Finite Group Databasebeta

← #16788225

Subgroups — #16788225

Order 3328.

412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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#13130735C211yesmaximal, characteristic
2C2×#11718286C211yesmaximal, characteristic
3#13129837C212yesmaximal, complemented
4#13129820C212yesmaximal, complemented
5#13120217C211yesmaximal, characteristic
6C2×C13×C22⋊Q8C411yescharacteristic
7C22×#6101C2211yescharacteristic
8#11719169—21no—
9C2×C22⋊#1057—21no—
10#11718890C411yescharacteristic
11#11718890C2211yescharacteristic
12C4×C22⋊Dic13C411yescharacteristic
13#11718641C411yescharacteristic
14#11718641C2211yescharacteristic
15#11718633—21no—
16#11718633—21no—
17#11718607—21no—
18#11718607—21no—
19#11718291—21no—
20#11718291C2212yes—
21#11718286—21no—
22#11718286C2212yes—
23C22×Q8×C13C2×C411yescharacteristic
24C23×C4×C13C2×C411yescharacteristic
25C23×Dic13D411yescharacteristic
26C23×Dic13D411yescharacteristic
27C2×#6122—22no—
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×#6101—21no—
36C2×#6101—21no—
37C2×#6101—21no—
38C2×#6101—21no—
39C2×#6101—41no—
40C2×#6101—41no—
41C2×#6101C2×C411yescharacteristic
42C2×#6101D411yescharacteristic
43C2×#6101D411yescharacteristic
44C2×#6101C2311yescharacteristic
45C2×C4×Dic13—21no—
46C22⋊#1057—22no—
47C22⋊#1057—41no—
48C22⋊#1057—41no—
49C22⋊#1057—41no—
50#88071—21no—
51#88071—21no—
52#88071—21no—
53#88071—21no—
54#88071C2×C411yescharacteristic
55#88071C2×C411yescharacteristic
56#11854—131nomaximal, Sylow
57C24×C13C22⋊C411yescharacteristic
58C2×Q8×C13—41no—
59C2×Q8×C13—81no—
60C22×C4×C13—21no—
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13—21no—
64C22×C4×C13—81no—
65C22×C4×C13C22⋊C411yescharacteristic
66C22×C4×C13C22×C411yescharacteristic
67C22×Dic13—21no—
68C22×Dic13—21no—
69C22×Dic13—21no—
70C22×Dic13—21no—
71C22×Dic13—21no—
72C22×Dic13—21no—
73C22×Dic13—22no—
74C22×Dic13—41no—
75C22×Dic13—81no—
76C22×Dic13C22⋊C411yescharacteristic
77C22×Dic13C22⋊C411yescharacteristic
78C22×Dic13C2×D411yescharacteristic
79C22×Dic13C2×D411yescharacteristic
80#6122—42no—
81#6122—42no—
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—
90#6101—41no—
91#6101—41no—
92#6101—41no—
93#6101—41no—
94#6101—42no—
95#6101—42no—
96#6101—81no—
97#6101—81no—
98C4×Dic13—81no—
99#2583—131no—
100#1821—131no—
101C2×#324—131no—
102#1354—132no—
103#1346—132no—
104C23×C13—22no—
105C23×C13—21no—
106C23×C13—21no—
107C23×C13—41no—
108C23×C13#9911yescharacteristic
109C23×C13#9911yescharacteristic
110C23×C13C2×C22⋊C411yescharacteristic
111Q8×C13—82no—
112C2×C4×C13—21no—
113C2×C4×C13—21no—
114C2×C4×C13—41no—
115C2×C4×C13—41no—
116C2×C4×C13—41no—
117C2×C4×C13—41no—
118C2×C4×C13—41no—
119C2×C4×C13—41no—
120C2×C4×C13—41no—
121C2×C4×C13—41no—
122C2×C4×C13—81no—
123C2×C4×C13—81no—
124C2×Dic13—21no—
125C2×Dic13—21no—
126C2×Dic13—22no—
127C2×Dic13—22no—
128C2×Dic13—21no—
129C2×Dic13—21no—
130C2×Dic13—21no—
131C2×Dic13—21no—
132C2×Dic13—21no—
133C2×Dic13—42no—
134C2×Dic13—42no—
135C2×Dic13—41no—
136C2×Dic13—41no—
137C2×Dic13—42no—
138C2×Dic13—41no—
139C2×Dic13—41no—
140C2×Dic13—41no—
141C2×Dic13—41no—
142C2×Dic13—81no—
143C2×Dic13#13512yes—
144#1057—82no—
145#1057—82no—
146C2×C22⋊Q8#26111yescharacteristic
147C22×C4⋊C4—131no—
148#407—261no—
149C2×C22⋊C8—261no—
150#398—131no—
151#398—131no—
152#395—131no—
153#395—131no—
154#389—261no—
155#389—261no—
156#385—261no—
157#385—261no—
158C4×C22⋊C4—131no—
159#343—132no—
160#343—261no—
161#324—132no—
162#324—261no—
163C22×C13—21no—
164C22×C13—21no—
165C22×C13—21no—
166C22×C13—21no—
167C22×C13—21no—
168C22×C13—21no—
169C22×C13—21no—
170C22×C13—42no—
171C22×C13—42no—
172C22×C13#35612yes—
173C22×C13C2×#9911yescharacteristic
174C22×C13#41611yescharacteristic
175C22×C13#42111yescharacteristic
176C4×C13—41no—
177C4×C13—41no—
178C4×C13—81no—
179C4×C13—81no—
180Dic13—22no—
181Dic13—42no—
182Dic13—42no—
183Dic13—41no—
184Dic13—41no—
185Dic13—41no—
186C22×Q8C2×#26111yescharacteristic
187C23×C4—131no—
188C23×C4—131no—
189C23×C4C2×#26111yescharacteristic
190C2×M16—262no—
191C22×C8—521no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—131no—
196C2×C4⋊C4—131no—
197C2×C4⋊C4—131no—
198C2×C4⋊C4—131no—
199C2×C4⋊C4—261no—
200C2×C4⋊C4—261no—
201C2×C4⋊C4—261no—
202C2×C4⋊C4—261no—
203C2×C4⋊C4—521no—
204C2×C4⋊C4—521no—
205C2×C4⋊C4C2×#26111yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—261no—
208C2×C42—261no—
209C22⋊C8—262no—
210C22⋊C8—521no—
211C22⋊C8—521no—
212C22⋊C8—521no—
213#95—131no—
214#95—131no—
215#95—261no—
216#95—261no—
217#95—261no—
218#95—261no—
219C2×C13—21no—
220C2×C13—21no—
221C2×C13—22no—
222C2×C13—41no—
223C2×C13#151611yescharacteristic
224C2×C13#212111yescharacteristic
225C2×C13C2×#35611yescharacteristic
226C24C22⋊#26111yescharacteristic
227C2×Q8—41no—
228C2×Q8—81no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—81no—
234C22×C4—131no—
235C22×C4—131no—
236C22×C4—131no—
237C22×C4—131no—
238C22×C4—262no—
239C22×C4—261no—
240C22×C4—261no—
241C22×C4—261no—
242C22×C4—261no—
243C22×C4—261no—
244C22×C4—261no—
245C22×C4—521no—
246C22×C4—1041no—
247C22×C4C22⋊#26111yescharacteristic
248C22×C4C22×#26111yescharacteristic
249M16—522no—
250M16—522no—
251C2×C8—522no—
252C2×C8—522no—
253C2×C8—1042no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—521no—
257C4⋊C4—521no—
258C4⋊C4—521no—
259C4⋊C4—521no—
260C4⋊C4—522no—
261C4⋊C4—522no—
262C4⋊C4—1041no—
263C4⋊C4—1041no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—264no—
267C42—1041no—
268C13#1185411yescharacteristic, complemented, Sylow
269C23—22no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23#8811911yescharacteristic
274C23#8811911yescharacteristic
275C23C2×C22⋊#26111yescharacteristic
276Q8—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—132no—
290C2×C4—261no—
291C2×C4—261no—
292C2×C4—262no—
293C2×C4—262no—
294C2×C4—261no—
295C2×C4—261no—
296C2×C4—261no—
297C2×C4—261no—
298C2×C4—261no—
299C2×C4—522no—
300C2×C4—522no—
301C2×C4—521no—
302C2×C4—521no—
303C2×C4—522no—
304C2×C4—521no—
305C2×C4—521no—
306C2×C4—521no—
307C2×C4—521no—
308C2×C4—1041no—
309C8—1042no—
310C8—1042no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1171830112yes—
321C22#1171916511yescharacteristic
322C22#1171916611yescharacteristic
323C22C2×#8811911yescharacteristic
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—262no—
329C4—522no—
330C4—522no—
331C4—521no—
332C4—521no—
333C4—521no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#1313026211yescharacteristic
339C2#1313109111yescharacteristic
340C2C2×#1171830111yescharacteristic
subgroupG/H
centerC22C2×#88119
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC2×C13×C22⋊Q8C4
socleC22×C13C2×#99
prime radicalC24×C13C22⋊C4
cosocleC2×#6101C23
hypercenterC2×C22⋊Q8#261
hypocenterC13#11854
LCSG ▷ C22×C4×C13 ▷ C23×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. #13129837C2semidirect
4. #13129820C2semidirect
268. C13#11854semidirect

Maximal subgroups

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

subgroupindex
#131307352
C2×#117182862
#131298372
#131298202
#131202172
#1185413