Finite Group Databasebeta

← #75593

Subgroups — #75593

Order 384.

576 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#888C211yesmaximal, characteristic, complemented
2C2×#885C211yesmaximal, characteristic
3C2×#885C211yesmaximal, characteristic
4C2×C4×Dic6C211yesmaximal, characteristic, complemented
5C3×#388C211yesmaximal, characteristic, complemented
6#4994C211yesmaximal, characteristic, complemented
7#4957C211yesmaximal, characteristic
8#4948C211yesmaximal, characteristic, complemented
9#4948C211yesmaximal, characteristic, complemented
10#4944C211yesmaximal, characteristic, complemented
11#4940C211yesmaximal, characteristic, complemented
12#4676C211yesmaximal, characteristic, complemented
13#4670C211yesmaximal, characteristic, complemented
14#4649C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2545—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C42C2211yescharacteristic
25C2×#217C2211yescharacteristic
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×(C2×D3)⋊C4C2211yescharacteristic
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×C4×Dic3C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46#888—24no—
47#885—24no—
48#885—24no—
49C4×Dic6—24no—
50C3×#95C2211yescharacteristic
51C3×#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×C4⋊Q8—31no—
57C2×#124—31no—
58C2×#124—31no—
59C2×C4×Q8—31no—
60#396—31no—
61#396—31no—
62#396—31no—
63#396—31no—
64#391—31no—
65#390—31no—
66#388—31no—
67#388—31no—
68#388—31no—
69#388D311yescharacteristic, complemented
70C4×C4⋊C4—31no—
71C24×C3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—22no—
80C2×#67—22no—
81C2×#67—22no—
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C2×Dic6—22no—
90C2×Dic6—22no—
91C2×Dic6—22no—
92C2×Dic6—22no—
93C2×Dic6—22no—
94C2×Dic6D414yes—
95C3×C4⋊C4—22no—
96C3×C22⋊C4—22no—
97C3×C22⋊C4—22no—
98C3×C22⋊C4—22no—
99C3×C42—22no—
100#217—22no—
101#217—22no—
102#217—22no—
103(C2×D3)⋊C4—22no—
104(C2×D3)⋊C4—22no—
105(C2×D3)⋊C4—22no—
106(C2×D3)⋊C4—22no—
107(C2×D3)⋊C4—22no—
108(C2×D3)⋊C4—22no—
109C4⋊Dic3—22no—
110C4⋊Dic3—22no—
111Dic3⋊C4—22no—
112Dic3⋊C4—22no—
113Dic3⋊C4—22no—
114Dic3⋊C4—22no—
115Dic3⋊C4—22no—
116Dic3⋊C4—22no—
117C4×Dic3—22no—
118C4×Dic3—22no—
119C4×Dic3—22no—
120C4×Dic3D414yes—
121C22×Q8—31no—
122C22×Q8—31no—
123C22×D4—31no—
124C4⋊Q8—64no—
125#124—64no—
126#124—64no—
127C4×Q8—64no—
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⋊C4—31no—
135C2×C4⋊C4—31no—
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⋊C4—31no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4C2×D311yescharacteristic
147C2×C22⋊C4C2×D311yescharacteristic
148C2×C22⋊C4C2×D311yescharacteristic
149C2×C42—31no—
150C2×C42—31no—
151C2×C42—31no—
152C2×C42—31no—
153C2×C42C2×D311yescharacteristic
154#95—31no—
155#95—31no—
156#95—31no—
157#95—31no—
158#95C2×D311yescharacteristic
159#95C2×D311yescharacteristic
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3C2411yescharacteristic
168C22×D3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—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×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4C2×D412yes—
191C2×C3×C4C2×D412yes—
192C2×C3×C4#4112yes—
193#67—44no—
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—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3C2×D412yes—
210C2×Dic3C2×D412yes—
211C2×Dic3C2×D412yes—
212C2×Dic3C2×D412yes—
213C2×Dic3#4112yes—
214C2×Dic3#4112yes—
215Dic6—28no—
216Dic6—44no—
217C24—31no—
218C24C22×D311yescharacteristic
219C2×Q8—34no—
220C2×Q8—62no—
221C2×Q8—62no—
222C2×Q8—62no—
223C2×Q8—62no—
224C2×Q8—62no—
225C2×D4—62no—
226C2×D4—62no—
227C2×D4—62no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237C22×C4C22×D311yescharacteristic
238C22×C4C22×D311yescharacteristic
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241C4⋊C4—22no—
242C4⋊C4—62no—
243C4⋊C4—62no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—62no—
254C22⋊C4—62no—
255C22⋊C4—62no—
256C22⋊C4—62no—
257C22⋊C4—62no—
258C22⋊C4—62no—
259C22⋊C4—62no—
260C22⋊C4—62no—
261C22⋊C4—62no—
262C42—22no—
263C42—34no—
264C42—62no—
265C42—62no—
266C42—62no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3C22×D411yescharacteristic
275C22×C3C22×D411yescharacteristic
276C22×C3C2×#4111yescharacteristic
277C22×C3C2×#4111yescharacteristic
278C22×C3C2×#4111yescharacteristic
279C22×C32−1+411yescharacteristic
280C22×C32−1+411yescharacteristic
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C3×C4—22no—
289C3×C4—22no—
290C3×C4—22no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294Dic3—22no—
295Dic3—22no—
296Dic3—22no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300Dic3#12114yes—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—61no—
309C23—61no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23—61no—
314C23—61no—
315C23C23×D311yescharacteristic
316Q8—68no—
317Q8—124no—
318D4—124no—
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—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—32no—
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—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4#23512yes—
356C2×C4D3×D412yes—
357C2×C4D3×D412yes—
358C2×C3—81no—
359C2×C3C2×#12111yescharacteristic
360C2×C3#53211yescharacteristic
361C2×C3#53611yescharacteristic
362C2×C3#54811yescharacteristic
363C2×C3#54811yescharacteristic
364C2×C3#55611yescharacteristic
365C2×C3#55611yescharacteristic
366D3—81no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22C2×#23511yescharacteristic
382C22C2×D3×D411yescharacteristic
383C22C2×D3×D411yescharacteristic
384C22D3×#4111yescharacteristic
385C22D3×#4111yescharacteristic
386C22#101011yescharacteristic
387C22#101011yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—34no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—62no—
396C4—62no—
397C4—62no—
398C4—121no—
399C4—121no—
400C4—121no—
401C3#254511yescharacteristic, complemented, Sylow
402C2—81no—
403C2—241no—
404C2#553311yescharacteristic
405C2#554811yescharacteristic
406C2#555211yescharacteristic
407C2D3×#12111yescharacteristic
408C2#564811yescharacteristic
409C2#567711yescharacteristic
410C2#570211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#388C2
prime radicalC22×#67C22
hypercenter#388D3
hypocenterC3#2545
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#888C2semidirect
4. C2×C4×Dic6C2semidirect
5. C3×#388C2semidirect
6. #4994C2semidirect
8. #4948C2semidirect
9. #4948C2semidirect
10. #4944C2semidirect
11. #4940C2semidirect
12. #4676C2semidirect
13. #4670C2semidirect
14. #4649C2semidirect
15. #4646C2semidirect
69. #388D3semidirect
401. C3#2545semidirect

Maximal subgroups

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

subgroupindex
C2×#8882
C2×#8852
C2×C4×Dic62
C3×#3882
#49942
#49572
#49482
#49442
#49402
#46762
#46702
#46492
#46462
#25453