Finite Group Databasebeta

← #16785094

Subgroups — #16785094

Order 3328.

654 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C4×#88111C211yesmaximal, characteristic
2#13130014C211yesmaximal, characteristic
3#13129709C214yesmaximal
4#13123189C211yesmaximal, characteristic
5C22×C4⋊#261—21no—
6C2×#88151—21no—
7C2×C42×D13C2211yescharacteristic
8C2×#88123C411yescharacteristic, complemented
9C2×(C2×#261)⋊C4—21no—
10C2×(C2×#261)⋊C4C2212yes—
11C2×#88111C2212yes—
12C2×C4×#1057—21no—
13C13×#384C411yescharacteristic, complemented
14#11718648—21no—
15#11718648C2212yes—
16#11718646—21no—
17#11718646C412yescomplemented
18#11718635—21no—
19#11718290—21no—
20C23×#261—22no—
21C22×C4×D13D411yescharacteristic
22C22×C4×D13D411yescharacteristic
23C22×C4×D13C2311yescharacteristic
24C22×Dic26—21no—
25C22×#1057—22no—
26C2×C4⋊#261—24no—
27C2×C4⋊#261—42no—
28C2×C4⋊#261—42no—
29C2×C13×C4⋊C4—21no—
30C2×C13×C4⋊C4—21no—
31C2×C13×C4⋊C4C2×C412yes—
32C2×C42×C13C2×C411yescharacteristic
33C2×#6103—22no—
34C2×C4⋊Dic13—21no—
35C2×C4⋊Dic13—22no—
36C2×C4⋊Dic13C2×C412yes—
37C2×C4×Dic13C2×C411yescharacteristic
38C2×C4×Dic13D411yescharacteristic
39C2×C4×Dic13D411yescharacteristic
40#88151—42no—
41#88151—42no—
42C42×D13—42no—
43#88123—42no—
44(C2×#261)⋊C4—24no—
45(C2×#261)⋊C4—42no—
46#88111—22no—
47#88111Q814yescomplemented
48C4×#1057—42no—
49#88071—22no—
50#88071—22no—
51#9885—131nomaximal, Sylow
52C23×D13C2×D411yescharacteristic
53C22×#261—24no—
54C22×#261—42no—
55C22×#261—42no—
56C22×#261—42no—
57C22×#261—42no—
58C22×#261—42no—
59C22×#261—42no—
60C22×C4×C13—22no—
61C22×C4×C13C22⋊C411yescharacteristic
62C22×C4×C13C22⋊C411yescharacteristic
63C22×C4×C13C22×C411yescharacteristic
64C22×Dic13—22no—
65C22×Dic13C22⋊C411yescharacteristic
66C22×Dic13C22⋊C411yescharacteristic
67C22×Dic13C2×D411yescharacteristic
68C2×C4×D13—22no—
69C2×C4×D13—22no—
70C2×C4×D13—41no—
71C2×C4×D13—41no—
72C2×C4×D13—41no—
73C2×C4×D13—41no—
74C2×C4×D13D812yes—
75C2×C4×D13SD1612yes—
76C2×C4×D13SD1612yes—
77C2×C4×D13Q1612yes—
78C2×C4×D13C2×Q812yes—
79C2×C4×D13#4112yes—
80C2×Dic26—42no—
81C2×Dic26—42no—
82C2×Dic26—42no—
83C2×#1057—24no—
84C2×#1057—42no—
85C2×#1057—42no—
86C4⋊#261—48no—
87C13×C4⋊C4—42no—
88C13×C4⋊C4—42no—
89C13×C4⋊C4—42no—
90C42×C13—22no—
91#6103—42no—
92#6103—42no—
93C4⋊Dic13—42no—
94C4⋊Dic13—44no—
95C4⋊Dic13—42no—
96C4×Dic13—22no—
97C4×Dic13—41no—
98C4×Dic13—41no—
99#2857—131no—
100C4×C22⋊C8—131no—
101C4⋊#95—131no—
102#1294—134no—
103C23×C13C2×C22⋊C411yescharacteristic
104C22×D13—21no—
105C22×D13—21no—
106C22×D13—21no—
107C22×D13—21no—
108C22×D13#12411yescharacteristic
109C22×D13C4⋊Q811yescharacteristic
110C22×D13C2×D811yescharacteristic
111C22×D13C2×SD1611yescharacteristic
112C22×D13C2×SD1611yescharacteristic
113C22×D13C2×Q1611yescharacteristic
114C2×#261—44no—
115C2×#261—44no—
116C2×#261—44no—
117C2×#261—82no—
118C2×#261—82no—
119C2×#261—82no—
120C2×#261—82no—
121C2×C4×C13—21no—
122C2×C4×C13—21no—
123C2×C4×C13—21no—
124C2×C4×C13—21no—
125C2×C4×C13—42no—
126C2×C4×C13—42no—
127C2×C4×C13—42no—
128C2×C4×C13#9912yes—
129C2×C4×C13#10212yes—
130C2×C4×C13#10312yes—
131C2×C4×C13#10412yes—
132C2×C4×C13C4×Q812yes—
133C2×Dic13—21no—
134C2×Dic13—21no—
135C2×Dic13—21no—
136C2×Dic13—21no—
137C2×Dic13—21no—
138C2×Dic13—21no—
139C2×Dic13—42no—
140C2×Dic13—42no—
141C2×Dic13—42no—
142C2×Dic13#10212yes—
143C2×Dic13#10312yes—
144C2×Dic13C22⋊Q812yes—
145C4×D13—24no—
146C4×D13—24no—
147C4×D13—42no—
148C4×D13—42no—
149C4×D13—42no—
150C4×D13—42no—
151Dic26—84no—
152#1057—44no—
153C2×C4⋊Q8—131no—
154C2×C22⋊Q8—261no—
155C22×C4⋊C4—261no—
156C22×C42—131no—
157C2×C22⋊C8—132no—
158C2×C4×C8—261no—
159#391—261no—
160#389—132no—
161#389—261no—
162#384#26111yescharacteristic, complemented
163#380—132no—
164#380—261no—
165C2×#95—132no—
166C2×#95—261no—
167#336—261no—
168C22×C13#39111yescharacteristic
169C22×C13C2×#9911yescharacteristic
170C22×C13#41011yescharacteristic
171C22×C13C2×#10211yescharacteristic
172C22×C13C2×#10311yescharacteristic
173C22×C13C2×#10411yescharacteristic
174C22×C13#42111yescharacteristic
175C2×D13—21no—
176C2×D13—21no—
177C2×D13—21no—
178C2×D13—21no—
179C2×D13—21no—
180C2×D13—21no—
181C2×D13—21no—
182C2×D13—21no—
183C2×D13—21no—
184C2×D13—21no—
185C2×D13—21no—
186C2×D13—21no—
187C2×D13#48611yescharacteristic
188C2×D13#48711yescharacteristic
189C2×D13#49811yescharacteristic
190C2×D13#50011yescharacteristic
191#261—84no—
192C4×C13—22no—
193C4×C13—22no—
194C4×C13—82no—
195C4×C13#32714yes—
196Dic13—22no—
197Dic13—22no—
198Dic13—22no—
199Dic13—82no—
200C22×Q8—261no—
201C23×C4—131no—
202C23×C4—131no—
203C23×C4—131no—
204C23×C4—262no—
205C22×C8—262no—
206C4⋊Q8—522no—
207C22⋊Q8—522no—
208C22⋊Q8—522no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—132no—
212C2×C4⋊C4—262no—
213C2×C4⋊C4—264no—
214C2×C4⋊C4—261no—
215C2×C4⋊C4—522no—
216C2×C4⋊C4—522no—
217C2×C4⋊C4C2×#26112yes—
218C2×C22⋊C4—262no—
219C2×C42—131no—
220C2×C42—131no—
221C2×C42—131no—
222C2×C42—522no—
223C2×C42C2×#26111yescharacteristic
224C22⋊C8—134no—
225C22⋊C8—262no—
226C4×C8—522no—
227#95—262no—
228#95—262no—
229#95—264no—
230#95—522no—
231C2×C13C2×#32711yescharacteristic
232C2×C13#150311yescharacteristic
233C2×C13#151711yescharacteristic
234C2×C13#152011yescharacteristic
235C2×C13#198611yescharacteristic
236C2×C13#199011yescharacteristic
237C2×C13#199611yescharacteristic
238D13—21no—
239D13—21no—
240D13—21no—
241D13—21no—
242C24—131no—
243C2×Q8—522no—
244C2×Q8—522no—
245C2×Q8—522no—
246C22×C4—22no—
247C22×C4—131no—
248C22×C4—131no—
249C22×C4—131no—
250C22×C4—132no—
251C22×C4—132no—
252C22×C4—132no—
253C22×C4—132no—
254C22×C4—132no—
255C22×C4—132no—
256C22×C4—262no—
257C22×C4—262no—
258C22×C4—264no—
259C22×C4—262no—
260C22×C4—521no—
261C22×C4—521no—
262C22×C4—521no—
263C22×C4—521no—
264C22×C4—522no—
265C22×C4—522no—
266C22×C4—522no—
267C22×C4—522no—
268C22×C4—522no—
269C22×C4—522no—
270C22×C4C22⋊#26111yescharacteristic
271C22×C4C22⋊#26111yescharacteristic
272C22×C4C22×#26111yescharacteristic
273C2×C8—264no—
274C2×C8—522no—
275C2×C8—522no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—522no—
280C4⋊C4—522no—
281C4⋊C4—524no—
282C4⋊C4—528no—
283C22⋊C4—522no—
284C22⋊C4—522no—
285C42—22no—
286C42—262no—
287C42—521no—
288C42—521no—
289C13#988511yescharacteristic, complemented, Sylow
290C23—131no—
291C23—131no—
292C23—131no—
293C23—131no—
294C23—131no—
295C23—131no—
296C23—261no—
297C23—261no—
298C23—261no—
299C23—261no—
300C23C2×C22⋊#26111yescharacteristic
301Q8—1044no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—132no—
310C2×C4—132no—
311C2×C4—132no—
312C2×C4—261no—
313C2×C4—261no—
314C2×C4—261no—
315C2×C4—261no—
316C2×C4—261no—
317C2×C4—261no—
318C2×C4—264no—
319C2×C4—264no—
320C2×C4—522no—
321C2×C4—522no—
322C2×C4—522no—
323C2×C4—522no—
324C2×C4—524no—
325C2×C4—524no—
326C2×C4—524no—
327C2×C4—522no—
328C2×C4—522no—
329C2×C4—522no—
330C2×C4—1042no—
331C2×C4—1042no—
332C2×C4—1042no—
333C2×C4—1042no—
334C2×C4#8810712yes—
335C2×C4#8811512yes—
336C2×C4#8811712yes—
337C2×C4#8811812yes—
338C2×C4Q8×#26112yes—
339C8—524no—
340C22—131no—
341C22—131no—
342C22—131no—
343C22—131no—
344C22—261no—
345C22—261no—
346C22—261no—
347C22—261no—
348C22—261no—
349C22—261no—
350C22—261no—
351C22—261no—
352C22—261no—
353C22—261no—
354C22—261no—
355C22—261no—
356C22C2×#8810711yescharacteristic
357C22C2×#8811511yescharacteristic
358C22#1171914011yescharacteristic
359C22#1171914211yescharacteristic
360C22C2×#8811711yescharacteristic
361C22C2×#8811811yescharacteristic
362C22(C2×#261)⋊Q811yescharacteristic
363C4—22no—
364C4—22no—
365C4—82no—
366C4—262no—
367C4—262no—
368C4—262no—
369C4—1044no—
370C4—1042no—
371C4#1171824014yes—
372C2—261no—
373C2—261no—
374C2—261no—
375C2—261no—
376C2C2×#1171824011yescharacteristic
377C2#1313022011yescharacteristic
378C2#1313026511yescharacteristic
379C2#1313027511yescharacteristic
380C2#1313101111yescharacteristic
381C2#1313101711yescharacteristic
382C2#1313102611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC13×#384C4
socleC23×C13C2×C22⋊C4
prime radicalC23×D13C2×D4
cosocleC22×C4×D13C23
hypercenter#384#261
hypocenterC13#9885
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
8. C2×#88123C4semidirect
13. C13×#384C4semidirect
17. #11718646C4semidirect
47. #88111Q8semidirect
162. #384#261semidirect
289. C13#9885semidirect

Maximal subgroups

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

subgroupindex
C4×#881112
#131300142
#131297092
#131231892
#988513