Finite Group Databasebeta

← #15781566

Subgroups — #15781566

Order 3328.

612 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13118201C211yesmaximal, characteristic
2C4×#88073C211yesmaximal, characteristic
3#13112953C214yesmaximal
4C13×#1799C211yesmaximal, characteristic
5C23×Dic26C411yescharacteristic, complemented
6C2×C4×D4×C13C2211yescharacteristic
7C2×#88178—21no—
8C2×C13×#99C2212yes—
9C13×#380—21no—
10#11718881C2212yes—
11C2×#88073C2212yes—
12#11718873C411yescharacteristic, complemented
13#11718640—21no—
14#11718608—21no—
15#11718607C412yescomplemented
16#11718590—21no—
17C4×#6099—21no—
18C13×#342—21no—
19#11718122—21no—
20C22×D4×C13C2311yescharacteristic
21C23×C4×C13C2×C411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×Dic13C2×C412yes—
24C22×Dic26—21no—
25C22×Dic26—21no—
26C22×Dic26—44no—
27C22×Dic26—41no—
28C22×Dic26—41no—
29C4×D4×C13—41no—
30C4×D4×C13—41no—
31C2×C13×C4⋊C4—41no—
32C2×C13×C4⋊C4D411yescharacteristic
33C2×C13×C22⋊C4—22no—
34C2×C13×C22⋊C4C2×C411yescharacteristic
35C2×C13×C22⋊C4D411yescharacteristic
36C2×C42×C13D411yescharacteristic
37C2×C22⋊Dic13—22no—
38C2×C22⋊Dic13C2×C412yes—
39C2×C4⋊Dic13—21no—
40#88178—44no—
41C2×#6101—22no—
42C2×#6099—22no—
43C13×#99—24no—
44C13×#95—41no—
45#88073—22no—
46#88073Q814yescomplemented
47#88071—22no—
48#88071—21no—
49#88071—22no—
50#88043—41no—
51C4×#1055—41no—
52#13293—131nomaximal, Sylow
53C24×C13C22×C411yescharacteristic
54C24×C13C2×D411yescharacteristic
55C2×D4×C13—41no—
56C2×D4×C13—41no—
57C2×D4×C13C2×Q812yes—
58C2×D4×C13#4112yes—
59C22×C4×C13—21no—
60C22×C4×C13—21no—
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13—42no—
64C22×C4×C13—42no—
65C22×C4×C13—42no—
66C22×C4×C13C22⋊C411yescharacteristic
67C22×C4×C13C22⋊C411yescharacteristic
68C22×C4×C13C22⋊C411yescharacteristic
69C22×C4×C13C22⋊C411yescharacteristic
70C22×C4×C13C2×D411yescharacteristic
71C22×Dic13—22no—
72C22×Dic13—22no—
73C22×Dic13—44no—
74C22×Dic13—42no—
75C2×Dic26—24no—
76C2×Dic26—44no—
77C2×Dic26—42no—
78C2×Dic26—42no—
79C2×Dic26—48no—
80C2×Dic26—44no—
81C2×Dic26—41no—
82C2×Dic26—41no—
83C2×Dic26—44no—
84C13×C4⋊C4—41no—
85C13×C4⋊C4—82no—
86C13×C22⋊C4—41no—
87C13×C22⋊C4—41no—
88C13×C22⋊C4—42no—
89C13×C22⋊C4—42no—
90C42×C13—21no—
91C42×C13—21no—
92C22⋊Dic13—44no—
93C22⋊Dic13—42no—
94C4⋊Dic13—42no—
95#6101—44no—
96#6099—24no—
97#6099—42no—
98C2×#1055—42no—
99#2409—131no—
100#1799D1311yescharacteristic
101C4×#100—131no—
102#1390—134no—
103C23×C13—21no—
104C23×C13—21no—
105C23×C13—21no—
106C23×C13—21no—
107C23×C13—41no—
108C23×C13—41no—
109C23×C13—41no—
110C23×C13—41no—
111C23×C13C2×C22⋊C411yescharacteristic
112C23×C13C4×Q812yes—
113C23×C13#12411yescharacteristic
114C23×C13C4⋊Q811yescharacteristic
115D4×C13—42no—
116D4×C13—42no—
117C2×C4×C13—21no—
118C2×C4×C13—21no—
119C2×C4×C13—22no—
120C2×C4×C13—21no—
121C2×C4×C13—21no—
122C2×C4×C13—21no—
123C2×C4×C13—42no—
124C2×C4×C13—42no—
125C2×C4×C13—42no—
126C2×C4×C13—41no—
127C2×C4×C13—41no—
128C2×C4×C13—41no—
129C2×C4×C13—81no—
130C2×C4×C13—82no—
131C2×C4×C13—82no—
132C2×C4×C13—82no—
133C2×C4×C13#9911yescharacteristic
134C2×C4×C13#9911yescharacteristic
135C2×C4×C13C22⋊Q812yes—
136C2×Dic13—24no—
137C2×Dic13—42no—
138C2×Dic13—44no—
139C2×Dic13—42no—
140C2×Dic13—44no—
141C2×Dic13—42no—
142C2×Dic13—42no—
143C2×Dic13—42no—
144Dic26—44no—
145Dic26—48no—
146Dic26—44no—
147#1055—44no—
148C23×Q8—131no—
149C2×C22⋊Q8—261no—
150C2×C4×D4C2×D1311yescharacteristic
151C2×#100—132no—
152C2×#99C2×D1312yes—
153C4×M16—261no—
154#395—261no—
155#393—261no—
156#391—261no—
157#386—132no—
158#385—132no—
159#380—21no—
160#380—131no—
161#343—261no—
162#342—21no—
163C22×C13—21no—
164C22×C13—21no—
165C22×C13—21no—
166C22×C13—22no—
167C22×C13—21no—
168C22×C13—42no—
169C22×C13—41no—
170C22×C13—41no—
171C22×C13—42no—
172C22×C13—41no—
173C22×C13—41no—
174C22×C13—41no—
175C22×C13—41no—
176C22×C13—81no—
177C22×C13#39111yescharacteristic
178C22×C13C2×#9911yescharacteristic
179C22×C13#41011yescharacteristic
180C4×C13—21no—
181C4×C13—22no—
182C4×C13—41no—
183C4×C13—41no—
184C4×C13—41no—
185C4×C13—162no—
186C4×C13AΣL(1,𝔽16)12yes—
187Dic13—44no—
188Dic13—82no—
189C22×Q8—261no—
190C22×Q8—261no—
191C22×Q8—521no—
192C22×Q8—521no—
193C22×Q8—524no—
194C22×D4C22×D1311yescharacteristic
195C23×C4—132no—
196C23×C4C4×D1311yescharacteristic
197C23×C4C22⋊D1311yescharacteristic
198C2×M16—262no—
199C22⋊Q8—524no—
200C4×D4—41no—
201C4×D4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—262no—
204C2×C4⋊C4—261no—
205C2×C4⋊C4D5211yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—132no—
208C2×C22⋊C4—262no—
209C2×C22⋊C4C4×D1311yescharacteristic
210C2×C22⋊C4C22⋊D1311yescharacteristic
211C2×C42D5211yescharacteristic
212#100—134no—
213#100—262no—
214#99—24no—
215#97—521no—
216C4×C8—521no—
217#95—41no—
218#95—261no—
219#95—262no—
220#95—262no—
221C2×C13—21no—
222C2×C13—21no—
223C2×C13—41no—
224C2×C13—42no—
225C2×C13—41no—
226C2×C13#199611yescharacteristic
227C2×C13C2×AΣL(1,𝔽16)11yescharacteristic
228C2×C13#212311yescharacteristic
229C24C2×C4×D1311yescharacteristic
230C24C2×C22⋊D1311yescharacteristic
231C2×Q8—264no—
232C2×Q8—522no—
233C2×Q8—522no—
234C2×Q8—524no—
235C2×Q8—524no—
236C2×Q8—528no—
237C2×Q8—524no—
238C2×Q8—521no—
239C2×Q8—521no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4#612912yes—
243C2×D4Q8×D1312yes—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—262no—
252C22×C4—262no—
253C22×C4—524no—
254C22×C4—522no—
255C22×C4#610311yescharacteristic
256C22×C4#610311yescharacteristic
257C22×C4#610311yescharacteristic
258C22×C4#610311yescharacteristic
259C22×C4C2×D5211yescharacteristic
260M16—264no—
261M16—522no—
262C2×C8—522no—
263C4⋊C4—41no—
264C4⋊C4—82no—
265C4⋊C4—522no—
266C4⋊C4—524no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—524no—
272C22⋊C4—522no—
273C42—21no—
274C42—21no—
275C13#1329311yescharacteristic, complemented, Sylow
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#8814112yes—
285C23C2×#610311yescharacteristic
286C23#8819011yescharacteristic
287C23#8819811yescharacteristic
288Q8—524no—
289Q8—528no—
290Q8—524no—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—264no—
310C2×C4—522no—
311C2×C4—524no—
312C2×C4—522no—
313C2×C4—524no—
314C2×C4—522no—
315C2×C4—522no—
316C2×C4—522no—
317C2×C4#8804611yescharacteristic
318C2×C4#8804611yescharacteristic
319C2×C4#8815112yes—
320C8—524no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22C2×#8804611yescharacteristic
336C22#1171863511yescharacteristic
337C22#1171866711yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—162no—
344C4—524no—
345C4—1042no—
346C4#1171806312yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—42no—
351C2—41no—
352C2#1311582011yescharacteristic
353C2C2×#1171806311yescharacteristic
354C2#1311648911yescharacteristic
subgroupG/H
centerC22C2×#88046
commutatorC24×C13C22×C4
FrattiniC22×D4C22×D13
FittingC13×#1799C2
socleC22×C13C2×#99
prime radical, cosocleC22×D4×C13C23
hypercenter#1799D13
hypocenterC13#13293
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
5. C23×Dic26C4semidirect
12. #11718873C4semidirect
15. #11718607C4semidirect
46. #88073Q8semidirect
275. C13#13293semidirect

Maximal subgroups

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

subgroupindex
#131182012
C4×#880732
#131129532
C13×#17992
#1329313