Finite Group Databasebeta

← #83572

Subgroups — #83572

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
1C3×#550C211yesmaximal, characteristic, complemented
2#5565C211yesmaximal, characteristic, complemented
3#5238C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic, complemented
5#5233C211yesmaximal, characteristic, complemented
6#5232C211yesmaximal, characteristic, complemented
7#5072C211yesmaximal, characteristic, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5039C211yesmaximal, characteristic, complemented
10#5038C211yesmaximal, characteristic, complemented
11#5035C211yesmaximal, characteristic, complemented
12#5033C211yesmaximal, characteristic, complemented
13#5026C211yesmaximal, characteristic, complemented
14#5022C211yesmaximal, characteristic, complemented
15#5011C211yesmaximal, characteristic, complemented
16#3309—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic
19C3×#127—21no—
20C3×#127C2211yescharacteristic
21C3×#121—21no—
22C3×#121—21no—
23C3×#121C2211yescharacteristic
24C3×#121C2211yescharacteristic
25C3×C4×Q8C2211yescharacteristic
26C3×C4×D4—21no—
27C3×C4×D4C2211yescharacteristic
28C2×#215C2211yescharacteristic
29#939—21no—
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930C2211yescharacteristic
33#930C2211yescharacteristic
34#926C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×#208C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic
38#898—21no—
39#883—21no—
40#67⋊C4—21no—
41C4×D12C2211yescharacteristic
42#870C2211yescharacteristic
43#835C2211yescharacteristic
44#835C2211yescharacteristic
45#835C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#832C2211yescharacteristic
52#830C2211yescharacteristic
53#830C2211yescharacteristic
54#809C2211yescharacteristic
55#808C2211yescharacteristic
56#807C2211yescharacteristic
57#807C2211yescharacteristic
58C4⋊(C3⋊C8)C2211yescharacteristic
59#803C2211yescharacteristic
60#553—31no—
61#550D311yescharacteristic, complemented
62#490—31no—
63#482—31no—
64#480—31no—
65#479—31no—
66#469—31no—
67#466—31no—
68#459—31no—
69#449—31no—
70#449—31no—
71#440—31no—
72#435—31no—
73#417—31no—
74#416—31no—
75C3×#41—22no—
76C3×#41—22no—
77C3×#41—41no—
78C3×#41—41no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4D411yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C2×#67—21no—
94C2×#67—21no—
95C22×Dic3—21no—
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4D411yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4D411yescharacteristic
104C3×C22⋊C4D411yescharacteristic
105C3×C42—21no—
106C3×C42C2311yescharacteristic
107#217—21no—
108#215—22no—
109#213—41no—
110#213—41no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120C4×Dic3—21no—
121#208—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×#41C2×D311yescharacteristic
128C2×#41C2×D311yescharacteristic
129C2×SD16—31no—
130C2×D8—31no—
131C2×D8—31no—
132C2×M16—31no—
133C22×C8—31no—
134#127—21no—
135#127C2×D311yescharacteristic
136#126—61no—
137#125—31no—
138#123—61no—
139#121—21no—
140#121—21no—
141#121—31no—
142#121—31no—
143#121—61no—
144#121C2×D311yescharacteristic
145#121C2×D311yescharacteristic
146C4×Q8C2×D311yescharacteristic
147C4×D4—21no—
148C4×D4—31no—
149C4×D4—61no—
150C4×D4C2×D311yescharacteristic
151#117—31no—
152C2×C4⋊C4—31no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8—31no—
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#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
173C23×C3C2×D411yescharacteristic
174C22×D3—21no—
175C3×Q8—21no—
176C3×Q8#4112yes—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—41no—
182C3×D4—42no—
183C3×D4—42no—
184C3×D4—42no—
185C3×D4—81no—
186C3×D4—81no—
187C3×D4—81no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—42no—
202#67—81no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—81no—
207D12—21no—
208D12—21no—
209C4×D3—81no—
210C3⋊C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214#41—22no—
215#41—22no—
216#41—41no—
217#41—41no—
218C2×Q8C22×D311yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4#6711yescharacteristic
229C2×D4C22×D311yescharacteristic
230C2×D4C22×D311yescharacteristic
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237SD16—62no—
238D8—121no—
239D8—121no—
240M16—121no—
241C2×C8—31no—
242C2×C8—31no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—121no—
246C4⋊C4—21no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4#6711yescharacteristic
254C4⋊C4C22×D311yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4#6711yescharacteristic
262C22⋊C4#6711yescharacteristic
263C42—21no—
264C42—61no—
265C42C22×D311yescharacteristic
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3C22×D411yescharacteristic
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—22no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4C2×#4111yescharacteristic
289C3×C4H(5,𝔽2)11yescharacteristic
290Dic3—41no—
291Dic3—41no—
292Dic3—41no—
293C23—21no—
294C23—21no—
295C23—61no—
296C23C2×#6711yescharacteristic
297C23C2×#6711yescharacteristic
298Q8—21no—
299Q8#23712yes—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—41no—
305D4—42no—
306D4—42no—
307D4—42no—
308D4—61no—
309D4—61no—
310D4—81no—
311D4—81no—
312D4—81no—
313D4—122no—
314D4—241no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4C2×#6711yescharacteristic
329C2×C4C2×#6711yescharacteristic
330C2×C4C2×#6711yescharacteristic
331C2×C4C2×#6711yescharacteristic
332C2×C4C23×D311yescharacteristic
333C8—61no—
334C8—61no—
335C8—61no—
336C8—61no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—81no—
340C2×C3—81no—
341C2×C3#54611yescharacteristic
342C2×C3#57511yescharacteristic
343C2×C3#57611yescharacteristic
344D3—81no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22C22×#6711yescharacteristic
361C4—21no—
362C4—21no—
363C4—22no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—121no—
368C4—121no—
369C4—121no—
370C4C2×#23711yescharacteristic
371C4#100411yescharacteristic
372C3#330911yescharacteristic, complemented, Sylow
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—241no—
378C2#580611yescharacteristic
379C2#582211yescharacteristic
380C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#550C2
socleC22×C3C22×D4
hypercenter#550D3
hypocenterC3#3309
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C3×#550C2semidirect
2. #5565C2semidirect
3. #5238C2semidirect
4. #5238C2semidirect
5. #5233C2semidirect
6. #5232C2semidirect
7. #5072C2semidirect
8. #5071C2semidirect
9. #5039C2semidirect
10. #5038C2semidirect
11. #5035C2semidirect
12. #5033C2semidirect
13. #5026C2semidirect
14. #5022C2semidirect
15. #5011C2semidirect
61. #550D3semidirect
372. C3#3309semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
#55652
#52382
#52332
#52322
#50722
#50712
#50392
#50382
#50352
#50332
#50262
#50222
#50112
#33093