Finite Group Databasebeta

← #16785071

Subgroups — #16785071

Order 3328.

680 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13130690C211yesmaximal, characteristic
2C4×C22⋊#1057C211yesmaximal, characteristic
3#13129674C214yesmaximal
4C2×#11718891C211yesmaximal, characteristic
5C23×Q8×C13C411yescharacteristic
6C22×C4⋊Dic13C411yescharacteristic
7C22×C4×Dic13C2211yescharacteristic
8C2×C22⋊#1057C2212yes—
9C2×C22⋊#1057C2212yes—
10C2×C4⋊#1057—21no—
11C2×C4×#1057—21no—
12#11718891—21no—
13#11718891—21no—
14#11718891—21no—
15#11718891—21no—
16C2×#88071C412yes—
17C2×#88071C2212yes—
18#11718290—21no—
19#11718290—21no—
20C22×Q8×C13—21no—
21C22×Q8×C13—21no—
22C22×Q8×C13—22no—
23C22×Q8×C13—42no—
24C22×Q8×C13—42no—
25C22×Q8×C13C814yescomplemented
26C23×C4×C13C2×C412yes—
27C23×C4×C13C2×C411yescharacteristic
28C23×Dic13C2×C411yescharacteristic
29C23×Dic13C2×C412yes—
30C23×Dic13C2311yescharacteristic
31C22×#1057—22no—
32C22×#1057—22no—
33C2×C4⋊Dic13—21no—
34C2×C4⋊Dic13—21no—
35C2×C4⋊Dic13—42no—
36C2×C4×Dic13—42no—
37C2×C4×Dic13D411yescharacteristic
38C2×C4×Dic13D411yescharacteristic
39C2×C4×Dic13D411yescharacteristic
40C2×C4×Dic13D411yescharacteristic
41C22⋊#1057—22no—
42C22⋊#1057—24no—
43C22⋊#1057Q814yescomplemented
44C4⋊#1057—42no—
45C4×#1057—42no—
46#88071—22no—
47#88071—22no—
48#88071—22no—
49#88071—22no—
50#9878—131nomaximal, Sylow
51C24×C13C22×C411yescharacteristic
52C2×Q8×C13—24no—
53C2×Q8×C13—24no—
54C2×Q8×C13—48no—
55C2×Q8×C13—42no—
56C2×Q8×C13—42no—
57C2×Q8×C13—48no—
58C2×Q8×C13—42no—
59C2×Q8×C13—42no—
60C22×C4×C13—22no—
61C22×C4×C13—22no—
62C22×C4×C13—22no—
63C22×C4×C13—22no—
64C22×C4×C13—42no—
65C22×C4×C13—42no—
66C22×C4×C13C22⋊C411yescharacteristic
67C22×C4×C13C22⋊C411yescharacteristic
68C22×C4×C13C2×C812yes—
69C22×C4×C13C2×C814yes—
70C22×C4×C13M1612yes—
71C22×Dic13—22no—
72C22×Dic13—22no—
73C22×Dic13—22no—
74C22×Dic13—24no—
75C22×Dic13—41no—
76C22×Dic13—41no—
77C22×Dic13—41no—
78C22×Dic13—42no—
79C22×Dic13—42no—
80C22×Dic13C22⋊C411yescharacteristic
81C22×Dic13C22⋊C411yescharacteristic
82C22×Dic13C2×D411yescharacteristic
83C22×Dic13C2×D411yescharacteristic
84C22×Dic13C2×Q812yes—
85C22×Dic13#4112yes—
86C2×#1057—24no—
87C2×#1057—42no—
88C2×#1057—42no—
89C2×#1057—42no—
90C2×#1057—42no—
91C2×#1057—42no—
92C4⋊Dic13—42no—
93C4⋊Dic13—42no—
94C4×Dic13—22no—
95C4×Dic13—22no—
96C4×Dic13—41no—
97C4×Dic13—41no—
98C2×#391—131no—
99#1797—131no—
100C4×C22⋊C8—131no—
101#1280—134no—
102C23×C13—21no—
103C23×C13—21no—
104C23×C13—21no—
105C23×C13—21no—
106C23×C13C2×C22⋊C411yescharacteristic
107C23×C13C4×Q812yes—
108C23×C13C22×C811yescharacteristic
109C23×C13C2×M1611yescharacteristic
110C23×C13#13112yes—
111Q8×C13—48no—
112Q8×C13—48no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—24no—
116C2×C4×C13—24no—
117C2×C4×C13—24no—
118C2×C4×C13—44no—
119C2×C4×C13—42no—
120C2×C4×C13—42no—
121C2×C4×C13—44no—
122C2×C4×C13C22⋊C812yes—
123C2×C4×C13C22⋊C812yes—
124C2×C4×C13#9912yes—
125C2×C4×C13#10012yes—
126C2×Dic13—21no—
127C2×Dic13—21no—
128C2×Dic13—21no—
129C2×Dic13—21no—
130C2×Dic13—21no—
131C2×Dic13—21no—
132C2×Dic13—21no—
133C2×Dic13—21no—
134C2×Dic13—21no—
135C2×Dic13—42no—
136C2×Dic13—42no—
137C2×Dic13—44no—
138C2×Dic13—42no—
139C2×Dic13—42no—
140C2×Dic13—42no—
141C2×Dic13—84no—
142C2×Dic13#10112yes—
143C2×Dic13C22⋊Q812yes—
144C2×Dic13#12411yescharacteristic
145C2×Dic13C4⋊Q811yescharacteristic
146#1057—44no—
147#1057—82no—
148C23×Q8#26111yescharacteristic
149C22×C4⋊C4—131no—
150C22×C42—131no—
151C2×C4⋊C8—261no—
152C2×C22⋊C8—132no—
153C2×C22⋊C8—132no—
154C2×C4×C8—261no—
155#391—261no—
156#391—261no—
157#391—261no—
158#391—261no—
159C2×#95—132no—
160C2×#95—132no—
161#336—261no—
162#336—261no—
163C22×C13—21no—
164C22×C13—21no—
165C22×C13—21no—
166C22×C13—21no—
167C22×C13—22no—
168C22×C13—22no—
169C22×C13—22no—
170C22×C13—22no—
171C22×C13#39111yescharacteristic
172C22×C13C2×C22⋊C811yescharacteristic
173C22×C13#40811yescharacteristic
174C22×C13C2×#9911yescharacteristic
175C22×C13#41011yescharacteristic
176C22×C13C2×#10011yescharacteristic
177C22×C13C2×#10111yescharacteristic
178C22×C13C8×Q812yes—
179C22×C13#44612yes—
180C4×C13—22no—
181C4×C13—44no—
182C4×C13#32314yes—
183Dic13—22no—
184Dic13—22no—
185Dic13—21no—
186Dic13—21no—
187Dic13—41no—
188Dic13—82no—
189C22×Q8—22no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8#105714yescomplemented
195C23×C4—131no—
196C23×C4—131no—
197C23×C4—132no—
198C23×C4C2×#26112yes—
199C23×C4C2×#26111yescharacteristic
200C22×C8—262no—
201C22×C8—262no—
202C2×C4⋊C4—261no—
203C2×C4⋊C4—261no—
204C2×C4⋊C4—522no—
205C2×C42—131no—
206C2×C42—131no—
207C2×C42—131no—
208C2×C42—131no—
209C2×C42—522no—
210C4⋊C8—522no—
211C22⋊C8—134no—
212C22⋊C8—262no—
213C22⋊C8—264no—
214C4×C8—522no—
215#95—262no—
216#95—262no—
217#95—262no—
218#95—262no—
219C2×C13—22no—
220C2×C13—22no—
221C2×C13C2×#32311yescharacteristic
222C2×C13#146111yescharacteristic
223C2×C13#146911yescharacteristic
224C2×C13#147211yescharacteristic
225C2×C13#198411yescharacteristic
226C2×C13#199611yescharacteristic
227C2×C13#199711yescharacteristic
228C24C22×#26111yescharacteristic
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—48no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—48no—
235C2×Q8—42no—
236C2×Q8—42no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—131no—
244C22×C4—131no—
245C22×C4—131no—
246C22×C4—132no—
247C22×C4—132no—
248C22×C4—131no—
249C22×C4—262no—
250C22×C4—262no—
251C22×C4—262no—
252C22×C4—264no—
253C22×C4—521no—
254C22×C4—521no—
255C22×C4—521no—
256C22×C4—522no—
257C22×C4—522no—
258C22×C4C2×#105712yes—
259C22×C4C2×#105714yes—
260C22×C4#612212yes—
261C22×C4C22⋊#26111yescharacteristic
262C22×C4C22⋊#26111yescharacteristic
263C2×C8—264no—
264C2×C8—522no—
265C2×C8—522no—
266C2×C8—522no—
267C2×C8—522no—
268C2×C8—522no—
269C4⋊C4—522no—
270C4⋊C4—522no—
271C42—262no—
272C42—262no—
273C42—521no—
274C42—521no—
275C13#987811yescharacteristic, complemented, Sylow
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#8824012yes—
281C23Q8×#26112yes—
282C23C22×#105711yescharacteristic
283C23C2×#612211yescharacteristic
284C23C2×C22⋊#26111yescharacteristic
285Q8—48no—
286Q8—48no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—132no—
297C2×C4—132no—
298C2×C4—131no—
299C2×C4—131no—
300C2×C4—261no—
301C2×C4—261no—
302C2×C4—261no—
303C2×C4—261no—
304C2×C4—261no—
305C2×C4—261no—
306C2×C4—261no—
307C2×C4—261no—
308C2×C4—261no—
309C2×C4—522no—
310C2×C4—522no—
311C2×C4—524no—
312C2×C4—522no—
313C2×C4—522no—
314C2×C4—522no—
315C2×C4—1044no—
316C2×C4#8811912yes—
317C2×C4C22⋊#105712yes—
318C2×C4C22⋊#105712yes—
319C2×C4#8812112yes—
320C8—524no—
321C8—1042no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22#1171915611yescharacteristic
331C22Q8×#105712yes—
332C22#1171915812yes—
333C22#1171915911yescharacteristic
334C22D13⋊#10111yescharacteristic
335C22(C2×#261)⋊Q811yescharacteristic
336C22C2×#8811911yescharacteristic
337C22C2×C22⋊#105711yescharacteristic
338C22C2×#8812111yescharacteristic
339C4—22no—
340C4—44no—
341C4—262no—
342C4—261no—
343C4—261no—
344C4—262no—
345C4—521no—
346C4—1042no—
347C4#1171830514yes—
348C2—22no—
349C2—22no—
350C2C2×#1171830511yescharacteristic
351C2#1313007911yescharacteristic
352C2#1313010711yescharacteristic
353C2#1313011611yescharacteristic
354C2#1313100511yescharacteristic
355C2#1313102511yescharacteristic
356C2#1313102811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC23×C13C22×C8
FrattiniC24C22×#261
FittingC23×Q8×C13C4
socleC23×C13C2×C22⋊C4
prime radicalC24×C13C22×C4
cosocleC23×Dic13C23
hypercenterC23×Q8#261
hypocenterC13#9878
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
25. C22×Q8×C13C8semidirect
43. C22⋊#1057Q8semidirect
194. C22×Q8#1057semidirect
275. C13#9878semidirect

Maximal subgroups

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

subgroupindex
#131306902
C4×C22⋊#10572
#131296742
C2×#117188912
#987813