Finite Group Databasebeta

← #77742

Subgroups — #77742

Order 384.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#896C211yesmaximal, characteristic, complemented
2C2×#896C211yesmaximal, characteristic, complemented
3C2×#893C211yesmaximal, characteristic
4C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×#398C211yesmaximal, characteristic, complemented
6#4994C211yesmaximal, characteristic, complemented
7#4994C211yesmaximal, characteristic, complemented
8#4987C211yesmaximal, characteristic, complemented
9#4982C211yesmaximal, characteristic, complemented
10#4667C211yesmaximal, characteristic, complemented
11#4667C211yesmaximal, characteristic, complemented
12#4665C211yesmaximal, characteristic
13#4665C211yesmaximal, characteristic
14#4647C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2417—31nomaximal, Sylow
17C22×D12C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C4⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2211yescharacteristic
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×(C2×D3)⋊C4C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44#896—24no—
45#896—24no—
46#893—24no—
47D3×C4⋊C4—24no—
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3×#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×C22⋊Q8—31no—
57C2×C22⋊Q8—31no—
58C2×C4×D4—31no—
59C22×C4⋊C4—31no—
60#398—31no—
61#398D311yescharacteristic, complemented
62#396—31no—
63#396—31no—
64#391—31no—
65#389—31no—
66#386—31no—
67#386—31no—
68#380—31no—
69#380—31no—
70#380—31no—
71C23×D3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C2×D12—22no—
87C2×D12—22no—
88C2×D12Q814yescomplemented
89C2×C4×D3—22no—
90C2×C4×D3—22no—
91C2×C4×D3—22no—
92C2×C4×D3—22no—
93C2×C4×D3—22no—
94C2×C4×D3—22no—
95C2×C4×D3—22no—
96C2×C4×D3—22no—
97C2×C4×D3—22no—
98C2×C4×D3—41no—
99C2×C4×D3—41no—
100C2×C4×D3D414yes—
101C2×Dic6—22no—
102C2×Dic6—22no—
103C2×Dic6—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108(C2×D3)⋊C4—22no—
109(C2×D3)⋊C4—22no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—22no—
113(C2×D3)⋊C4—22no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116C4⋊Dic3—22no—
117C4⋊Dic3—41no—
118Dic3⋊C4—22no—
119Dic3⋊C4—22no—
120Dic3⋊C4—22no—
121Dic3⋊C4—22no—
122Dic3⋊C4—41no—
123Dic3⋊C4D414yescomplemented
124C4×Dic3—22no—
125C22×Q8—31no—
126C22×D4—31no—
127C23×C4—31no—
128C23×C4—31no—
129C23×C4—31no—
130C23×C4—31no—
131C22⋊Q8—64no—
132C22⋊Q8—64no—
133C4×D4—64no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—64no—
143C2×C4⋊C4C2×D311yescharacteristic
144C2×C4⋊C4C2×D311yescharacteristic
145C2×C4⋊C4C2×D311yescharacteristic
146C2×C4⋊C4C2×D311yescharacteristic
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—31no—
155C2×C42—31no—
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95C2×D311yescharacteristic
162#95C2×D311yescharacteristic
163#95C2×D311yescharacteristic
164C23×C3C2411yescharacteristic
165C22×D3—21no—
166C22×D3—21no—
167C22×D3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3C2×D412yes—
176C22×D3C2×Q812yes—
177C22×D3C2×Q812yes—
178C22×D3#4112yes—
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×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4C2×D412yes—
198C2×C3×C4C2×D412yes—
199C2×C3×C4C2×Q812yes—
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×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3C2×D412yes—
215C2×Dic3C2×D412yes—
216C2×Dic3C2×D412yes—
217C2×Dic3#4112yes—
218D12—28no—
219C4×D3—28no—
220C4×D3—44no—
221C4×D3—44no—
222C4×D3—44no—
223Dic6—44no—
224C24—31no—
225C24—31no—
226C2×Q8—62no—
227C2×Q8—62no—
228C2×Q8—62no—
229C2×D4—34no—
230C2×D4—62no—
231C2×D4—62no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—34no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—121no—
249C22×C4—121no—
250C22×C4C22×D311yescharacteristic
251C22×C4C22×D311yescharacteristic
252C22×C4C22×D311yescharacteristic
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D311yescharacteristic
255C22×C4C22×D311yescharacteristic
256C22×C4C22×D311yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—34no—
262C4⋊C4—62no—
263C4⋊C4—62no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—62no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C22⋊C4—62no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C22⋊C4—62no—
274C22⋊C4—62no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C42—62no—
278C22×C3C22×D411yescharacteristic
279C22×C3C22×D411yescharacteristic
280C22×C3C22×Q811yescharacteristic
281C22×C3C2×#4111yescharacteristic
282C22×C3C2×#4111yescharacteristic
283C22×C3H(5,𝔽2)11yescharacteristic
284C22×C32−1+411yescharacteristic
285C2×D3—22no—
286C2×D3—22no—
287C2×D3—22no—
288C2×D3—22no—
289C2×D3—22no—
290C2×D3—22no—
291C2×D3—22no—
292C2×D3—22no—
293C2×D3—22no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3C22⋊Q814yescomplemented
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306Dic3—22no—
307Dic3—41no—
308Dic3—41no—
309Dic3—41no—
310Dic3—41no—
311Dic3#12114yes—
312C23—32no—
313C23—32no—
314C23—32no—
315C23—32no—
316C23—61no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23—61no—
322C23—61no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23C23×D311yescharacteristic
327Q8—124no—
328D4—68no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—32no—
351C2×C4—68no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—124no—
367C2×C4—124no—
368C2×C4—124no—
369C2×C4D3×D412yes—
370C2×C4D3×D412yes—
371C2×C4D3×Q812yes—
372C2×C3C2×#12111yescharacteristic
373C2×C3C2×C22⋊Q811yescharacteristic
374C2×C3#53711yescharacteristic
375C2×C3#54611yescharacteristic
376C2×C3D4×Q811yescharacteristic
377C2×C3#55211yescharacteristic
378C2×C3#55411yescharacteristic
379D3—24no—
380D3—42no—
381C22—34no—
382C22—62no—
383C22—62no—
384C22—62no—
385C22—62no—
386C22—62no—
387C22—62no—
388C22—62no—
389C22—62no—
390C22—62no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22C2×D3×D411yescharacteristic
396C22C2×D3×D411yescharacteristic
397C22C2×D3×Q811yescharacteristic
398C22#100711yescharacteristic
399C22D3×#4111yescharacteristic
400C22D3×#4111yescharacteristic
401C22#100911yescharacteristic
402C4—22no—
403C4—22no—
404C4—22no—
405C4—34no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—62no—
411C4—121no—
412C4—121no—
413C4—121no—
414C4—121no—
415C3#241711yescharacteristic, complemented, Sylow
416C2—64no—
417C2—122no—
418C2D3×#12111yescharacteristic
419C2D3×C22⋊Q811yescharacteristic
420C2#562911yescharacteristic
421C2#565411yescharacteristic
422C2#569011yescharacteristic
423C2#569311yescharacteristic
424C2#572711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC22×D12C22
hypercenter#398D3
hypocenterC3#2417
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
2. C2×#896C2semidirect
4. C2×D3×C4⋊C4C2semidirect
5. C3×#398C2semidirect
6. #4994C2semidirect
7. #4994C2semidirect
8. #4987C2semidirect
9. #4982C2semidirect
10. #4667C2semidirect
11. #4667C2semidirect
14. #4647C2semidirect
15. #4646C2semidirect
61. #398D3semidirect
88. C2×D12Q8semidirect
123. Dic3⋊C4D4semidirect
298. C2×D3C22⋊Q8semidirect
415. C3#2417semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8932
C2×D3×C4⋊C42
C3×#3982
#49942
#49872
#49822
#46672
#46652
#46472
#46462
#24173