Finite Group Databasebeta

← #16787186

Subgroups — #16787186

Order 3328.

414 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 72 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#13130084C211yesmaximal, characteristic
2C2×#11718285C211yesmaximal, characteristic
3#13129779C212yesmaximal, complemented
4#13129766C212yesmaximal, complemented
5#13119315C211yesmaximal, characteristic
6C2×C13×C22⋊Q8C411yescharacteristic
7C22×C4⋊Dic13C2211yescharacteristic
8#11719169—21no—
9C2×C22⋊#1057—21no—
10#11718891C411yescharacteristic
11#11718891C2211yescharacteristic
12C4×C22⋊Dic13C411yescharacteristic
13#11718644—21no—
14#11718644—21no—
15#11718637C411yescharacteristic
16#11718637C2211yescharacteristic
17#11718607—21no—
18#11718607—21no—
19#11718285—21no—
20#11718285—21no—
21#11718285C2212yes—
22#11718285C2212yes—
23C22×Q8×C13C2×C411yescharacteristic
24C23×C4×C13C2×C411yescharacteristic
25C23×Dic13D411yescharacteristic
26C23×Dic13D411yescharacteristic
27C2×#6122—41no—
28C22×#1057—41no—
29C13×C22⋊Q8—41no—
30C13×C22⋊Q8—41no—
31C2×C13×C4⋊C4—21no—
32C2×C13×C4⋊C4C2×C411yescharacteristic
33C2×C13×C22⋊C4—21no—
34C2×C22⋊Dic13—21no—
35C2×C4⋊Dic13—41no—
36C2×C4⋊Dic13—41no—
37C2×C4⋊Dic13C2×C411yescharacteristic
38C2×C4⋊Dic13D411yescharacteristic
39C2×C4⋊Dic13D411yescharacteristic
40C2×C4⋊Dic13C2311yescharacteristic
41C2×#6101—21no—
42C2×#6101—21no—
43C2×#6101—21no—
44C2×#6101—21no—
45C2×C4×Dic13C2×C411yescharacteristic
46C2×C4×Dic13C2×C411yescharacteristic
47C22⋊#1057—22no—
48C22⋊#1057—22no—
49C22⋊#1057—41no—
50C22⋊#1057—41no—
51#88071—21no—
52#88071—21no—
53#88071—21no—
54#88071—21no—
55#88071—21no—
56#11555—131nomaximal, Sylow
57C24×C13C22⋊C411yescharacteristic
58C2×Q8×C13—22no—
59C2×Q8×C13—41no—
60C2×Q8×C13—41no—
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13—22no—
64C22×C4×C13—21no—
65C22×C4×C13—21no—
66C22×C4×C13—41no—
67C22×C4×C13C22⋊C411yescharacteristic
68C22×C4×C13C22×C411yescharacteristic
69C22×Dic13—21no—
70C22×Dic13—21no—
71C22×Dic13—21no—
72C22×Dic13—21no—
73C22×Dic13—21no—
74C22×Dic13—21no—
75C22×Dic13—81no—
76C22×Dic13—81no—
77C22×Dic13C22⋊C411yescharacteristic
78C22×Dic13C22⋊C411yescharacteristic
79C22×Dic13C2×D411yescharacteristic
80C22×Dic13C2×D411yescharacteristic
81#6122—82no—
82C2×#1057—42no—
83C2×#1057—42no—
84C2×#1057—82no—
85C13×C4⋊C4—41no—
86C13×C4⋊C4—42no—
87C13×C22⋊C4—41no—
88C13×C22⋊C4—41no—
89C22⋊Dic13—24no—
90C4⋊Dic13—41no—
91C4⋊Dic13—41no—
92C4⋊Dic13—41no—
93C4⋊Dic13—41no—
94#6101—81no—
95#6101—81no—
96#6101—81no—
97#6101—81no—
98C4×Dic13—22no—
99C4×Dic13—41no—
100#2502—131no—
101#1463—131no—
102C2×#324—131no—
103#1325—132no—
104#1319—132no—
105C23×C13—22no—
106C23×C13—21no—
107C23×C13—21no—
108C23×C13—41no—
109C23×C13#9911yescharacteristic
110C23×C13#9911yescharacteristic
111C23×C13C2×C22⋊C411yescharacteristic
112Q8×C13—82no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—21no—
116C2×C4×C13—21no—
117C2×C4×C13—21no—
118C2×C4×C13—42no—
119C2×C4×C13—42no—
120C2×C4×C13—41no—
121C2×C4×C13—41no—
122C2×C4×C13—41no—
123C2×C4×C13—41no—
124C2×C4×C13—41no—
125C2×C4×C13—41no—
126C2×C4×C13#13112yes—
127C2×Dic13—22no—
128C2×Dic13—22no—
129C2×Dic13—21no—
130C2×Dic13—21no—
131C2×Dic13—21no—
132C2×Dic13—21no—
133C2×Dic13—41no—
134C2×Dic13—41no—
135C2×Dic13—42no—
136C2×Dic13—42no—
137C2×Dic13—41no—
138C2×Dic13—41no—
139C2×Dic13—41no—
140C2×Dic13—41no—
141C2×Dic13—41no—
142C2×Dic13—41no—
143C2×Dic13—81no—
144C2×Dic13—81no—
145#1057—82no—
146#1057—82no—
147C2×C22⋊Q8#26111yescharacteristic
148C22×C4⋊C4—131no—
149#407—261no—
150C2×C22⋊C8—261no—
151#398—261no—
152#398—261no—
153#391—131no—
154#391—131no—
155#385—261no—
156#385—261no—
157#384—131no—
158#384—131no—
159C4×C22⋊C4—131no—
160#324—132no—
161#324—132no—
162#324—261no—
163#324—261no—
164C22×C13—21no—
165C22×C13—21no—
166C22×C13—21no—
167C22×C13—21no—
168C22×C13—21no—
169C22×C13—21no—
170C22×C13—21no—
171C22×C13—42no—
172C22×C13—42no—
173C22×C13#35612yes—
174C22×C13#40811yescharacteristic
175C22×C13C2×#9911yescharacteristic
176C22×C13#41311yescharacteristic
177C4×C13—22no—
178C4×C13—41no—
179C4×C13—81no—
180C4×C13—81no—
181Dic13—42no—
182Dic13—42no—
183Dic13—41no—
184Dic13—41no—
185Dic13—41no—
186Dic13—41no—
187C22×Q8C2×#26111yescharacteristic
188C23×C4—131no—
189C23×C4—131no—
190C23×C4C2×#26111yescharacteristic
191C2×M16—521no—
192C22×C8—521no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—131no—
197C2×C4⋊C4—131no—
198C2×C4⋊C4—131no—
199C2×C4⋊C4—131no—
200C2×C4⋊C4—261no—
201C2×C4⋊C4—261no—
202C2×C4⋊C4—261no—
203C2×C4⋊C4—261no—
204C2×C4⋊C4—521no—
205C2×C4⋊C4—521no—
206C2×C4⋊C4C2×#26111yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—261no—
209C2×C42—131no—
210C2×C42—131no—
211C22⋊C8—262no—
212C22⋊C8—262no—
213C22⋊C8—521no—
214C22⋊C8—521no—
215#95—261no—
216#95—261no—
217#95—261no—
218#95—261no—
219#95—261no—
220C2×C13—21no—
221C2×C13—21no—
222C2×C13—22no—
223C2×C13—41no—
224C2×C13#146811yescharacteristic
225C2×C13#212711yescharacteristic
226C2×C13C2×#35611yescharacteristic
227C24C22⋊#26111yescharacteristic
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—131no—
238C22×C4—131no—
239C22×C4—131no—
240C22×C4—131no—
241C22×C4—261no—
242C22×C4—261no—
243C22×C4—261no—
244C22×C4—261no—
245C22×C4—261no—
246C22×C4—261no—
247C22×C4—1041no—
248C22×C4—1041no—
249C22×C4C22⋊#26111yescharacteristic
250C22×C4C22×#26111yescharacteristic
251M16—1042no—
252C2×C8—522no—
253C2×C8—522no—
254C2×C8—1042no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—521no—
258C4⋊C4—521no—
259C4⋊C4—521no—
260C4⋊C4—521no—
261C4⋊C4—1041no—
262C4⋊C4—1041no—
263C4⋊C4—1041no—
264C4⋊C4—1041no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—264no—
268C42—262no—
269C42—521no—
270C13#1155511yescharacteristic, complemented, Sylow
271C23—22no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23#8811911yescharacteristic
276C23#8811911yescharacteristic
277C23C2×C22⋊#26111yescharacteristic
278Q8—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—262no—
293C2×C4—262no—
294C2×C4—261no—
295C2×C4—261no—
296C2×C4—261no—
297C2×C4—261no—
298C2×C4—521no—
299C2×C4—521no—
300C2×C4—522no—
301C2×C4—522no—
302C2×C4—521no—
303C2×C4—521no—
304C2×C4—521no—
305C2×C4—521no—
306C2×C4—521no—
307C2×C4—521no—
308C2×C4—1041no—
309C2×C4—1041no—
310C2×C4#8823812yes—
311C8—1042no—
312C8—1042no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—42no—
321C22—42no—
322C22#1171830112yes—
323C22#1171914611yescharacteristic
324C22#1171914811yescharacteristic
325C22C2×#8811911yescharacteristic
326C4—22no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—522no—
331C4—522no—
332C4—521no—
333C4—521no—
334C4—521no—
335C4—521no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—41no—
340C2#1313010111yescharacteristic
341C2#1313111111yescharacteristic
342C2C2×#1171830111yescharacteristic
subgroupG/H
centerC22C2×#88119
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC2×C13×C22⋊Q8C4
socleC22×C13C2×#99
prime radicalC24×C13C22⋊C4
cosocleC2×C4⋊Dic13C23
hypercenterC2×C22⋊Q8#261
hypocenterC13#11555
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #13129779C2semidirect
4. #13129766C2semidirect
270. C13#11555semidirect

Maximal subgroups

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

subgroupindex
#131300842
C2×#117182852
#131297792
#131297662
#131193152
#1155513