharacteristic
253C22×C3H(5,𝔽2)11yescharacteristic
254C22×C32−1+411yescharacteristic
255C22×C32−1+411yescharacteristic
256C2×D3—41no—
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—41no—
261C2×D3—41no—
262C2×D3—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269Dic3—41no—
270Dic3—41no—
271Dic3—41no—
272Dic3—41no—
273Dic3—41no—
274Dic3—41no—
275Dic3—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—61no—
283C23—61no—
284C23—61no—
285C23—61no—
286C23—61no—
287C23—61no—
288C23—61no—
289C23#23512yes—
290C23C23×D311yescharacteristic
291D4—124no—
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—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—44no—
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—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C3—42no—
333C2×C3#53811yescharacteristic
334C2×C3#53911yescharacteristic
335C2×C3#53911yescharacteristic
336C2×C3#56011yescharacteristic
337C2×C3#56211yescharacteristic
338C2×C3#56211yescharacteristic
339C2×C3#56311yescharacteristic
340D3—81no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22C2×#23511yescharacteristic
356C22#100411yescharacteristic
357C22#100411yescharacteristic
358C22#100411yescharacteristic
359C22#100711yescharacteristic
360C22#100911yescharacteristic
361C22#101011yescharacteristic
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—121no—
369C4—121no—
370C4—121no—
371C4—121no—
372C4—121no—
373C4—121no—
374C4—121no—
375C3#282711yescharacteristic, complemented, Sylow
376C2—42no—
377C2—241no—
378C2#551511yescharacteristic
379C2#556011yescharacteristic
380C2#556411yescharacteristic
381C2#559511yescharacteristic
382C2#564611yescharacteristic
383C2#564711yescharacteristic
384C2#566711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC22×#67C22
hypercenter#385D3
hypocenterC3#2827
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C3×#385C2semidirect
3. #5210C2semidirect
4. #4993C2semidirect
5. #4992C2semidirect
6. #4982C2semidirect
7. #4981C2semidirect
10. #4952C2semidirect
11. #4948C2semidirect
12. #4676C2semidirect
13. #4674C2semidirect
14. #4660C2semidirect
15. #4658C2semidirect
66. #385D3semidirect
375. C3#2827semidirect

Maximal subgroups

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

subgroupindex
C3×#3852
#52132
#52102
#49932
#49922
#49822
#49812
#49612
#49592
#49522
#49482
#46762
#46742
#46602
#46582
#28273