Finite Group Databasebeta

← #16785072

Subgroups — #16785072

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×#11718635C211yesmaximal, characteristic
5C23×Dic26C411yescharacteristic
6C22×C13×C4⋊C4C411yescharacteristic
7C22×C4×Dic13C2211yescharacteristic
8C2×C22⋊#1057C2212yes—
9C2×C22⋊#1057C2212yes—
10C2×C4⋊#1057—21no—
11C2×C4×#1057—21no—
12C2×#88071C412yes—
13C2×#88071C2212yes—
14#11718635—21no—
15#11718635—21no—
16#11718635—21no—
17#11718635—21no—
18#11718290—21no—
19#11718290—21no—
20C23×C4×C13C2×C412yes—
21C23×C4×C13C2×C411yescharacteristic
22C23×Dic13C2×C411yescharacteristic
23C23×Dic13C2×C412yes—
24C23×Dic13C2311yescharacteristic
25C22×Dic26—21no—
26C22×Dic26—21no—
27C22×Dic26—22no—
28C22×Dic26—42no—
29C22×Dic26—42no—
30C22×Dic26C814yescomplemented
31C22×#1057—22no—
32C22×#1057—22no—
33C2×C13×C4⋊C4—21no—
34C2×C13×C4⋊C4—21no—
35C2×C13×C4⋊C4—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
52C22×C4×C13—22no—
53C22×C4×C13—22no—
54C22×C4×C13—22no—
55C22×C4×C13—24no—
56C22×C4×C13—42no—
57C22×C4×C13—42no—
58C22×C4×C13C22⋊C411yescharacteristic
59C22×C4×C13C22⋊C411yescharacteristic
60C22×C4×C13C2×C812yes—
61C22×C4×C13M1612yes—
62C22×Dic13—22no—
63C22×Dic13—22no—
64C22×Dic13—22no—
65C22×Dic13—22no—
66C22×Dic13—41no—
67C22×Dic13—41no—
68C22×Dic13—41no—
69C22×Dic13—42no—
70C22×Dic13—42no—
71C22×Dic13C22⋊C411yescharacteristic
72C22×Dic13C22⋊C411yescharacteristic
73C22×Dic13C2×C814yes—
74C22×Dic13C2×D411yescharacteristic
75C22×Dic13C2×D411yescharacteristic
76C22×Dic13C2×Q812yes—
77C22×Dic13#4112yes—
78C2×Dic26—24no—
79C2×Dic26—24no—
80C2×Dic26—42no—
81C2×Dic26—42no—
82C2×Dic26—48no—
83C2×Dic26—42no—
84C2×Dic26—42no—
85C2×Dic26—48no—
86C2×#1057—24no—
87C2×#1057—42no—
88C2×#1057—42no—
89C2×#1057—42no—
90C2×#1057—42no—
91C2×#1057—42no—
92C13×C4⋊C4—42no—
93C13×C4⋊C4—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—
111C2×C4×C13—21no—
112C2×C4×C13—21no—
113C2×C4×C13—24no—
114C2×C4×C13—24no—
115C2×C4×C13—42no—
116C2×C4×C13—42no—
117C2×C4×C13—42no—
118C2×C4×C13—84no—
119C2×C4×C13C22⋊C812yes—
120C2×C4×C13C22⋊C812yes—
121C2×C4×C13#9912yes—
122C2×C4×C13#10012yes—
123C2×Dic13—21no—
124C2×Dic13—21no—
125C2×Dic13—21no—
126C2×Dic13—21no—
127C2×Dic13—21no—
128C2×Dic13—21no—
129C2×Dic13—24no—
130C2×Dic13—21no—
131C2×Dic13—21no—
132C2×Dic13—21no—
133C2×Dic13—42no—
134C2×Dic13—42no—
135C2×Dic13—44no—
136C2×Dic13—44no—
137C2×Dic13—42no—
138C2×Dic13—42no—
139C2×Dic13—44no—
140C2×Dic13#10112yes—
141C2×Dic13C22⋊Q812yes—
142C2×Dic13#12411yescharacteristic
143C2×Dic13C4⋊Q811yescharacteristic
144Dic26—48no—
145Dic26—48no—
146#1057—44no—
147#1057—82no—
148C23×Q8—131no—
149C22×C4⋊C4#26111yescharacteristic
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—82no—
182C4×C13#32314yes—
183Dic13—22no—
184Dic13—22no—
185Dic13—21no—
186Dic13—21no—
187Dic13—44no—
188Dic13—41no—
189C22×Q8—134no—
190C22×Q8—262no—
191C22×Q8—261no—
192C22×Q8—261no—
193C22×Q8—522no—
194C22×Q8—522no—
195C23×C4—131no—
196C23×C4—131no—
197C23×C4—132no—
198C23×C4C2×#26112yes—
199C23×C4C2×#26111yescharacteristic
200C22×C8—262no—
201C22×C8—262no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—42no—
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—264no—
230C2×Q8—264no—
231C2×Q8—528no—
232C2×Q8—522no—
233C2×Q8—522no—
234C2×Q8—528no—
235C2×Q8—522no—
236C2×Q8—522no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
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—134no—
250C22×C4—262no—
251C22×C4—262no—
252C22×C4—262no—
253C22×C4—262no—
254C22×C4—521no—
255C22×C4—521no—
256C22×C4—521no—
257C22×C4—522no—
258C22×C4—522no—
259C22×C4D13⋊C812yes—
260C22×C4#611812yes—
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—42no—
270C4⋊C4—42no—
271C42—262no—
272C42—262no—
273C42—521no—
274C42—521no—
275C13#987811yescharacteristic, complemented, Sylow
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23C2×D13⋊C811yescharacteristic
281C23C2×#611811yescharacteristic
282C23#8823812yes—
283C23Q8×#26112yes—
284C23C2×C22⋊#26111yescharacteristic
285Q8—528no—
286Q8—528no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—84no—
295C2×C4—132no—
296C2×C4—132no—
297C2×C4—131no—
298C2×C4—131no—
299C2×C4—261no—
300C2×C4—261no—
301C2×C4—261no—
302C2×C4—261no—
303C2×C4—261no—
304C2×C4—261no—
305C2×C4—261no—
306C2×C4—264no—
307C2×C4—261no—
308C2×C4—261no—
309C2×C4—522no—
310C2×C4—522no—
311C2×C4—524no—
312C2×C4—524no—
313C2×C4—522no—
314C2×C4—522no—
315C2×C4—524no—
316C2×C4#8810712yes—
317C2×C4#8811112yes—
318C2×C4#8811112yes—
319C2×C4#8812112yes—
320C8—524no—
321C8—1042no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22#1171907412yes—
331C22#1171908012yes—
332C22C2×#8810711yescharacteristic
333C22C2×#8811111yescharacteristic
334C22#1171914211yescharacteristic
335C22#1171914711yescharacteristic
336C22D13⋊#10111yescharacteristic
337C22(C2×#261)⋊Q811yescharacteristic
338C22C2×#8812111yescharacteristic
339C4—22no—
340C4—82no—
341C4—262no—
342C4—261no—
343C4—261no—
344C4—262no—
345C4—524no—
346C4—521no—
347C4#1171828714yes—
348C2—22no—
349C2—22no—
350C2C2×#1171828711yescharacteristic
351C2#1313008011yescharacteristic
352C2#1313010611yescharacteristic
353C2#1313011411yescharacteristic
354C2#1313100611yescharacteristic
355C2#1313102611yescharacteristic
356C2#1313102811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC23×C13C22×C8
FrattiniC24C22×#261
FittingC22×C13×C4⋊C4C4
socleC23×C13C2×C22⋊C4
prime radicalC24×C13C22×C4
cosocleC23×Dic13C23
hypercenterC22×C4⋊C4#261
hypocenterC13#9878
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
30. C22×Dic26C8semidirect
43. C22⋊#1057Q8semidirect
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×#117186352
#987813