Finite Group Databasebeta

← #80810

Subgroups — #80810

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×#915C211yesmaximal, characteristic, complemented
2C2×#907C211yesmaximal, characteristic, complemented
3#5628C211yesmaximal, characteristic, complemented
4#5608C211yesmaximal, characteristic
5C3×#468C211yesmaximal, characteristic, complemented
6#5171C211yesmaximal, characteristic, complemented
7#5137C211yesmaximal, characteristic, complemented
8#5051C211yesmaximal, characteristic, complemented
9#5041C211yesmaximal, characteristic, complemented
10#4887C211yesmaximal, characteristic, complemented
11#4881C211yesmaximal, characteristic, complemented
12#4816C211yesmaximal, characteristic, complemented
13D3×#103C211yesmaximal, characteristic, complemented
14#4780C211yesmaximal, characteristic, complemented
15#4772C211yesmaximal, characteristic, complemented
16#3063—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic
25C2×#216C2211yescharacteristic
26#939—21no—
27#938—21no—
28#938C2211yescharacteristic
29C2×#214C2211yescharacteristic
30#924C2211yescharacteristic
31#924C2211yescharacteristic
32C22×C3⋊C8C2211yescharacteristic
33#915—24no—
34#907—24no—
35C2×Dic12C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37C2×ΓL(1,𝔽25)C2211yescharacteristic
38C2×D3×C8C2211yescharacteristic
39C2×D3×C8C2211yescharacteristic
40#897C2211yescharacteristic
41#897C2211yescharacteristic
42#894—21no—
43#892C2211yescharacteristic
44D3×C4⋊C4C2211yescharacteristic
45#888—21no—
46#885—21no—
47#884—21no—
48#883—21no—
49#878—21no—
50C3×#107C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#102C2211yescharacteristic
53#835C2211yescharacteristic
54#832C2211yescharacteristic
55#817C2211yescharacteristic
56#816C2211yescharacteristic
57#816C2211yescharacteristic
58#810C2211yescharacteristic
59#810C2211yescharacteristic
60#808C2211yescharacteristic
61#807C2211yescharacteristic
62C2×#137—31no—
63C2×#131—31no—
64#537—31no—
65#536—31no—
66#470—31no—
67#470—31no—
68#468—31no—
69#468D311yescharacteristic, complemented
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×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67D411yescharacteristic
84C2×#67D411yescharacteristic
85C22×Dic3—21no—
86D3×Q8—22no—
87D3×Q8—22no—
88#237—22no—
89#237—22no—
90#235—22no—
91#235—22no—
92C2×D12D411yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—21no—
98C2×Dic6D411yescharacteristic
99C2×Dic6C2311yescharacteristic
100C2×Dic6C2311yescharacteristic
101C3×SD16—22no—
102C3×M16—22no—
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C3×C4⋊C4—21no—
106C3×C4⋊C4C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110#217—21no—
111#217—21no—
112#217—21no—
113#216—22no—
114#214—22no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3—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
131Dic12—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×Q16—31no—
142C2×SD16—31no—
143C2×SD16C2×D311yescharacteristic
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—31no—
160C22⋊Q8—61no—
161C22⋊Q8C2×D311yescharacteristic
162#121—31no—
163#121—61no—
164#121—61no—
165#121—61no—
166#121C2×D311yescharacteristic
167#117—31no—
168C2×C4⋊C4—31no—
169#107—31no—
170#107C2×D311yescharacteristic
171#106—31no—
172#106—31no—
173#103—31no—
174#103—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#103C2×D311yescharacteristic
179#102—31no—
180#102C2×D311yescharacteristic
181C23×C3—21no—
182C23×C3C2×D411yescharacteristic
183C22×D3C2×D411yescharacteristic
184C22×D3C2×D411yescharacteristic
185C3×Q8—21no—
186C3×Q8—21no—
187C3×D4—21no—
188C3×D4—21no—
189C3×D4—81no—
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—42no—
200#67—81no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—42no—
206C2×Dic3C2×D411yescharacteristic
207C2×Dic3C2×D411yescharacteristic
208D12—22no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—42no—
212C4×D3—81no—
213C4×D3C2×D412yes—
214C4×D3#4112yes—
215Dic6—22no—
216Dic6—21no—
217Dic6—21no—
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—31no—
232C2×Q8—61no—
233C2×Q8—62no—
234C2×Q8—62no—
235C2×Q8C22×D311yescharacteristic
236C2×D4—21no—
237C2×D4—31no—
238C2×D4—31no—
239C2×D4—31no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4C22×D311yescharacteristic
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4C22×D311yescharacteristic
249Q16—62no—
250Q16—62no—
251SD16—22no—
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—31no—
265C4⋊C4—31no—
266C4⋊C4—61no—
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×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3C22×D411yescharacteristic
291C2×D3—21no—
292C2×D3—21no—
293C2×D3—21no—
294C2×D3—21no—
295C2×D3—21no—
296C2×D3#12112yes—
297C3×C4—21no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4C22×D411yescharacteristic
302C3×C4C2×#4111yescharacteristic
303Dic3—21no—
304Dic3—41no—
305Dic3—41no—
306Dic3—41no—
307Dic3—41no—
308Dic3#12112yes—
309C23—21no—
310C23—31no—
311C23—31no—
312C23D3×D411yescharacteristic
313Q8—21no—
314Q8—21no—
315Q8—62no—
316Q8—61no—
317Q8—61no—
318Q8—61no—
319Q8—61no—
320Q8—122no—
321D4—21no—
322D4—21no—
323D4—62no—
324D4—62no—
325D4—62no—
326D4—81no—
327D4—122no—
328D4—241no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—31no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—31no—
338C2×C4—61no—
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×C3—81no—
355C2×C3C2×#12111yescharacteristic
356C2×C3#57711yescharacteristic
357C2×C3#57811yescharacteristic
358D3—22no—
359D3—41no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—32no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—61no—
368C22—61no—
369C22—61no—
370C22—61no—
371C22—61no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—32no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—61no—
379C4—121no—
380C4—121no—
381C4—121no—
382C4—121no—
383C4C2×D3×D411yescharacteristic
384C4D3×#4111yescharacteristic
385C3#306311yescharacteristic, complemented, Sylow
386C2—41no—
387C2—62no—
388C2—81no—
389C2—121no—
390C2D3×#12111yescharacteristic
391C2#577511yescharacteristic
392C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
prime radical#5608C2
hypercenter#468D3
hypocenterC3#3063
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#915C2semidirect
2. C2×#907C2semidirect
3. #5628C2semidirect
5. C3×#468C2semidirect
6. #5171C2semidirect
7. #5137C2semidirect
8. #5051C2semidirect
9. #5041C2semidirect
10. #4887C2semidirect
11. #4881C2semidirect
12. #4816C2semidirect
13. D3×#103C2semidirect
14. #4780C2semidirect
15. #4772C2semidirect
69. #468D3semidirect
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×#9152
C2×#9072
#56282
#56082
C3×#4682
#51712
#51372
#50512
#50412
#48872
#48812
#48162
D3×#1032
#47802
#47722
#30633