Finite Group Databasebeta

← #15867397

Subgroups — #15867397

Order 3328.

1162 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719175C211yesmaximal, characteristic
2C2×#11718780C211yesmaximal, characteristic
3C2×#11718783C211yesmaximal, characteristic
4#13115774C214yesmaximal, complemented
5#13115770C214yesmaximal, complemented
6#13114534C211yesmaximal, characteristic
7#13114534C211yesmaximal, characteristic
8C2×#11718116C211yesmaximal, characteristic
9C13×#1728C211yesmaximal, characteristic
10C22×C13×M16C2211yescharacteristic
11C22×C13×M16C2211yescharacteristic
12C22×C42×C13C2211yescharacteristic
13C2×#88180C2211yescharacteristic
14C2×#88180C2211yescharacteristic
15C2×#88178C414yes—
16C23×#1055C2211yescharacteristic
17C23×#1055C2211yescharacteristic
18#11719175—24no—
19#11719175—24no—
20C2×#88124C412yes—
21C2×#88123C412yes—
22C2×C4×Dic26C2214yes—
23#11718783—24no—
24#11718780—24no—
25C2×#88055C2212yes—
26C2×#88054C2214yes—
27C2×C8×Dic13C2212yes—
28C13×#336C2214yes—
29#11718142C2212yes—
30#11718142C2214yes—
31#11718142C2212yes—
32#11718116C2212yes—
33#11718116C2212yes—
34C23×C4×C13C2×C412yes—
35C23×C4×C13C2311yescharacteristic
36C22×Dic26C2×C414yes—
37C2×C13×M16—22no—
38C2×C13×M16—22no—
39C2×C13×M16—22no—
40C2×C13×M16—22no—
41C2×C13×M16—22no—
42C2×C13×M16Q814yescomplemented
43C22×C8×C13C2312yes—
44C22×C8×C13C2312yes—
45C2×C42×C13—22no—
46C2×C42×C13—22no—
47C2×C42×C13C2×C411yescharacteristic
48C2×C42×C13C2×C411yescharacteristic
49C2×C42×C13C2312yes—
50C2×C22⋊Dic13C2×C412yes—
51C2×C22⋊Dic13C2×C412yes—
52#88180—22no—
53#88180—22no—
54#88180D414yes—
55#88180D414yes—
56C2×C4⋊Dic13C2×C414yes—
57C2×C4⋊Dic13C2×C412yes—
58C2×C4⋊Dic13C2×C412yes—
59#88178—216no—
60C2×#6101C2×C418yes—
61C2×C4×Dic13C2312yes—
62C2×C4×Dic13C2312yes—
63C22×#1055—22no—
64C22×#1055—22no—
65C22×#1055—22no—
66C22×#1055—22no—
67C22×#1055—22no—
68C22×#1055Q814yescomplemented
69C22×#1055C2312yes—
70C22×#1055C2311yescharacteristic
71C22×#1055C2311yescharacteristic
72#88124—28no—
73#88123—28no—
74C4×Dic26—28no—
75C4×Dic26—28no—
76#88055—24no—
77#88054—28no—
78C8×Dic13—24no—
79#21304—131nomaximal, Sylow
80C24×C13C22×C411yescharacteristic
81C22×C4×C13—24no—
82C22×C4×C13—24no—
83C22×C4×C13—42no—
84C22×C4×C13C22×C412yes—
85C22×C4×C13C22×C412yes—
86C22×C4×C13C22×C411yescharacteristic
87C22×C4×C13C2×D412yes—
88C22×C4×C13C2×D412yes—
89C22×C4×C13C2×Q812yes—
90C22×C4×C13C2×Q812yes—
91C22×C4×C13#4112yes—
92C22×C4×C13#4112yes—
93C22×C4×C13C2411yescharacteristic
94C22×Dic13C22×C414yes—
95C22×Dic13C22×C414yes—
96C2×Dic26—28no—
97C2×Dic26—28no—
98C2×Dic26—44no—
99C13×M16—28no—
100C13×M16—44no—
101C2×C8×C13—22no—
102C2×C8×C13—22no—
103C2×C8×C13—22no—
104C2×C8×C13—22no—
105C2×C8×C13—22no—
106C2×C8×C13C2×Q814yes—
107C42×C13—22no—
108C42×C13—22no—
109C42×C13—22no—
110C42×C13—22no—
111C22⋊Dic13—24no—
112C22⋊Dic13C22⋊C418yes—
113C4⋊Dic13—24no—
114C4⋊Dic13—28no—
115C4⋊Dic13C22⋊C418yes—
116#6101—216no—
117C4×Dic13—22no—
118C4×Dic13—22no—
119C4×Dic13C2×D414yes—
120C4×Dic13C2×D414yes—
121C2×#1055—28no—
122C2×#1055—21no—
123C2×#1055—21no—
124C2×#1055—21no—
125C2×#1055—21no—
126C2×#1055—22no—
127C2×#1055—22no—
128C2×#1055—22no—
129C2×#1055—44no—
130C2×#1055C2×Q812yes—
131C2×#1055C2×Q812yes—
132C2×#533—131no—
133C2×#424—131no—
134C2×#405—131no—
135#1985—134no—
136#1984—134no—
137#1729—131no—
138#1729—131no—
139#1728D1311yescharacteristic
140C2×#336—131no—
141C23×C13—22no—
142C23×C13—22no—
143C23×C13C2×C22⋊C412yes—
144C23×C13C4×Q812yes—
145C23×C13C4×Q812yes—
146C23×C13C23×C411yescharacteristic
147C2×C4×C13—22no—
148C2×C4×C13—22no—
149C2×C4×C13—22no—
150C2×C4×C13—22no—
151C2×C4×C13—22no—
152C2×C4×C13—48no—
153C2×C4×C13C2×C22⋊C412yes—
154C2×C4×C13C4×Q812yes—
155C2×C4×C13C4×Q812yes—
156C2×C4×C13C22⋊Q814yes—
157C2×C4×C13C22⋊Q814yes—
158C2×C4×C13#12414yes—
159C2×C4×C13C4⋊Q814yes—
160C2×C4×C13#13114yes—
161C2×C4×C13C22×D411yescharacteristic
162C2×C4×C13C22×D411yescharacteristic
163C2×C4×C13C22×Q811yescharacteristic
164C2×C4×C13C22×Q811yescharacteristic
165C2×C4×C13C2×#4111yescharacteristic
166C2×C4×C13C2×#4111yescharacteristic
167C2×Dic13—24no—
168C2×Dic13—24no—
169C2×Dic13—24no—
170C2×Dic13—24no—
171C2×Dic13—24no—
172C2×Dic13C2×C22⋊C418yes—
173Dic26—416no—
174C8×C13—24no—
175C8×C13—42no—
176#1055—22no—
177#1055—22no—
178#1055—42no—
179C22×M16C2×D1311yescharacteristic
180C22×M16C2×D1311yescharacteristic
181C23×C8—131no—
182C23×C8—131no—
183#533—264no—
184#533—264no—
185C2×C4⋊Q8—132no—
186C2×#125—132no—
187C2×C22⋊Q8—134no—
188C2×C4×Q8—134no—
189C2×#117—131no—
190C2×#117—131no—
191C22×C42C2×D1311yescharacteristic
192#424—264no—
193C2×C4⋊C8—134no—
194#405—264no—
195C2×#97—132no—
196C2×C4×C8—132no—
197#336—132no—
198#336—134no—
199#336—132no—
200#336—132no—
201#336—132no—
202#336C2×D1314yes—
203C22×C13—24no—
204C22×C13—22no—
205C22×C13—22no—
206C22×C13—24no—
207C22×C13#39114yes—
208C22×C13C22×C22⋊C411yescharacteristic
209C22×C13C2×C4×Q811yescharacteristic
210C22×C13C2×C4×Q811yescharacteristic
211C22×C13C2×#13112yes—
212C22×C13#56812yes—
213C4×C13—24no—
214C4×C13—42no—
215C4×C13#39114yes—
216C4×C13C2×C22⋊Q811yescharacteristic
217C4×C13C2×C22⋊Q811yescharacteristic
218C4×C13C2×#12411yescharacteristic
219C4×C13C2×C4⋊Q811yescharacteristic
220Dic13—28no—
221Dic13—44no—
222C22×Q8—134no—
223C23×C4C4×D1312yes—
224C23×C4C22×D1311yescharacteristic
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—22no—
229C2×M16—22no—
230C2×M16Dic2614yescomplemented
231C22×C8—134no—
232C22×C8—131no—
233C22×C8—131no—
234C22×C8—132no—
235C22×C8—262no—
236C22×C8—262no—
237C22×C8—262no—
238C22×C8—262no—
239C22×C8—262no—
240C22×C8C22×D1312yes—
241C22×C8C22×D1312yes—
242C4⋊Q8—268no—
243#125—268no—
244C22⋊Q8—2616no—
245C4×Q8—268no—
246C4×Q8—268no—
247#117—134no—
248#117—134no—
249#117—262no—
250#117—262no—
251C2×C4⋊C4—132no—
252C2×C4⋊C4—132no—
253C2×C4⋊C4—138no—
254C2×C4⋊C4—134no—
255C2×C22⋊C4—132no—
256C2×C22⋊C4—132no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—132no—
260C2×C42—132no—
261C2×C42C4×D1311yescharacteristic
262C2×C42C4×D1311yescharacteristic
263C2×C42C22×D1312yes—
264C4⋊C8—268no—
265#97—264no—
266C4×C8—264no—
267C2×C13—24no—
268C2×C13C2×#39111yescharacteristic
269C2×C13#288012yes—
270C2×C13#296011yescharacteristic
271C2×C13#296111yescharacteristic
272C2×C13#296212yes—
273C24C2×C4×D1311yescharacteristic
274C2×Q8—268no—
275C2×Q8—268no—
276C2×Q8—524no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—42no—
280C22×C4—134no—
281C22×C4—134no—
282C22×C4C2×Dic2612yes—
283C22×C4C2×C4×D1312yes—
284C22×C4C2×C4×D1312yes—
285C22×C4C2×C4×D1311yescharacteristic
286C22×C4#612712yes—
287C22×C4D4×D1312yes—
288C22×C4D4×D1312yes—
289C22×C4#612912yes—
290C22×C4Q8×D1312yes—
291C22×C4C23×D1311yescharacteristic
292M16—28no—
293M16—44no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—132no—
300C2×C8—132no—
301C2×C8—268no—
302C2×C8—261no—
303C2×C8—261no—
304C2×C8—261no—
305C2×C8—261no—
306C2×C8—262no—
307C2×C8—262no—
308C2×C8—262no—
309C2×C8—524no—
310C2×C8C2×Dic2614yes—
311C4⋊C4—138no—
312C4⋊C4—2616no—
313C4⋊C4—268no—
314C4⋊C4—264no—
315C22⋊C4—138no—
316C22⋊C4—264no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—134no—
322C42—134no—
323C42—262no—
324C42—262no—
325C13#2130411yescharacteristic, complemented, Sylow
326C23—22no—
327C23—22no—
328C23C4×Dic2612yes—
329C23C22⋊C4×D1312yes—
330C23#8814112yes—
331C23C22×C4×D1311yescharacteristic
332Q8—5216no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—48no—
339C2×C4—138no—
340C2×C4—264no—
341C2×C4—264no—
342C2×C4—264no—
343C2×C4—264no—
344C2×C4—264no—
345C2×C4C4×Dic2612yes—
346C2×C4#8813214yes—
347C2×C4C22⋊C4×D1312yes—
348C2×C4#8813914yes—
349C2×C4#8814112yes—
350C2×C4#8814214yes—
351C2×C4#8815014yes—
352C2×C4#8816114yes—
353C2×C4C22×Dic2611yescharacteristic
354C2×C4C2×#612711yescharacteristic
355C2×C4C2×D4×D1311yescharacteristic
356C2×C4C2×D4×D1311yescharacteristic
357C2×C4C2×#612911yescharacteristic
358C2×C4C2×Q8×D1311yescharacteristic
359C8—24no—
360C8—42no—
361C8—262no—
362C8—262no—
363C8—522no—
364C22—22no—
365C22—22no—
366C22—24no—
367C22—24no—
368C22#1171830714yes—
369C22C2×C4×Dic2611yescharacteristic
370C22C2×C22⋊C4×D1311yescharacteristic
371C22C2×#8814111yescharacteristic
372C22C2×#8816112yes—
373C22#1171945012yes—
374C4—24no—
375C4—42no—
376C4—268no—
377C4—524no—
378C4#1171830714yes—
379C4C2×#8813211yescharacteristic
380C4C2×#8813911yescharacteristic
381C4C2×#8814211yescharacteristic
382C4C2×#8815011yescharacteristic
383C2—24no—
384C2C2×#1171830711yescharacteristic
385C2#1312328612yes—
386C2#1312376611yescharacteristic
387C2#1312377311yescharacteristic
388C2#1312378312yes—
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC13×#1728C2
prime radicalC24×C13C22×C4
cosocleC22×C4×C13C24
hypercenter#1728D13
hypocenterC13#21304
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ #1728

Complements

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

NH ≅ G/Ntype
4. #13115774C2semidirect
5. #13115770C2semidirect
42. C2×C13×M16Q8semidirect
68. C22×#1055Q8semidirect
230. C2×M16Dic26semidirect
325. C13#21304semidirect

Maximal subgroups

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

subgroupindex
C2×#117191752
C2×#117187802
C2×#117187832
#131157742
#131157702
#131145342
C2×#117181162
C13×#17282
#2130413