Finite Group Databasebeta

← #80346

Subgroups — #80346

Order 384.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5719C211yesmaximal, characteristic, complemented
2#5561C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5159C211yesmaximal, characteristic, complemented
5#5157C211yesmaximal, characteristic, complemented
6#5153C211yesmaximal, characteristic, complemented
7#5153C211yesmaximal, characteristic, complemented
8#5075C211yesmaximal, characteristic, complemented
9#5009C211yesmaximal, characteristic, complemented
10#4839C211yesmaximal, characteristic, complemented
11#4829C211yesmaximal, characteristic, complemented
12#4780C211yesmaximal, characteristic, complemented
13#4778C211yesmaximal, characteristic, complemented
14#4761C211yesmaximal, characteristic, complemented
15#4759C211yesmaximal, characteristic, complemented
16#3257—31nomaximal, Sylow
17C2×C3×D8C2211yescharacteristic
18C2×C3×D8C2211yescharacteristic
19C3×#127C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21#939—21no—
22#939—21no—
23#938C2211yescharacteristic
24#938C2211yescharacteristic
25#938C2211yescharacteristic
26#936C2211yescharacteristic
27#935—21no—
28#935—21no—
29D4×Dic3C2211yescharacteristic
30D4×Dic3C2211yescharacteristic
31C2×#214C2211yescharacteristic
32C2×#214C2211yescharacteristic
33#929—21no—
34#924—21no—
35C2×ΓL(1,𝔽25)C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37#897C2211yescharacteristic
38#888C2211yescharacteristic
39#885—21no—
40#879—21no—
41#872C2211yescharacteristic
42#870C2211yescharacteristic
43C4×Dic6C2211yescharacteristic
44C3×C4⋊C8C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#816C2211yescharacteristic
54#816C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#542—31no—
61#540—31no—
62#479—31no—
63#478—31no—
64#476—31no—
65#474—31no—
66#469—31no—
67#468—31no—
68#466—31no—
69#465—31no—
70#463—31no—
71#463—31no—
72#460—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—
84C22×Dic3—21no—
85C22×Dic3—21no—
86C2×C4×D3C2311yescharacteristic
87C2×Dic6—21no—
88C2×Dic6C2311yescharacteristic
89C3×D8—41no—
90C3×D8—41no—
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C22⋊C4—21no—
95C3×C42C2311yescharacteristic
96#217—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#214—41no—
104#214—41no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4D411yescharacteristic
107(C2×D3)⋊C4D411yescharacteristic
108C4⋊Dic3—21no—
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122ΓL(1,𝔽25)—41no—
123D3×C8—41no—
124C2×SD16—31no—
125C2×SD16—31no—
126C2×D8C2×D311yescharacteristic
127C2×D8C2×D311yescharacteristic
128C2×M16—31no—
129C22×C8—31no—
130C4⋊Q8—31no—
131#127C2×D311yescharacteristic
132#126—61no—
133#125—31no—
134#124—31no—
135#124—61no—
136#123—61no—
137#123—61no—
138#123—61no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141#121—31no—
142#121—31no—
143#121—31no—
144#121—61no—
145#121—61no—
146C4×Q8—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#103—31no—
160#103—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×D3C2×D411yescharacteristic
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—21no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—21no—
177C3×D4—81no—
178C3×D4—81no—
179C2×C3×C4—21no—
180C2×C3×C4—81no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184#67—81no—
185#67—81no—
186#67—81no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C4×D3—21no—
197C4×D3—21no—
198Dic6—21no—
199Dic6—21no—
200Dic6—81no—
201C3×C8—21no—
202C3×C8—21no—
203C3⋊C8—21no—
204C3⋊C8—21no—
205C2×Q8—31no—
206C2×Q8—61no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—61no—
210C2×D4—61no—
211C2×D4—61no—
212C2×D4C22×D311yescharacteristic
213C2×D4C22×D311yescharacteristic
214C2×D4C22×D311yescharacteristic
215C22×C4—21no—
216C22×C4—31no—
217C22×C4—61no—
218C22×C4—61no—
219SD16—121no—
220SD16—121no—
221D8—41no—
222D8—41no—
223M16—121no—
224C2×C8—31no—
225C2×C8—31no—
226C2×C8—121no—
227C2×C8C22×D311yescharacteristic
228C2×C8C22×D311yescharacteristic
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4C22×D311yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—31no—
242C22⋊C4—31no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C42—31no—
252C42—61no—
253C42C22×D311yescharacteristic
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3C22×D411yescharacteristic
264C2×D3—21no—
265C2×D3—21no—
266C2×D3—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4H(5,𝔽2)11yescharacteristic
271C3×C42−1+411yescharacteristic
272Dic3—21no—
273Dic3—21no—
274Dic3—21no—
275Dic3—41no—
276Dic3—41no—
277Dic3—41no—
278Dic3—41no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—31no—
283Q8—61no—
284Q8—61no—
285Q8—241no—
286D4—21no—
287D4—21no—
288D4—21no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—81no—
294D4—241no—
295D4—241no—
296D4—241no—
297C2×C4—21no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—81no—
308C2×C4—241no—
309C2×C4—241no—
310C2×C4D3×D411yescharacteristic
311C2×C4D3×D411yescharacteristic
312C2×C4C23×D311yescharacteristic
313C8—21no—
314C8—21no—
315C8—61no—
316C8—61no—
317C2×C3—81no—
318C2×C3—81no—
319C2×C3—81no—
320C2×C3#53711yescharacteristic
321C2×C3#57611yescharacteristic
322C2×C3#57711yescharacteristic
323D3—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—61no—
334C22—61no—
335C22—61no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—21no—
339C4—41no—
340C4—61no—
341C4—61no—
342C4—61no—
343C4—121no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4#100411yescharacteristic
348C4#101011yescharacteristic
349C3#325711yescharacteristic, complemented, Sylow
350C2—81no—
351C2—81no—
352C2—81no—
353C2—121no—
354C2#560811yescharacteristic
355C2#575711yescharacteristic
356C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3257
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. #5561C2semidirect
3. C3×#459C2semidirect
4. #5159C2semidirect
5. #5157C2semidirect
6. #5153C2semidirect
7. #5153C2semidirect
8. #5075C2semidirect
9. #5009C2semidirect
10. #4839C2semidirect
11. #4829C2semidirect
12. #4780C2semidirect
13. #4778C2semidirect
14. #4761C2semidirect
15. #4759C2semidirect
73. #459D3semidirect
349. C3#3257semidirect

Maximal subgroups

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

subgroupindex
#57192
#55612
C3×#4592
#51592
#51572
#51532
#50752
#50092
#48392
#48292
#47802
#47782
#47612
#47592
#32573