Finite Group Databasebeta

← #80109

Subgroups — #80109

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5831C211yesmaximal, characteristic, complemented
2#5611C211yesmaximal, characteristic, complemented
3C3×#449C211yesmaximal, characteristic, complemented
4#5232C211yesmaximal, characteristic, complemented
5#5170C211yesmaximal, characteristic, complemented
6#5164C211yesmaximal, characteristic, complemented
7#5157C211yesmaximal, characteristic, complemented
8#5153C211yesmaximal, characteristic, complemented
9#5042C211yesmaximal, characteristic, complemented
10D3×#103C211yesmaximal, characteristic, complemented
11#4793C211yesmaximal, characteristic, complemented
12#4774C211yesmaximal, characteristic, complemented
13#4765C211yesmaximal, characteristic, complemented
14#4728C211yesmaximal, characteristic, complemented
15#4722C211yesmaximal, characteristic, complemented
16#3308—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×#237—21no—
20C2×C3×SD16C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic
22C3×#121C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24#938—21no—
25#938C2211yescharacteristic
26#938C2211yescharacteristic
27#936—21no—
28#936C2211yescharacteristic
29#935—21no—
30D4×Dic3C2211yescharacteristic
31C2×#214C2211yescharacteristic
32C4×#67—21no—
33C4×#67C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35#924—21no—
36#924C2211yescharacteristic
37C2×#208C2211yescharacteristic
38C2×D3×C8C2211yescharacteristic
39D3×C4⋊C4C2211yescharacteristic
40#888C2211yescharacteristic
41#883—21no—
42#878—21no—
43C3×#103C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#818C2211yescharacteristic
53#818C2211yescharacteristic
54#816C2211yescharacteristic
55#816C2211yescharacteristic
56#815C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58#810C2211yescharacteristic
59#807C2211yescharacteristic
60#548—31no—
61#547—31no—
62#489—31no—
63#484—31no—
64#480—31no—
65#477—31no—
66#468—31no—
67#466—31no—
68#463—31no—
69#449D311yescharacteristic, complemented
70#448—31no—
71#440—31no—
72#435—31no—
73#417—31no—
74C2×#103—31no—
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67D411yescharacteristic
87C22×Dic3—21no—
88C22×Dic3—21no—
89D3×Q8—22no—
90D3×Q8—41no—
91#237—44no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3C2311yescharacteristic
95C2×Dic6—21no—
96C2×Dic6—21no—
97C2×Dic6C2311yescharacteristic
98C3×SD16—22no—
99C3×D8—41no—
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104#217—21no—
105#217—21no—
106#217—21no—
107#217—21no—
108#217—21no—
109#217—21no—
110#217D411yescharacteristic
111#214—41no—
112(C2×D3)⋊C4D411yescharacteristic
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124#208—41no—
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127D3×C8—41no—
128C2×#41—61no—
129C2×#41C2×D311yescharacteristic
130C22×Q8—31no—
131C2×SD16—31no—
132C2×SD16C2×D311yescharacteristic
133C2×D8C2×D311yescharacteristic
134C2×M16—31no—
135C22×C8—31no—
136C4⋊Q8—31no—
137#124—31no—
138#124—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144#121—31no—
145#121—31no—
146#121—61no—
147#121C2×D311yescharacteristic
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—61no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154#107—31no—
155#107—31no—
156#107—31no—
157C4⋊C8—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168C22⋊C8—31no—
169C22⋊C8C2×D311yescharacteristic
170#97—31no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3C2×D411yescharacteristic
174C22×D3C2×D411yescharacteristic
175C3×Q8—21no—
176C3×Q8#4112yescomplemented
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—42no—
182C3×D4—42no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190#67—22no—
191#67—42no—
192#67—42no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C4×D3—21no—
203C4×D3—21no—
204C4×D3—42no—
205C4×D3—42no—
206Dic6—21no—
207Dic6—21no—
208Dic6—42no—
209Dic6—42no—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214#41—22no—
215#41—41no—
216#41—124no—
217C2×Q8—31no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—62no—
221C2×Q8—121no—
222C2×Q8C22×D311yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4C22×D311yescharacteristic
229C2×D4C22×D311yescharacteristic
230C22×C4—21no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4C22×D311yescharacteristic
237SD16—22no—
238SD16—121no—
239D8—41no—
240M16—121no—
241C2×C8—31no—
242C2×C8—31no—
243C2×C8—121no—
244C2×C8C22×D311yescharacteristic
245C2×C8C22×D311yescharacteristic
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4C22×D311yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—31no—
258C22⋊C4—31no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C42—31no—
266C42—61no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3C22×D411yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3AGL(1,ℤ/8)12yes—
280C3×C4—22no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4C2×#4111yescharacteristic
284C3×C4H(5,𝔽2)11yescharacteristic
285Dic3—21no—
286Dic3—21no—
287Dic3—21no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291Dic3—41no—
292C23—21no—
293C23—21no—
294C23—31no—
295C23D3×D411yescharacteristic
296Q8—21no—
297Q8—61no—
298Q8—61no—
299Q8—122no—
300Q8—122no—
301Q8#23712yescomplemented
302D4—21no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—42no—
308D4—62no—
309D4—122no—
310D4—122no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—42no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4D3×D411yescharacteristic
330C2×C4C23×D311yescharacteristic
331C8—21no—
332C8—21no—
333C8—61no—
334C8—61no—
335C2×C3—41no—
336C2×C3—81no—
337C2×C3—81no—
338C2×C3#54611yescharacteristic
339C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
340C2×C3#57811yescharacteristic
341D3—22no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—32no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—61no—
353C22—61no—
354C22C2×D3×D411yescharacteristic
355C4—21no—
356C4—22no—
357C4—41no—
358C4—61no—
359C4—61no—
360C4—61no—
361C4—121no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4C2×#23711yescharacteristic
366C4#100411yescharacteristic
367C3#330811yescharacteristic, complemented, Sylow
368C2—41no—
369C2—62no—
370C2—81no—
371C2—81no—
372C2#559111yescharacteristic
373C2#577111yescharacteristic
374C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#449C2
socleC22×C3C22×D4
hypercenter#449D3
hypocenterC3#3308
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #5831C2semidirect
2. #5611C2semidirect
3. C3×#449C2semidirect
4. #5232C2semidirect
5. #5170C2semidirect
6. #5164C2semidirect
7. #5157C2semidirect
8. #5153C2semidirect
9. #5042C2semidirect
10. D3×#103C2semidirect
11. #4793C2semidirect
12. #4774C2semidirect
13. #4765C2semidirect
14. #4728C2semidirect
15. #4722C2semidirect
69. #449D3semidirect
176. C3×Q8#41semidirect
301. Q8#237semidirect
367. C3#3308semidirect

Maximal subgroups

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

subgroupindex
#58312
#56112
C3×#4492
#52322
#51702
#51642
#51572
#51532
#50422
D3×#1032
#47932
#47742
#47652
#47282
#47222
#33083