Finite Group Databasebeta

← #15848255

Subgroups — #15848255

Order 3328.

622 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125977C211yesmaximal, characteristic
2C2×#11718692C211yesmaximal, characteristic
3#13123917C211yesmaximal, characteristic
4#13114264C214yesmaximal, complemented
5#13114220C214yesmaximal, complemented
6#13113608C211yesmaximal, characteristic
7#13113509C211yesmaximal, characteristic
8#13113493C211yesmaximal, characteristic
9C13×#3078C211yesmaximal, characteristic
10C13×#533C2211yescharacteristic
11C2×C4×Q8×C13C2211yescharacteristic
12C22×#6107—21no—
13C2×#88123C2211yescharacteristic
14C13×#479—22no—
15C13×#477C2212yes—
16C13×#475C2212yes—
17C13×#428—21no—
18C13×#423C2211yescharacteristic
19C13×#419—21no—
20#11718742C2212yes—
21#11718740C2212yes—
22#11718704—22no—
23#11718692—22no—
24#11718692C2214yes—
25Q8×#1055—22no—
26Q8×#1055C2214yes—
27#11718681—22no—
28C2×#88050—21no—
29#11718586—22no—
30#11718586C2211yescharacteristic
31#11718586C2211yescharacteristic
32C2×C4⋊#1055C2211yescharacteristic
33#11718139C2212yes—
34#11718138C2212yes—
35#11718074C2214yes—
36#11718072C2214yes—
37C22×Q8×C13D411yescharacteristic
38C22×Dic26—21no—
39C2×C13×M16—21no—
40C13×C4⋊Q8C2311yescharacteristic
41C13×C4⋊Q8C2311yescharacteristic
42C13×#125—21no—
43C13×C22⋊Q8—21no—
44C13×C22⋊Q8—21no—
45C4×Q8×C13—21no—
46C4×Q8×C13—21no—
47C4×Q8×C13—21no—
48C4×Q8×C13—21no—
49C4×Q8×C13D412yes—
50C4×Q8×C13C2312yes—
51C13×#117—21no—
52C2×C13×C4⋊C4—21no—
53C2×C13×C4⋊C4D411yescharacteristic
54C2×C42×C13—21no—
55C2×C42×C13C2311yescharacteristic
56C2×#6107—24no—
57C2×#6107—44no—
58C2×C4⋊Dic13—21no—
59C2×C4⋊Dic13—21no—
60C22×#1055—21no—
61#88123—21no—
62#88123—21no—
63#88123—21no—
64#88123C2312yes—
65C13×#107—22no—
66C13×#106—22no—
67C13×C4⋊C8—21no—
68C13×C4⋊C8C2312yes—
69C13×#103—22no—
70C13×#103—22no—
71#88075—24no—
72C22⋊#1055—21no—
73C22⋊#1055D412yes—
74#88050—24no—
75#88048—24no—
76C4⋊#1055—21no—
77C4⋊#1055—24no—
78C4⋊#1055D412yes—
79C4⋊#1055C2312yes—
80C4⋊#1055C2312yes—
81C4×#1055—21no—
82C4×#1055—24no—
83C4×#1055C2312yes—
84#19402—131nomaximal, Sylow
85C2×Q8×C13—21no—
86C2×Q8×C13—22no—
87C2×Q8×C13—21no—
88C2×Q8×C13—21no—
89C2×Q8×C13SD1614yes—
90C2×Q8×C13C2×D412yes—
91C22×C4×C13—21no—
92C22×C4×C13—21no—
93C22×C4×C13C2×D411yescharacteristic
94C22×C4×C13C2×D411yescharacteristic
95C22×C4×C13#4111yescharacteristic
96C22×Dic13—41no—
97C2×Dic26—22no—
98C2×Dic26—42no—
99C2×Dic26—42no—
100C2×Dic26—41no—
101C13×M16—42no—
102C2×C8×C13—22no—
103C13×C4⋊C4—22no—
104C13×C4⋊C4—21no—
105C13×C4⋊C4—22no—
106C13×C4⋊C4—21no—
107C13×C4⋊C4—21no—
108C13×C4⋊C4—21no—
109C13×C4⋊C4—21no—
110C13×C4⋊C4—41no—
111C13×C4⋊C4—41no—
112C13×C4⋊C4C2×D412yes—
113C13×C22⋊C4—41no—
114C42×C13—22no—
115C42×C13—22no—
116C42×C13—41no—
117C42×C13C2×D411yescharacteristic
118C42×C13C2×D411yescharacteristic
119C42×C13#4111yescharacteristic
120C42×C13C2411yescharacteristic
121#6107—48no—
122C4⋊Dic13—22no—
123C4⋊Dic13—22no—
124C4⋊Dic13—41no—
125C4⋊Dic13—41no—
126C2×#1055—22no—
127C2×#1055—22no—
128C2×#1055—42no—
129C2×#1055C2×D414yes—
130#3229—131no—
131#3078D1311yescharacteristic
132C2×#462—131no—
133#2982—131no—
134#1680—134no—
135#1672—134no—
136#1570—131no—
137#1546—131no—
138#1542—131no—
139C23×C13#12111yescharacteristic
140Q8×C13—22no—
141Q8×C13—24no—
142Q8×C13—42no—
143Q8×C13—81no—
144Q8×C13C2×SD1614yes—
145C2×C4×C13—22no—
146C2×C4×C13—24no—
147C2×C4×C13—22no—
148C2×C4×C13—22no—
149C2×C4×C13—41no—
150C2×C4×C13—41no—
151C2×C4×C13—41no—
152C2×C4×C13—81no—
153C2×C4×C13#12111yescharacteristic
154C2×C4×C13#12111yescharacteristic
155C2×C4×C13#12111yescharacteristic
156C2×C4×C13C2×SD1612yes—
157C2×C4×C13#13512yes—
158C2×C4×C13C22×D411yescharacteristic
159C2×C4×C13C22×D411yescharacteristic
160C2×C4×C13C2×#4111yescharacteristic
161C2×Dic13—42no—
162C2×Dic13—82no—
163Dic26—42no—
164Dic26—42no—
165Dic26—82no—
166C8×C13—42no—
167#1055—24no—
168#1055—42no—
169#1055—42no—
170C22×Q16—261no—
171#533C2×D1311yescharacteristic
172C2×C4⋊Q8—131no—
173C2×C4×Q8C2×D1311yescharacteristic
174#498—132no—
175#487—132no—
176#483—262no—
177#479—22no—
178#477C2×D1312yes—
179#475—262no—
180#475C2×D1312yes—
181#462—134no—
182#462—262no—
183C8×Q8—134no—
184C8×Q8—262no—
185#431—131no—
186#431—131no—
187#431—262no—
188#428—21no—
189#423C2×D1311yescharacteristic
190C2×C4⋊C8—131no—
191#419—21no—
192C2×#103—261no—
193#334—134no—
194#333—132no—
195#332—132no—
196#332—134no—
197C22×C13—21no—
198C22×C13—21no—
199C22×C13#46512yes—
200C22×C13C2×#12111yescharacteristic
201C4×C13—21no—
202C4×C13—24no—
203C4×C13—21no—
204C4×C13—21no—
205C4×C13—42no—
206C4×C13—81no—
207C4×C13—81no—
208C4×C13#46512yes—
209C4×C13C22×SD1611yescharacteristic
210C4×C13C2×#13511yescharacteristic
211C4×C13#57812yes—
212Dic13—82no—
213C22×Q8—261no—
214C22×Q8C22⋊D1311yescharacteristic
215C2×Q16—264no—
216C2×Q16—524no—
217C2×M16—21no—
218C22×C8—261no—
219C4⋊Q8—132no—
220C4⋊Q8—261no—
221C4⋊Q8—261no—
222C4⋊Q8—261no—
223C4⋊Q8C22×D1311yescharacteristic
224C4⋊Q8C22×D1311yescharacteristic
225#125—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8C22⋊D1312yes—
233C4×Q8C22×D1312yes—
234#117—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—261no—
237C2×C4⋊C4—261no—
238C2×C4⋊C4C22⋊D1311yescharacteristic
239C2×C42—21no—
240C2×C42C22×D1311yescharacteristic
241#107—22no—
242#106—22no—
243#106—264no—
244C4⋊C8—21no—
245C4⋊C8—132no—
246C4⋊C8—132no—
247C4⋊C8—132no—
248C4⋊C8—264no—
249C4⋊C8—261no—
250C4⋊C8C22×D1312yes—
251#103—22no—
252#103—22no—
253#103—264no—
254#103—264no—
255C22⋊C8—132no—
256C22⋊C8—261no—
257C4×C8—132no—
258C4×C8—264no—
259C4×C8—261no—
260C2×C13—22no—
261C2×C13#304411yescharacteristic
262C2×C13C2×#46511yescharacteristic
263C2×C13#306911yescharacteristic
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—262no—
269C2×Q8—521no—
270C2×Q8—522no—
271C2×Q8—522no—
272C2×Q8#610614yes—
273C2×Q8C2×C22⋊D1312yes—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—521no—
277C22×C4D4×D1311yescharacteristic
278C22×C4#612911yescharacteristic
279C22×C4C2×C22⋊D1311yescharacteristic
280Q16—528no—
281M16—42no—
282C2×C8—22no—
283C2×C8—134no—
284C2×C8—262no—
285C2×C8—262no—
286C2×C8—522no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—262no—
297C4⋊C4—262no—
298C4⋊C4—521no—
299C4⋊C4—521no—
300C4⋊C4C2×C22⋊D1312yes—
301C22⋊C4—41no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42D4×D1311yescharacteristic
306C42#612911yescharacteristic
307C42C2×C22⋊D1311yescharacteristic
308C42C23×D1311yescharacteristic
309C13#1940211yescharacteristic, complemented, Sylow
310C23#8819311yescharacteristic
311Q8—24no—
312Q8—22no—
313Q8—42no—
314Q8—81no—
315Q8—522no—
316Q8—522no—
317Q8—1042no—
318Q8C2×#610614yes—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—522no—
328C2×C4—1042no—
329C2×C4#8819311yescharacteristic
330C2×C4#8819311yescharacteristic
331C2×C4#8819311yescharacteristic
332C2×C4C2×#610612yes—
333C2×C4#8820412yes—
334C2×C4C2×D4×D1311yescharacteristic
335C2×C4C2×#612911yescharacteristic
336C2×C4C22×C22⋊D1311yescharacteristic
337C8—42no—
338C8—264no—
339C8—522no—
340C8—522no—
341C22—21no—
342C22—21no—
343C22#1171870912yes—
344C22C2×#8819311yescharacteristic
345C4—21no—
346C4—24no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—81no—
351C4—81no—
352C4—1042no—
353C4#1171870912yes—
354C4#1171948412yes—
355C4C22×#610611yescharacteristic
356C4C2×#8820411yescharacteristic
357C2—22no—
358C2#1312436811yescharacteristic
359C2C2×#1171870911yescharacteristic
360C2#1312457511yescharacteristic
subgroupG/H
centerC22C2×#88193
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3078C2
socleC22×C13C2×#121
prime radicalC23×C13#121
hypercenter#3078D13
hypocenterC13#19402
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
4. #13114264C2semidirect
5. #13114220C2semidirect
309. C13#19402semidirect

Maximal subgroups

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

subgroupindex
#131259772
C2×#117186922
#131239172
#131142642
#131142202
#131136082
#131135092
#131134932
C13×#30782
#1940213