Finite Group Databasebeta

← #80907

Subgroups — #80907

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#918C211yesmaximal, characteristic, complemented
2C2×#907C211yesmaximal, characteristic, complemented
3#5629C211yesmaximal, characteristic
4#5628C211yesmaximal, characteristic, complemented
5C3×#470C211yesmaximal, characteristic, complemented
6#5188C211yesmaximal, characteristic, complemented
7#5134C211yesmaximal, characteristic, complemented
8#5051C211yesmaximal, characteristic, complemented
9#5048C211yesmaximal, characteristic, complemented
10#4864C211yesmaximal, characteristic, complemented
11#4861C211yesmaximal, characteristic, complemented
12#4816C211yesmaximal, characteristic, complemented
13#4812C211yesmaximal, characteristic, complemented
14#4804C211yesmaximal, characteristic, complemented
15D3×#103C211yesmaximal, characteristic, complemented
16#3063—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24#946C2211yescharacteristic
25#946C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#215C2211yescharacteristic
28#930C2211yescharacteristic
29#924C2211yescharacteristic
30C22×C3⋊C8C2211yescharacteristic
31#918—24no—
32#907—24no—
33C2×#204C2211yescharacteristic
34C2×ΓL(1,𝔽25)C2211yescharacteristic
35C2×ΓL(1,𝔽25)C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37C2×D3×C8C2211yescharacteristic
38#897C2211yescharacteristic
39#896—21no—
40#895—21no—
41#895C2211yescharacteristic
42#894—21no—
43#892C2211yescharacteristic
44D3×C4⋊C4C2211yescharacteristic
45#888—21no—
46#885—21no—
47#884—21no—
48#884—21no—
49#883—21no—
50C3×#106C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53#835C2211yescharacteristic
54#835C2211yescharacteristic
55#821C2211yescharacteristic
56#818C2211yescharacteristic
57#816C2211yescharacteristic
58#810C2211yescharacteristic
59#809C2211yescharacteristic
60#808C2211yescharacteristic
61#807C2211yescharacteristic
62C2×#137—31no—
63C2×#131—31no—
64#537—31no—
65#536—31no—
66#470—31no—
67#470D311yescharacteristic, complemented
68#468—31no—
69#468—31no—
70#467—31no—
71#467—31no—
72#465—31no—
73#465—31no—
74#428—31no—
75#416—31no—
76C2×#103—31no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67D411yescharacteristic
82C2×#67D411yescharacteristic
83#239—22no—
84#239—22no—
85D3×Q8—22no—
86D3×Q8—22no—
87#235—22no—
88#235—22no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12D411yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3—21no—
96C2×C4×D3C2311yescharacteristic
97C2×C4×D3C2311yescharacteristic
98C2×Dic6—21no—
99C2×Dic6D411yescharacteristic
100C2×Dic6C2311yescharacteristic
101C3×Q16—22no—
102C3×M16—22no—
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111#217—21no—
112#216—22no—
113#215—22no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3C2311yescharacteristic
122C4⋊Dic3C2311yescharacteristic
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125Dic3⋊C4—21no—
126Dic3⋊C4—21no—
127C4×Dic3—21no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131#204—22no—
132ΓL(1,𝔽25)—22no—
133ΓL(1,𝔽25)D414yescomplemented
134D3×C8—22no—
135D3×C8—22no—
136C2×#41—31no—
137C2×#41—31no—
138C22×Q8—31no—
139#137—64no—
140C2×Q16—31no—
141C2×Q16C2×D311yescharacteristic
142C2×SD16—31no—
143C2×SD16—31no—
144#131—64no—
145C2×M16—31no—
146C2×M16—31no—
147C2×M16C2×D311yescharacteristic
148C22×C8—31no—
149C22×C8—31no—
150C22×C8—31no—
151C4⋊Q8—61no—
152#124—61no—
153#123—61no—
154#123—61no—
155C22⋊Q8—31no—
156C22⋊Q8—31no—
157C22⋊Q8—31no—
158C22⋊Q8—31no—
159C22⋊Q8—61no—
160C22⋊Q8C2×D311yescharacteristic
161C22⋊Q8C2×D311yescharacteristic
162#121—31no—
163#121—31no—
164#121—61no—
165#121—61no—
166#121—61no—
167#117—31no—
168C2×C4⋊C4—31no—
169#107—31no—
170#107—31no—
171#106—31no—
172#106C2×D311yescharacteristic
173#103—31no—
174#103—31no—
175#103—31no—
176#103—31no—
177#103C2×D311yescharacteristic
178#103C2×D311yescharacteristic
179#102—31no—
180#102—31no—
181C23×C3C2×D411yescharacteristic
182C22×D3—21no—
183C22×D3C2×D411yescharacteristic
184C22×D3C2×D411yescharacteristic
185C3×Q8—21no—
186C3×Q8—21no—
187C3×Q8—21no—
188C3×Q8—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×C4C2×D411yescharacteristic
196C2×C3×C4C2411yescharacteristic
197#67—22no—
198#67—22no—
199#67—81no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3C2×D411yescharacteristic
204C2×Dic3C2×D411yescharacteristic
205D12—22no—
206D12—21no—
207D12—21no—
208D12—42no—
209D12—81no—
210C4×D3—21no—
211C4×D3—21no—
212C4×D3—42no—
213C4×D3—42no—
214C4×D3—81no—
215C4×D3C2×D412yes—
216C4×D3#4112yes—
217Dic6—22no—
218Dic6—21no—
219Dic6—21no—
220Dic6—42no—
221C3×C8—21no—
222C3×C8C2×D412yes—
223C3⋊C8—21no—
224C3⋊C8C2×D412yes—
225#41—62no—
226#41—62no—
227#41—62no—
228#41—62no—
229C2×Q8—31no—
230C2×Q8—31no—
231C2×Q8—61no—
232C2×Q8—62no—
233C2×Q8—62no—
234C2×Q8C22×D311yescharacteristic
235C2×Q8C22×D311yescharacteristic
236C2×D4—31no—
237C2×D4—31no—
238C2×D4—31no—
239C2×D4—31no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4C22×D311yescharacteristic
249Q16—22no—
250Q16—62no—
251SD16—62no—
252SD16—62no—
253M16—22no—
254M16—34no—
255M16—62no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8C22×D311yescharacteristic
262C2×C8C22×D311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—31no—
266C4⋊C4—31no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4C22×D311yescharacteristic
273C4⋊C4C22×D311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—61no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C42—61no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3C22×D411yescharacteristic
288C2×D3—21no—
289C2×D3—21no—
290C2×D3—21no—
291C2×D3—21no—
292C2×D3—21no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3#12112yes—
297C3×C4—21no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4C22×D411yescharacteristic
303C3×C4C2×#4111yescharacteristic
304Dic3—21no—
305Dic3—41no—
306Dic3—41no—
307Dic3—41no—
308Dic3#12112yes—
309C23—31no—
310C23—31no—
311C23—61no—
312C23D3×D411yescharacteristic
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—21no—
317Q8—62no—
318Q8—61no—
319Q8—61no—
320Q8—122no—
321D4—62no—
322D4—62no—
323D4—62no—
324D4—61no—
325D4—61no—
326D4—122no—
327D4—241no—
328D4—241no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—31no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—31no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—241no—
347C2×C4D3×D411yescharacteristic
348C2×C4C23×D311yescharacteristic
349C8—21no—
350C8—32no—
351C8—61no—
352C8D3×D412yes—
353C2×C3—41no—
354C2×C3C2×#12111yescharacteristic
355C2×C3#57711yescharacteristic
356C2×C3#57811yescharacteristic
357D3—22no—
358D3—41no—
359D3—81no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—32no—
364C22—61no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—32no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—61no—
380C4—121no—
381C4—121no—
382C4—121no—
383C4C2×D3×D411yescharacteristic
384C4D3×#4111yescharacteristic
385C3#306311yescharacteristic, complemented, Sylow
386C2—41no—
387C2—62no—
388C2—121no—
389C2—241no—
390C2D3×#12111yescharacteristic
391C2#577811yescharacteristic
392C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
prime radical#5629C2
hypercenter#470D3
hypocenterC3#3063
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#918C2semidirect
2. C2×#907C2semidirect
4. #5628C2semidirect
5. C3×#470C2semidirect
6. #5188C2semidirect
7. #5134C2semidirect
8. #5051C2semidirect
9. #5048C2semidirect
10. #4864C2semidirect
11. #4861C2semidirect
12. #4816C2semidirect
13. #4812C2semidirect
14. #4804C2semidirect
15. D3×#103C2semidirect
67. #470D3semidirect
133. ΓL(1,𝔽25)D4semidirect
385. C3#3063semidirect

Maximal subgroups

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

subgroupindex
C2×#9182
C2×#9072
#56292
#56282
C3×#4702
#51882
#51342
#50512
#50482
#48642
#48612
#48162
#48122
#48042
D3×#1032
#30633