Finite Group Databasebeta

← #80347

Subgroups — #80347

Order 384.

370 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5719C211yesmaximal, characteristic, complemented
2#5562C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5159C211yesmaximal, characteristic, complemented
5#5157C211yesmaximal, characteristic, complemented
6#5150C211yesmaximal, characteristic, complemented
7#5150C211yesmaximal, characteristic, complemented
8#5072C211yesmaximal, characteristic, complemented
9#5009C211yesmaximal, characteristic, complemented
10#4847C211yesmaximal, characteristic, complemented
11#4839C211yesmaximal, characteristic, complemented
12#4782C211yesmaximal, characteristic, complemented
13#4775C211yesmaximal, characteristic, complemented
14#4766C211yesmaximal, characteristic, complemented
15#4763C211yesmaximal, characteristic, complemented
16#3259—31nomaximal, Sylow
17C2×C3×D8C2211yescharacteristic
18C2×C3×D8C2211yescharacteristic
19C3×#127C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21#940C2211yescharacteristic
22#939—21no—
23#939—21no—
24#938C2211yescharacteristic
25#938C2211yescharacteristic
26#938C2211yescharacteristic
27#935—21no—
28#935—21no—
29D4×Dic3C2211yescharacteristic
30D4×Dic3C2211yescharacteristic
31C2×#213C2211yescharacteristic
32C2×#213C2211yescharacteristic
33#930—21no—
34#929—21no—
35C2×ΓL(1,𝔽25)C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37#895C2211yescharacteristic
38#890C2211yescharacteristic
39#884—21no—
40#883—21no—
41C4×D12C2211yescharacteristic
42#872C2211yescharacteristic
43#870C2211yescharacteristic
44C3×C4⋊C8C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53#818C2211yescharacteristic
54#817C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#540—31no—
61#540—31no—
62#478—31no—
63#478—31no—
64#478—31no—
65#478—31no—
66#469—31no—
67#469—31no—
68#466—31no—
69#466—31no—
70#459—31no—
71#459—31no—
72#459—31no—
73#459D311yescharacteristic, complemented
74#424—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—
86C22×Dic3—21no—
87C22×Dic3—21no—
88C2×D12—21no—
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×C4×D3C2311yescharacteristic
92C3×D8—41no—
93C3×D8—41no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C22⋊C4—21no—
98C3×C42C2311yescharacteristic
99#217—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#217—21no—
104#213—41no—
105#213—41no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4D411yescharacteristic
109(C2×D3)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4D411yescharacteristic
118Dic3⋊C4D411yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122ΓL(1,𝔽25)—41no—
123D3×C8—41no—
124C2×D8—31no—
125C2×D8—31no—
126C2×D8C2×D311yescharacteristic
127C2×D8C2×D311yescharacteristic
128C2×M16—31no—
129C22×C8—31no—
130#127—31no—
131#127C2×D311yescharacteristic
132#125—31no—
133#125—31no—
134#123—61no—
135#123—61no—
136#123—61no—
137#123—61no—
138#121—31no—
139#121—31no—
140#121—31no—
141#121—31no—
142#121—61no—
143#121—61no—
144#121—61no—
145#121—61no—
146C4×D4—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4C2×D311yescharacteristic
150#117—31no—
151#107—31no—
152#107—31no—
153#106—31no—
154#106—31no—
155C4⋊C8—31no—
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8C2×D311yescharacteristic
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C22×D3—21no—
171C22×D3C2×D411yescharacteristic
172C3×D4—21no—
173C3×D4—21no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—21no—
177C3×D4—21no—
178C3×D4—81no—
179C3×D4—81no—
180C2×C3×C4—21no—
181C2×C3×C4—81no—
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185#67—81no—
186#67—81no—
187#67—81no—
188#67—81no—
189#67—81no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—81no—
194C2×Dic3—81no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198D12—21no—
199D12—21no—
200D12—81no—
201C4×D3—21no—
202C4×D3—21no—
203C4×D3—81no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—31no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4C22×D311yescharacteristic
218C2×D4C22×D311yescharacteristic
219C2×D4C22×D311yescharacteristic
220C22×C4—21no—
221C22×C4—31no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225D8—41no—
226D8—41no—
227D8—121no—
228D8—121no—
229M16—121no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—121no—
233C2×C8C22×D311yescharacteristic
234C2×C8C22×D311yescharacteristic
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—31no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4C22×D311yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—31no—
247C22⋊C4—31no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C42—31no—
256C42C22×D311yescharacteristic
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3C22×D411yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—21no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4H(5,𝔽2)11yescharacteristic
277C3×C42−1+411yescharacteristic
278Dic3—21no—
279Dic3—21no—
280Dic3—21no—
281Dic3—41no—
282Dic3—41no—
283Dic3—41no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—31no—
288C23—61no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—61no—
296D4—61no—
297D4—81no—
298D4—81no—
299D4—241no—
300D4—241no—
301D4—241no—
302D4—241no—
303D4—241no—
304D4—241no—
305C2×C4—21no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—81no—
315C2×C4—241no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4D3×D411yescharacteristic
319C2×C4D3×D411yescharacteristic
320C2×C4C23×D311yescharacteristic
321C8—21no—
322C8—21no—
323C8—61no—
324C8—61no—
325C2×C3—81no—
326C2×C3—81no—
327C2×C3—81no—
328C2×C3#53711yescharacteristic
329C2×C3#57611yescharacteristic
330C2×C3#57611yescharacteristic
331D3—41no—
332D3—81no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—61no—
343C22—61no—
344C22—61no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22C2×D3×D411yescharacteristic
349C4—21no—
350C4—21no—
351C4—41no—
352C4—61no—
353C4—61no—
354C4—61no—
355C4—121no—
356C4—121no—
357C4—121no—
358C4#100411yescharacteristic
359C4#101011yescharacteristic
360C3#325911yescharacteristic, complemented, Sylow
361C2—81no—
362C2—81no—
363C2—81no—
364C2—121no—
365C2—241no—
366C2#560811yescharacteristic
367C2#575711yescharacteristic
368C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3259
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. #5719C2semidirect
2. #5562C2semidirect
3. C3×#459C2semidirect
4. #5159C2semidirect
5. #5157C2semidirect
6. #5150C2semidirect
7. #5150C2semidirect
8. #5072C2semidirect
9. #5009C2semidirect
10. #4847C2semidirect
11. #4839C2semidirect
12. #4782C2semidirect
13. #4775C2semidirect
14. #4766C2semidirect
15. #4763C2semidirect
73. #459D3semidirect
360. C3#3259semidirect

Maximal subgroups

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

subgroupindex
#57192
#55622
C3×#4592
#51592
#51572
#51502
#50722
#50092
#48472
#48392
#47822
#47752
#47662
#47632
#32593