Finite Group Databasebeta

← #80354

Subgroups — #80354

Order 384.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5718C211yesmaximal, characteristic, complemented
2#5558C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5159C211yesmaximal, characteristic, complemented
5#5158C211yesmaximal, characteristic, complemented
6#5158C211yesmaximal, characteristic, complemented
7#5157C211yesmaximal, characteristic, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5018C211yesmaximal, characteristic, complemented
10#4836C211yesmaximal, characteristic, complemented
11#4829C211yesmaximal, characteristic, complemented
12#4779C211yesmaximal, characteristic, complemented
13#4774C211yesmaximal, characteristic, complemented
14#4772C211yesmaximal, characteristic, complemented
15#4771C211yesmaximal, characteristic, complemented
16#3309—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C3×#127C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic
23#940—21no—
24#939—21no—
25#939—21no—
26#938C2211yescharacteristic
27#938C2211yescharacteristic
28#938C2211yescharacteristic
29#938C2211yescharacteristic
30D4×Dic3—21no—
31C2×#214C2211yescharacteristic
32#929—21no—
33C4×#67—21no—
34C2×ΓL(1,𝔽25)C2211yescharacteristic
35C2×D3×C8C2211yescharacteristic
36#892C2211yescharacteristic
37D3×C4⋊C4C2211yescharacteristic
38#884—21no—
39#879—21no—
40C4×D12C2211yescharacteristic
41#870C2211yescharacteristic
42C4×Dic6C2211yescharacteristic
43C3×C4⋊C8C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×#102C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53#818C2211yescharacteristic
54#817C2211yescharacteristic
55#816C2211yescharacteristic
56#816C2211yescharacteristic
57#810C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#553—31no—
61#550—31no—
62#490—31no—
63#482—31no—
64#480—31no—
65#479—31no—
66#469—31no—
67#466—31no—
68#459D311yescharacteristic, complemented
69#449—31no—
70#449—31no—
71#440—31no—
72#435—31no—
73#417—31no—
74#416—31no—
75C2×C3×D4—21no—
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C22×Dic3—21no—
88C22×Dic3—21no—
89#237—22no—
90#237—22no—
91#237—41no—
92#237—41no—
93C2×D12D411yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C2×Dic6C2311yescharacteristic
98C3×D8—41no—
99C3×D8—41no—
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104C3×C42C2311yescharacteristic
105#217—21no—
106#217—21no—
107#217—21no—
108#217—21no—
109#214—22no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4D411yescharacteristic
112(C2×D3)⋊C4D411yescharacteristic
113C4⋊Dic3D411yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121C4×Dic3—21no—
122C4×Dic3—21no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125ΓL(1,𝔽25)—41no—
126D3×C8—41no—
127C2×#41—31no—
128C2×#41—31no—
129C2×SD16—31no—
130C2×D8C2×D311yescharacteristic
131C2×D8C2×D311yescharacteristic
132C2×M16—31no—
133C22×C8—31no—
134#127—61no—
135#127C2×D311yescharacteristic
136#126—61no—
137#125—31no—
138#123—61no—
139#121—31no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—61no—
144#121—61no—
145#121—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—61no—
149C4×D4—61no—
150C4×D4C2×D311yescharacteristic
151#117—31no—
152C2×C4⋊C4—31no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8C2×D311yescharacteristic
157#103—31no—
158#103—31no—
159#103—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C22×D3C2×D411yescharacteristic
174C22×D3C2×D411yescharacteristic
175C3×D4—21no—
176C3×D4—21no—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—42no—
182C3×D4—81no—
183C2×C3×C4—21no—
184C2×C3×C4—81no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2411yescharacteristic
188#67—42no—
189#67—42no—
190#67—42no—
191#67—81no—
192#67—81no—
193#67—81no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202D12—41no—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—21no—
206C4×D3—21no—
207C4×D3—81no—
208Dic6—21no—
209Dic6#4112yes—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214#41—62no—
215#41—62no—
216#41—121no—
217#41—121no—
218C2×Q8—31no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—31no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4C22×D311yescharacteristic
229C2×D4C22×D311yescharacteristic
230C2×D4C22×D311yescharacteristic
231C22×C4—21no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237SD16—62no—
238D8—41no—
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⋊C4C22×D311yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—31no—
257C22⋊C4—31no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C42—61no—
264C42—61no—
265C42C22×D311yescharacteristic
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3C22×D411yescharacteristic
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4C2×#4111yescharacteristic
286C3×C4H(5,𝔽2)11yescharacteristic
287Dic3—22no—
288Dic3—21no—
289Dic3—21no—
290Dic3—41no—
291Dic3—41no—
292Dic3—41no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—31no—
297C23—31no—
298Q8—32no—
299Q8—61no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—81no—
308D4—121no—
309D4—122no—
310D4—122no—
311D4—122no—
312D4—241no—
313D4—241no—
314D4—241no—
315C2×C4—21no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—81no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—241no—
330C2×C4D3×D411yescharacteristic
331C2×C4D3×D411yescharacteristic
332C2×C4C23×D311yescharacteristic
333C8—21no—
334C8—21no—
335C8—61no—
336C8—61no—
337C2×C3—81no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3#54611yescharacteristic
341C2×C3#57511yescharacteristic
342C2×C3#57611yescharacteristic
343D3—41no—
344D3—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—61no—
355C22—61no—
356C22—61no—
357C22—61no—
358C22—61no—
359C22—61no—
360C22C2×D3×D411yescharacteristic
361C4—21no—
362C4—21no—
363C4—41no—
364C4—61no—
365C4—61no—
366C4—62no—
367C4—121no—
368C4—121no—
369C4—121no—
370C4#100411yescharacteristic
371C4D3×#4111yescharacteristic
372C3#330911yescharacteristic, complemented, Sylow
373C2—81no—
374C2—81no—
375C2—81no—
376C2—121no—
377C2—121no—
378C2#561211yescharacteristic
379C2#575711yescharacteristic
380C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3309
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5718C2semidirect
2. #5558C2semidirect
3. C3×#459C2semidirect
4. #5159C2semidirect
5. #5158C2semidirect
6. #5158C2semidirect
7. #5157C2semidirect
8. #5071C2semidirect
9. #5018C2semidirect
10. #4836C2semidirect
11. #4829C2semidirect
12. #4779C2semidirect
13. #4774C2semidirect
14. #4772C2semidirect
15. #4771C2semidirect
68. #459D3semidirect
372. C3#3309semidirect

Maximal subgroups

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

subgroupindex
#57182
#55582
C3×#4592
#51592
#51582
#51572
#50712
#50182
#48362
#48292
#47792
#47742
#47722
#47712
#33093