Finite Group Databasebeta

← #75668

Subgroups — #75668

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#897C211yesmaximal, characteristic, complemented
2C2×#886C211yesmaximal, characteristic
3C2×#883C211yesmaximal, characteristic
4C2×#878C211yesmaximal, characteristic, complemented
5C3×#388C211yesmaximal, characteristic, complemented
6#4981C211yesmaximal, characteristic, complemented
7#4961C211yesmaximal, characteristic
8#4953C211yesmaximal, characteristic, complemented
9#4950C211yesmaximal, characteristic, complemented
10#4940C211yesmaximal, characteristic, complemented
11#4935C211yesmaximal, characteristic, complemented
12#4674C211yesmaximal, characteristic, complemented
13#4668C211yesmaximal, characteristic, complemented
14#4661C211yesmaximal, characteristic, complemented
15#4656C211yesmaximal, characteristic, complemented
16#2680—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C42C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊Dic3C2211yescharacteristic
38C2×C4⋊Dic3C2211yescharacteristic
39C2×C4⋊Dic3C2211yescharacteristic
40C2×C4⋊Dic3C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44#897—24no—
45#886—24no—
46#883—24no—
47#878—24no—
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#123—31no—
57C2×#123—31no—
58C2×C22⋊Q8—31no—
59C2×C22⋊Q8—31no—
60#398—31no—
61#398—31no—
62#397—31no—
63#397—31no—
64#395—31no—
65#394—31no—
66#388—31no—
67#388D311yescharacteristic, complemented
68#385—31no—
69#385—31no—
70#384—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—41no—
82C22×Dic3—22no—
83C22×Dic3—22no—
84C22×Dic3—41no—
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C2×C4×D3—22no—
93C2×C4×D3—22no—
94C2×C4×D3—41no—
95C2×Dic6—22no—
96C2×Dic6—22no—
97C2×Dic6—41no—
98C3×C4⋊C4—22no—
99C3×C22⋊C4—22no—
100C3×C22⋊C4—22no—
101C3×C22⋊C4D414yes—
102C3×C42—22no—
103#217—22no—
104#217—22no—
105(C2×D3)⋊C4—22no—
106(C2×D3)⋊C4—22no—
107(C2×D3)⋊C4—22no—
108(C2×D3)⋊C4—22no—
109(C2×D3)⋊C4D414yes—
110C4⋊Dic3—22no—
111C4⋊Dic3—22no—
112C4⋊Dic3—22no—
113C4⋊Dic3—22no—
114C4⋊Dic3—22no—
115C4⋊Dic3—22no—
116C4⋊Dic3—41no—
117C4⋊Dic3—41no—
118Dic3⋊C4—22no—
119Dic3⋊C4—22no—
120Dic3⋊C4—22no—
121C22×Q8—31no—
122C22×D4—31no—
123C23×C4—31no—
124C23×C4—31no—
125#123—64no—
126#123—64no—
127C22⋊Q8—64no—
128C22⋊Q8—64no—
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⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4C2×D311yescharacteristic
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4—31no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4C2×D311yescharacteristic
149C2×C22⋊C4C2×D311yescharacteristic
150C2×C22⋊C4C2×D311yescharacteristic
151C2×C42C2×D311yescharacteristic
152#95—31no—
153#95—31no—
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×C3C2×D412yes—
167C23×C3C2411yescharacteristic
168C22×D3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3C2×D412yes—
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×C4C2×D412yes—
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×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—44no—
215C2×Dic3C2×D412yes—
216C4×D3—44no—
217Dic6—44no—
218C24—31no—
219C24C22×D311yescharacteristic
220C2×Q8—62no—
221C2×Q8—62no—
222C2×Q8—121no—
223C2×D4—62no—
224C2×D4—62no—
225C2×D4—121no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—62no—
234C22×C4—62no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4—121no—
238C22×C4—121no—
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245C4⋊C4—22no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—34no—
260C22⋊C4—62no—
261C22⋊C4—62no—
262C22⋊C4—62no—
263C22⋊C4—62no—
264C22⋊C4—62no—
265C22⋊C4—62no—
266C22⋊C4D1214yes—
267C42—22no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3C22×D411yescharacteristic
276C22×C3C22×D411yescharacteristic
277C22×C3C2×#4111yescharacteristic
278C22×C3H(5,𝔽2)11yescharacteristic
279C22×C32−1+411yescharacteristic
280C22×C32−1+411yescharacteristic
281C22×C32−1+411yescharacteristic
282C2×D3—22no—
283C2×D3—22no—
284C2×D3—22no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C3×C4—22no—
290C3×C4—22no—
291C3×C4—22no—
292C3×C4—22no—
293C3×C4—41no—
294C3×C4—41no—
295Dic3—22no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300Dic3—41no—
301Dic3—41no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—32no—
309C23—61no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23—61no—
314C23—61no—
315C23C2×D1212yes—
316C23C23×D311yescharacteristic
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—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—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—124no—
355C2×C4—124no—
356C2×C4C2×D1212yes—
357C2×C4C2×D1212yes—
358C2×C4D3×D412yes—
359C2×C4#23712yes—
360C2×C3—42no—
361C2×C3#53611yescharacteristic
362C2×C3#53611yescharacteristic
363C2×C3#53911yescharacteristic
364C2×C3D4211yescharacteristic
365C2×C3#54711yescharacteristic
366C2×C3#54711yescharacteristic
367C2×C3#56311yescharacteristic
368D3—42no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—62no—
377C22—62no—
378C22—62no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22C22×D1211yescharacteristic
384C22C2×D3×D411yescharacteristic
385C22C2×#23711yescharacteristic
386C22#100911yescharacteristic
387C22#101011yescharacteristic
388C22#101011yescharacteristic
389C22#101011yescharacteristic
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—62no—
397C4—121no—
398C4—121no—
399C4—121no—
400C4—121no—
401C4—121no—
402C4—121no—
403C3#268011yescharacteristic, complemented, Sylow
404C2—42no—
405C2—122no—
406C2#552611yescharacteristic
407C2D4×D1211yescharacteristic
408C2#555511yescharacteristic
409C2#561111yescharacteristic
410C2#566511yescharacteristic
411C2#567511yescharacteristic
412C2#571211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#388C2
prime radicalC22×#67C22
hypercenter#388D3
hypocenterC3#2680
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#897C2semidirect
4. C2×#878C2semidirect
5. C3×#388C2semidirect
6. #4981C2semidirect
8. #4953C2semidirect
9. #4950C2semidirect
10. #4940C2semidirect
11. #4935C2semidirect
12. #4674C2semidirect
13. #4668C2semidirect
14. #4661C2semidirect
15. #4656C2semidirect
67. #388D3semidirect
403. C3#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×#8862
C2×#8832
C2×#8782
C3×#3882
#49812
#49612
#49532
#49502
#49402
#49352
#46742
#46682
#46612
#46562
#26803