Finite Group Databasebeta

← #80689

Subgroups — #80689

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5632C211yesmaximal, characteristic, complemented
2#5611C211yesmaximal, characteristic
3C3×#465C211yesmaximal, characteristic, complemented
4#5170C211yesmaximal, characteristic, complemented
5#5162C211yesmaximal, characteristic, complemented
6C8×#67C211yesmaximal, characteristic, complemented
7C2×#816C211yesmaximal, characteristic, complemented
8#5043C211yesmaximal, characteristic, complemented
9#5040C211yesmaximal, characteristic, complemented
10#4862C211yesmaximal, characteristic, complemented
11#4854C211yesmaximal, characteristic, complemented
12D3×#103C211yesmaximal, characteristic, complemented
13#4792C211yesmaximal, characteristic, complemented
14#4778C211yesmaximal, characteristic, complemented
15#4758C211yesmaximal, characteristic, complemented
16#3298—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237—21no—
19C22×Dic6C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C22×C3×C8C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#944C2211yescharacteristic
25#938C2211yescharacteristic
26#935—21no—
27D4×Dic3C2211yescharacteristic
28C2×#214C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C4×#67C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×D3×C8C2211yescharacteristic
33#897C2211yescharacteristic
34#896—21no—
35D3×C4⋊C4C2211yescharacteristic
36#888—21no—
37#888C2211yescharacteristic
38#887C2211yescharacteristic
39#883—21no—
40#67⋊C4—21no—
41#878—21no—
42#878—21no—
43C3×#106C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#102C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#830C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50#816C2211yescharacteristic
51#816C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#808C2211yescharacteristic
59#808C2211yescharacteristic
60D4×Q8—31no—
61#547—31no—
62#487—31no—
63#483—31no—
64#483—31no—
65#475—31no—
66#465—31no—
67#465D311yescharacteristic, complemented
68#461—31no—
69#448—31no—
70#448—31no—
71C4×SD16—31no—
72C8×D4—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67D411yescharacteristic
81C22×Dic3—21no—
82C22×Dic3—21no—
83C22×Dic3—21no—
84D3×Q8—22no—
85D3×Q8—41no—
86#237—44no—
87C2×C4×D3—21no—
88C2×C4×D3—21no—
89C2×C4×D3C2311yescharacteristic
90C2×Dic6—22no—
91C2×Dic6—21no—
92C2×Dic6—21no—
93C2×Dic6—21no—
94C2×Dic6—41no—
95C2×Dic6C2311yescharacteristic
96C2×Dic6C2311yescharacteristic
97C3×SD16—41no—
98C2×C3×C8—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C22⋊C4—21no—
105C3×C22⋊C4—21no—
106#217—21no—
107#217—21no—
108#217D411yescharacteristic
109#214—22no—
110#214—41no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4D411yescharacteristic
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123C4×Dic3—21no—
124C4×Dic3C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127D3×C8—41no—
128C2×#41—61no—
129C22×Q8—31no—
130C22×Q8—31no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16C2×D311yescharacteristic
134C22×C8—31no—
135C22×C8C2×D311yescharacteristic
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138C4⋊Q8—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C22⋊Q8C2×D311yescharacteristic
146#121—31no—
147#121C2×D311yescharacteristic
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—61no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154#106—31no—
155#106—31no—
156#106C2×D311yescharacteristic
157C4⋊C8—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103—31no—
166#103C2×D311yescharacteristic
167#102C2×D311yescharacteristic
168C22⋊C8—31no—
169C22⋊C8—31no—
170C4×C8—31no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
173C22×D3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×D4—21no—
177C3×D4—21no—
178C3×D4—42no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—42no—
187#67SD1614yescomplemented
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198C4×D3—21no—
199C4×D3—21no—
200C4×D3—42no—
201C4×D3—42no—
202Dic6—21no—
203Dic6—21no—
204Dic6—21no—
205Dic6—42no—
206Dic6—42no—
207Dic6—42no—
208Dic6#4112yescomplemented
209C3×C8—21no—
210C3×C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—124no—
214C2×Q8—31no—
215C2×Q8—31no—
216C2×Q8—61no—
217C2×Q8—61no—
218C2×Q8—62no—
219C2×Q8—61no—
220C2×Q8—62no—
221C2×Q8—121no—
222C2×Q8—121no—
223C2×Q8C22×D311yescharacteristic
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—61no—
227C2×D4C22×D311yescharacteristic
228C22×C4—31no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235SD16—41no—
236SD16—62no—
237SD16—121no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—41no—
241C2×C8—121no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4C22×D311yescharacteristic
256C4⋊C4C22×D311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—31no—
260C22⋊C4—31no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C42—31no—
265C42—61no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3C2×SD1612yes—
272C22×C3C22×D411yescharacteristic
273C2×D3—21no—
274C2×D3—21no—
275C2×D3C2×SD1612yes—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4C2×#4111yescharacteristic
281C3×C4H(5,𝔽2)11yescharacteristic
282Dic3—22no—
283Dic3—21no—
284Dic3—21no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288Dic3C2×SD1612yes—
289C23—21no—
290C23—31no—
291C23D3×D411yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—32no—
295Q8—61no—
296Q8—61no—
297Q8—61no—
298Q8—122no—
299Q8—122no—
300Q8—122no—
301D4—21no—
302D4—21no—
303D4—34no—
304D4—42no—
305D4—122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4D3×D411yescharacteristic
326C2×C4C23×D311yescharacteristic
327C8—21no—
328C8—21no—
329C8—61no—
330C8—61no—
331C2×C3—22no—
332C2×C3—81no—
333C2×C3#54611yescharacteristic
334C2×C3C22×SD1611yescharacteristic
335C2×C3#57811yescharacteristic
336D3—22no—
337C22—21no—
338C22—21no—
339C22—32no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—61no—
344C22—61no—
345C22D3×SD1612yes—
346C22C2×D3×D411yescharacteristic
347C4—21no—
348C4—32no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—61no—
353C4—61no—
354C4—62no—
355C4—121no—
356C4—121no—
357C4—121no—
358C4#100411yescharacteristic
359C4D3×#4111yescharacteristic
360C3#329811yescharacteristic, complemented, Sylow
361C2—22no—
362C2—62no—
363C2—81no—
364C2#561011yescharacteristic
365C2C2×D3×SD1611yescharacteristic
366C2#577111yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
prime radical#5611C2
hypercenter#465D3
hypocenterC3#3298
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5632C2semidirect
3. C3×#465C2semidirect
4. #5170C2semidirect
5. #5162C2semidirect
6. C8×#67C2semidirect
7. C2×#816C2semidirect
8. #5043C2semidirect
9. #5040C2semidirect
10. #4862C2semidirect
11. #4854C2semidirect
12. D3×#103C2semidirect
13. #4792C2semidirect
14. #4778C2semidirect
15. #4758C2semidirect
67. #465D3semidirect
187. #67SD16semidirect
208. Dic6#41semidirect
360. C3#3298semidirect

Maximal subgroups

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

subgroupindex
#56322
#56112
C3×#4652
#51702
#51622
C8×#672
C2×#8162
#50432
#50402
#48622
#48542
D3×#1032
#47922
#47782
#47582
#32983