Finite Group Databasebeta

← #16023122

Subgroups — #16023122

Order 3328.

2414 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117785C216yesmaximal, complemented
2Q8×C4⋊Dic13C212yesmaximal, complemented
3#13117729C216yesmaximal, complemented
4#13117677C212yesmaximal
5#13117479C216yesmaximal, complemented
6#13117243C213yesmaximal, complemented
7#13117234C213yesmaximal, complemented
8#13117201C212yesmaximal, complemented
9C2×C4×C13×#41C211yesmaximal, characteristic
10C22×C13×#41C2211yescharacteristic
11C4×C13×#41C4116yescomplemented
12C2×C4×Q8×C13C2211yescharacteristic
13C2×C4×Q8×C13C2211yescharacteristic
14C2×C4×D4×C13C2213yes—
15C2×C4×D4×C13C2213yes—
16C2×C13×#117C2213yes—
17C22×C42×C13C2213yes—
18C2×Q8×Dic13C2214yes—
19C2×#88180C2216yes—
20#11718882C22112yes—
21#11718873C2216yes—
22#11718873C2216yes—
23#11718873C2216yes—
24#11718872C2216yes—
25C4×C22⋊Dic13C2216yes—
26#11718646C22112yes—
27C4⋊C4×Dic13C22112yes—
28#11718612C22124yes—
29#11718598C2216yes—
30#11718597C2211yescharacteristic
31#11718597C2213yes—
32#11718596C2216yes—
33C4×C4⋊Dic13C2216yes—
34C4×C4⋊Dic13C2216yes—
35C4×C4⋊Dic13C2216yes—
36#11718593C2216yes—
37C2×C13×#41—28no—
38C2×C13×#41C2×C418yes—
39C22×Q8×C13C2311yescharacteristic
40C22×D4×C13C2313yes—
41C23×C4×C13C2313yes—
42C23×C4×C13C2313yes—
43C23×C4×C13C2313yes—
44C4×Q8×C13C2×C418yes—
45C4×Q8×C13C2×C418yes—
46C4×D4×C13C2×C4124yes—
47C4×D4×C13C2×C4124yes—
48C13×#117C2×C4124yes—
49C2×C13×C4⋊C4C2313yes—
50C2×C13×C4⋊C4C2313yes—
51C2×C13×C22⋊C4C2316yes—
52C2×C42×C13C2×C4124yes—
53C2×C42×C13C2311yescharacteristic
54C2×C42×C13C2313yes—
55C2×C42×C13C2313yes—
56C2×C42×C13C2313yes—
57C2×C22⋊Dic13C23112yes—
58C2×C22⋊Dic13C23112yes—
59Q8×Dic13—216no—
60#88180—412no—
61C2×C4⋊Dic13C2316yes—
62C2×C4⋊Dic13C23112yes—
63C2×C4⋊Dic13C2312yes—
64C2×C4⋊Dic13C23112yes—
65C2×C4×Dic13C23112yes—
66C2×C4×Dic13C23112yes—
67#88071C2312yes—
68#88071C2316yes—
69#88071C23112yes—
70#88071C2316yes—
71#88071C23112yes—
72#88071C2312yes—
73#34193—131nomaximal, Sylow
74C24×C13C2413yes—
75C13×#41—232no—
76C2×Q8×C13C22×C414yes—
77C2×Q8×C13#4118yes—
78C2×D4×C13—212no—
79C2×D4×C13C22×C4112yes—
80C22×C4×C13—212no—
81C22×C4×C13—26no—
82C22×C4×C13—26no—
83C22×C4×C13—46no—
84C22×C4×C13C22×C4112yes—
85C22×C4×C13C22×C4112yes—
86C22×C4×C13C22×C4112yes—
87C22×C4×C13C2411yescharacteristic
88C22×C4×C13C2411yescharacteristic
89C22×C4×C13C2413yes—
90C22×C4×C13C2413yes—
91C22×C4×C13C2413yes—
92C22×C4×C13C2411yescharacteristic
93C22×Dic13C2414yes—
94C22×Dic13C24112yes—
95C13×C4⋊C4C22×C4112yes—
96C13×C4⋊C4C22×C4112yes—
97C13×C22⋊C4C22×C4124yes—
98C42×C13C22×C414yes—
99C42×C13C22×C4112yes—
100C42×C13C22×C4112yes—
101C42×C13C22×C4112yes—
102C22⋊Dic13—412no—
103C22⋊Dic13—412no—
104C4⋊Dic13—224no—
105C4⋊Dic13—212no—
106C4⋊Dic13—24no—
107C4⋊Dic13—412no—
108C4×Dic13—224no—
109C4×Dic13—412no—
110C2×C4×#41D1311yescharacteristic
111#2354—136no—
112Q8×C4⋊C4—132no—
113#2347—136no—
114#2341—132no—
115#2319—136no—
116#2285—133no—
117#2283—133no—
118#2276—132no—
119C23×C13—26no—
120C23×C13—23no—
121C23×C13—23no—
122C23×C13—26no—
123C23×C13C23×C416yes—
124C23×C13C2511yescharacteristic
125Q8×C13#117116yescomplemented
126D4×C13—224no—
127C2×C4×C13—22no—
128C2×C4×C13—23no—
129C2×C4×C13—23no—
130C2×C4×C13—26no—
131C2×C4×C13—22no—
132C2×C4×C13—224no—
133C2×C4×C13—224no—
134C2×C4×C13—412no—
135C2×C4×C13C23×C412yes—
136C2×C4×C13C23×C412yes—
137C2×C4×C13C23×C416yes—
138C2×C4×C13C23×C416yes—
139C2×C4×C13C23×C416yes—
140C2×C4×C13C23×C412yes—
141C2×C4×C13C2×#41112yes—
142C2×C4×C13H(5,𝔽2)12yes—
143C2×C4×C132−1+412yes—
144C2×Dic13—24no—
145C2×Dic13—212no—
146C2×Dic13—212no—
147C2×Dic13—24no—
148C2×Dic13—212no—
149C2×Dic13—24no—
150C22×#41C2×D1311yescharacteristic
151C4×#41Dic13116yescomplemented
152C2×C4×Q8—134no—
153C2×C4×Q8C2×D1311yescharacteristic
154C2×C4×Q8C2×D1311yescharacteristic
155C2×C4×D4C2×D1313yes—
156C2×C4×D4C2×D1313yes—
157C2×#117—136no—
158C2×#117C2×D1313yes—
159C22×C42C2×D1313yes—
160#390—1312no—
161#389—1312no—
162#388—1324no—
163#384—136no—
164#383—136no—
165#382—131no—
166#382—133no—
167#381—136no—
168#380—136no—
169#380—136no—
170#380—136no—
171C4×C4⋊C4—1312no—
172C4×C4⋊C4—136no—
173C4×C4⋊C4—136no—
174C4×C4⋊C4—136no—
175C4×C22⋊C4—136no—
176#376—136no—
177C22×C13—26no—
178C22×C13—26no—
179C22×C13—26no—
180C22×C13—46no—
181C22×C13—46no—
182C22×C13C24×C411yescharacteristic
183C22×C13C22×#4112yes—
184C22×C13C2×H(5,𝔽2)11yescharacteristic
185C22×C13C2×2−1+411yescharacteristic
186C22×C13#58512yes—
187C4×C13—22no—
188C4×C13—26no—
189C4×C13—22no—
190C4×C13—26no—
191C4×C13C2×#117112yes—
192C4×C13#52014yes—
193Dic13—412no—
194Dic13—44no—
195C2×#41—28no—
196C2×#41C2×Dic1318yes—
197C22×Q8C22×D1311yescharacteristic
198C22×D4C22×D1313yes—
199C23×C4C22×D1313yes—
200C23×C4C22×D1313yes—
201C23×C4C22×D1313yes—
202C4×Q8—2616no—
203C4×Q8C2×Dic1318yes—
204C4×Q8C2×Dic1318yes—
205C4×D4C2×Dic13124yes—
206C4×D4C2×Dic13124yes—
207#117—5212no—
208#117C2×Dic13124yes—
209C2×C4⋊C4—136no—
210C2×C4⋊C4—1312no—
211C2×C4⋊C4—132no—
212C2×C4⋊C4—1312no—
213C2×C4⋊C4C22×D1313yes—
214C2×C4⋊C4C22×D1313yes—
215C2×C22⋊C4—1312no—
216C2×C22⋊C4—1312no—
217C2×C22⋊C4C22×D1316yes—
218C2×C42—1312no—
219C2×C42—1312no—
220C2×C42C2×Dic13124yes—
221C2×C42C22×D1313yes—
222C2×C42C22×D1313yes—
223C2×C42C22×D1313yes—
224C2×C42C22×D1311yescharacteristic
225#95—132no—
226#95—136no—
227#95—1312no—
228#95—132no—
229#95—136no—
230#95—1312no—
231C2×C13—46no—
232C2×C13C22×#11711yescharacteristic
233C2×C13C2×#52011yescharacteristic
234C2×C13#342811yescharacteristic
235C2×C13#351712yes—
236C2×C13#351812yes—
237C24C23×D1313yes—
238#41—232no—
239C2×Q8#612718yes—
240C2×Q8C22×Dic1314yes—
241C2×D4—212no—
242C2×D4C22×Dic13112yes—
243C22×C4—212no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—46no—
247C22×C4—134no—
248C22×C4—1312no—
249C22×C4C22×Dic13112yes—
250C22×C4C22×Dic13112yes—
251C22×C4C22×Dic13112yes—
252C22×C4C23×D1311yescharacteristic
253C22×C4C23×D1311yescharacteristic
254C22×C4C23×D1313yes—
255C22×C4C23×D1313yes—
256C22×C4C23×D1313yes—
257C22×C4C23×D1311yescharacteristic
258C4⋊C4—2624no—
259C4⋊C4—2612no—
260C4⋊C4—264no—
261C4⋊C4—5212no—
262C4⋊C4C22×Dic13112yes—
263C4⋊C4C22×Dic13112yes—
264C22⋊C4—5212no—
265C22⋊C4—5212no—
266C22⋊C4C22×Dic13124yes—
267C42—2624no—
268C42—5212no—
269C42C22×Dic1314yes—
270C42C22×Dic13112yes—
271C42C22×Dic13112yes—
272C42C22×Dic13112yes—
273C13#3419311yescharacteristic, complemented, Sylow
274C23—26no—
275C23—23no—
276C23—23no—
277C23—26no—
278C23C23×Dic1316yes—
279C23C24×D1311yescharacteristic
280Q8#88180116yescomplemented
281D4—224no—
282C2×C4—22no—
283C2×C4—23no—
284C2×C4—23no—
285C2×C4—26no—
286C2×C4—22no—
287C2×C4—224no—
288C2×C4—224no—
289C2×C4—412no—
290C2×C4—264no—
291C2×C4—2612no—
292C2×C4—2612no—
293C2×C4—264no—
294C2×C4—264no—
295C2×C4—2612no—
296C2×C4C2×#6127112yes—
297C2×C4#8825612yes—
298C2×C4#8825712yes—
299C2×C4C23×Dic1312yes—
300C2×C4C23×Dic1312yes—
301C2×C4C23×Dic1316yes—
302C2×C4C23×Dic1316yes—
303C2×C4C23×Dic1316yes—
304C2×C4C23×Dic1312yes—
305C22—26no—
306C22—26no—
307C22—26no—
308C22—46no—
309C22—46no—
310C22C22×#612712yes—
311C22C2×#8825611yescharacteristic
312C22C2×#8825711yescharacteristic
313C22#1171964812yes—
314C22C24×Dic1311yescharacteristic
315C4—22no—
316C4—26no—
317C4—22no—
318C4—26no—
319C4—5212no—
320C4—524no—
321C4C2×#88180112yes—
322C4#1171953214yes—
323C2—46no—
324C2C22×#8818011yescharacteristic
325C2C2×#1171953211yescharacteristic
326C2#1312803911yescharacteristic
327C2#1312884212yes—
328C2#1312885312yes—
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingC2×C4×C13×#41C2
socle, cosocleC23×C13C25
prime radicalC22×C13×#41C22
hypercenterC2×C4×#41D13
hypocenterC13#34193
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #13117785C2semidirect
2. Q8×C4⋊Dic13C2semidirect
3. #13117729C2semidirect
5. #13117479C2semidirect
6. #13117243C2semidirect
7. #13117234C2semidirect
8. #13117201C2semidirect
11. C4×C13×#41C4semidirect
125. Q8×C13#117semidirect
151. C4×#41Dic13semidirect
273. C13#34193semidirect
280. Q8#88180semidirect

Maximal subgroups

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

subgroupindex
#131177852
Q8×C4⋊Dic132
#131177292
#131176772
#131174792
#131172432
#131172342
#131172012
C2×C4×C13×#412
#3419313