Finite Group Databasebeta

← #78023

Subgroups — #78023

Order 384.

548 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#889C211yesmaximal, characteristic, complemented
2C2×#67⋊C4C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic, complemented
4#4994C211yesmaximal, characteristic, complemented
5#4993C211yesmaximal, characteristic, complemented
6#4987C211yesmaximal, characteristic, complemented
7#4976C211yesmaximal, characteristic, complemented
8Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
9#4957C211yesmaximal, characteristic
10#4957C211yesmaximal, characteristic
11#4950C211yesmaximal, characteristic, complemented
12#4948C211yesmaximal, characteristic, complemented
13#4942C211yesmaximal, characteristic, complemented
14#4668C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2454—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46#889—24no—
47#67⋊C4—24no—
48C3×#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#125—31no—
55C2×C4×D4—31no—
56#399—31no—
57#399D311yescharacteristic, complemented
58#396—31no—
59#396—31no—
60#390—31no—
61#390—31no—
62#389—31no—
63#389—31no—
64#385—31no—
65#385—31no—
66#380—31no—
67#380—31no—
68C4×C4⋊C4—31no—
69C24×C3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C2×#67—22no—
78C2×#67—22no—
79C2×#67Q814yes—
80C22×Dic3—22no—
81C22×Dic3—22no—
82C22×Dic3—41no—
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C2×C4×D3—22no—
91C2×C4×D3—22no—
92C2×C4×D3—41no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—41no—
96C3×C22⋊C4—22no—
97C3×C22⋊C4—22no—
98C3×C22⋊C4—41no—
99#217—41no—
100#217—41no—
101(C2×D3)⋊C4—22no—
102(C2×D3)⋊C4—22no—
103(C2×D3)⋊C4—41no—
104(C2×D3)⋊C4—41no—
105(C2×D3)⋊C4—41no—
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—
115Dic3⋊C4D414yes—
116C4×Dic3—22no—
117C4×Dic3—22no—
118C4×Dic3—22no—
119C22×D4—31no—
120C23×C4—31no—
121C23×C4—31no—
122#125—64no—
123C4×D4—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×C4⋊C4C2×D311yescharacteristic
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4C2×D311yescharacteristic
145C2×C22⋊C4C2×D311yescharacteristic
146C2×C22⋊C4C2×D311yescharacteristic
147C2×C42—31no—
148C2×C42—31no—
149C2×C42—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95C2×D311yescharacteristic
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3C2×Q812yes—
163C23×C3C2411yescharacteristic
164C22×D3—21no—
165C22×D3—21no—
166C22×D3—21no—
167C22×D3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3C2×Q812yes—
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—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4C2×D412yes—
188C2×C3×C4#4112yes—
189#67—28no—
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—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—44no—
207C2×Dic3C2×D412yes—
208C2×Dic3C2×D412yes—
209C2×Dic3C2×Q812yes—
210C2×Dic3#4112yes—
211C2×Dic3#4112yes—
212C4×D3—44no—
213C24—31no—
214C24C22×D311yescharacteristic
215C2×D4—34no—
216C2×D4—62no—
217C2×D4—62no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—121no—
230C22×C4—121no—
231C22×C4C22×D311yescharacteristic
232C22×C4C22×D311yescharacteristic
233C22×C4C22×D311yescharacteristic
234C22×C4C22×D311yescharacteristic
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—34no—
240C4⋊C4—41no—
241C4⋊C4—62no—
242C4⋊C4—62no—
243C4⋊C4—62no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—41no—
253C22⋊C4—62no—
254C22⋊C4—62no—
255C22⋊C4—121no—
256C22⋊C4—121no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C42—62no—
261C42—62no—
262C42—62no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3C22×D411yescharacteristic
271C22×C3C22×Q811yescharacteristic
272C22×C3C2×#4111yescharacteristic
273C22×C3C2×#4111yescharacteristic
274C22×C3C2×#4111yescharacteristic
275C22×C3H(5,𝔽2)11yescharacteristic
276C22×C3H(5,𝔽2)11yescharacteristic
277C2×D3—22no—
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290Dic3—22no—
291Dic3—22no—
292Dic3—41no—
293Dic3—41no—
294Dic3—41no—
295Dic3—41no—
296Dic3C22⋊Q814yes—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—32no—
304C23—61no—
305C23—61no—
306C23—61no—
307C23—61no—
308C23—61no—
309C23—61no—
310C23D3×Q812yes—
311C23C23×D311yescharacteristic
312D4—68no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—124no—
351C2×C4—124no—
352C2×C4D3×D412yes—
353C2×C4#23712yes—
354C2×C3—42no—
355C2×C3C2×C22⋊Q811yescharacteristic
356C2×C3#52911yescharacteristic
357C2×C3#54011yescharacteristic
358C2×C3#54611yescharacteristic
359C2×C3#55011yescharacteristic
360C2×C3#55411yescharacteristic
361C2×C3#55411yescharacteristic
362D3—42no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—62no—
371C22—62no—
372C22—62no—
373C22—121no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22C2×D3×D411yescharacteristic
378C22C2×#23711yescharacteristic
379C22#100411yescharacteristic
380C22C2×D3×Q811yescharacteristic
381C22D3×#4111yescharacteristic
382C22D3×#4111yescharacteristic
383C22#100911yescharacteristic
384C4—22no—
385C4—22no—
386C4—34no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—62no—
392C4—62no—
393C4—121no—
394C4—121no—
395C4—121no—
396C4—121no—
397C3#245411yescharacteristic, complemented, Sylow
398C2—42no—
399C2—122no—
400C2D3×C22⋊Q811yescharacteristic
401C2#563511yescharacteristic
402C2#563611yescharacteristic
403C2#564911yescharacteristic
404C2#565411yescharacteristic
405C2#566311yescharacteristic
406C2#571811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC22×#67C22
hypercenter#399D3
hypocenterC3#2454
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#889C2semidirect
3. C3×#399C2semidirect
4. #4994C2semidirect
5. #4993C2semidirect
6. #4987C2semidirect
7. #4976C2semidirect
8. Dic3×C4⋊C4C2semidirect
11. #4950C2semidirect
12. #4948C2semidirect
13. #4942C2semidirect
14. #4668C2semidirect
15. #4647C2semidirect
57. #399D3semidirect
397. C3#2454semidirect

Maximal subgroups

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

subgroupindex
C2×#8892
C2×#67⋊C42
C3×#3992
#49942
#49932
#49872
#49762
Dic3×C4⋊C42
#49572
#49502
#49482
#49422
#46682
#46472
#24543