Finite Group Databasebeta

← #16790326

Subgroups — #16790326

Order 3328.

426 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C2×Q16×#261C211yesmaximal, characteristic
2C2×#11718222C211yesmaximal, characteristic
3C2×#11718280C211yesmaximal, characteristic
4#13130940C211yesmaximal, characteristic
5#13129893C211yesmaximal, characteristic
6(C2×#261)⋊C16C211yesmaximal, characteristic
7D13×#358C211yesmaximal, characteristic
8C2×Q8×#261—21no—
9C2×Q16×D13C2211yescharacteristic
10C2×#88117—21no—
11Q16×#261—22no—
12Q16×#261—22no—
13C2×#88102C2211yescharacteristic
14C2×#88102C2211yescharacteristic
15C2×C8×#261C2211yescharacteristic
16C2×D13⋊C16C2211yescharacteristic
17#11719083—21no—
18C2×C16×D13C2211yescharacteristic
19D13×#107C2211yescharacteristic
20#11718280—22no—
21#11718246—21no—
22#11718222—22no—
23C13×#358C411yescharacteristic, complemented
24#11718156C411yescharacteristic, complemented
25#11718095C411yescharacteristic, complemented
26#11718084C411yescharacteristic, complemented
27C2×Q8×D13—21no—
28Q8×#261—24no—
29Q8×#261—42no—
30C2×C4⋊#261—21no—
31C2×C4⋊#261—21no—
32C2×C4⋊#261—41no—
33C2×C4⋊#261—41no—
34C2×C4×#261—41no—
35C2×C4×#261D411yescharacteristic
36C2×D13⋊C8D411yescharacteristic
37C2×C13×Q16C2×C411yescharacteristic
38C2×#6107—21no—
39Q16×D13—21no—
40Q16×D13—21no—
41Q16×D13D412yes—
42Q16×D13Q812yes—
43C2×Dic52C2×C411yescharacteristic
44C2×C8×D13C2311yescharacteristic
45C4⋊C4×D13—21no—
46#88117—24no—
47(C2×#261)⋊C4—41no—
48C2×#6092—21no—
49#88102—21no—
50#88102—21no—
51#88102—21no—
52#88102D412yes—
53C8×#261—21no—
54C8×#261—21no—
55D13⋊C16—21no—
56D13⋊C16—21no—
57C2×C13×C16C2×C411yescharacteristic
58C13×#107C2×C411yescharacteristic
59#88058C2×C411yescharacteristic
60C2×#6090C2×C411yescharacteristic
61#88047—21no—
62C16×D13—21no—
63C16×D13—21no—
64#12938—131nomaximal, Sylow
65C22×#261—41no—
66C22×#261—41no—
67C22×#261D811yescharacteristic
68C22×#261D811yescharacteristic
69C2×Q8×C13—21no—
70Q8×D13—22no—
71Q8×D13—22no—
72Q8×D13—41no—
73Q8×D13—41no—
74C2×C4×D13—41no—
75C2×C4×D13—41no—
76C2×C4×D13C2×D411yescharacteristic
77C2×Dic26—21no—
78C2×Dic26—41no—
79C2×#1057—21no—
80C4⋊#261—22no—
81C4⋊#261—44no—
82C4⋊#261—41no—
83C4⋊#261—41no—
84C4⋊#261—41no—
85C4⋊#261—82no—
86C4×#261—21no—
87C4×#261—21no—
88C4×#261—44no—
89D13⋊C8—21no—
90D13⋊C8—21no—
91C13×Q16—21no—
92C13×Q16C4⋊C412yescomplemented
93C2×C8×C13C22×C411yescharacteristic
94C13×C4⋊C4—21no—
95#6107—22no—
96#6107—22no—
97C4⋊Dic13—21no—
98#6101—41no—
99C2×#1055#4111yescharacteristic
100Dic52—21no—
101Dic52C4⋊C412yescomplemented
102C8×D13C2×D411yescharacteristic
103C8×D13C2×D411yescharacteristic
104C8×D13C2×Q811yescharacteristic
105C8×D13#4111yescharacteristic
106#6092—41no—
107C13×C16—21no—
108#6090—21no—
109C2×C4×Q16—131no—
110C2×#367—131no—
111C2×#358—131no—
112C2×#358—131no—
113#1942—131no—
114#1380—131no—
115#1374—131no—
116C22×D13C2×D811yescharacteristic
117C2×#261—21no—
118C2×#261—21no—
119C2×#261—21no—
120C2×#261—21no—
121C2×#261—21no—
122C2×#261—42no—
123C2×#261—81no—
124C2×#261—81no—
125C2×#261—81no—
126C2×#261—81no—
127C2×#261—81no—
128C2×#261SD3212yes—
129Q8×C13—22no—
130Q8×C13—41no—
131C2×C4×C13—41no—
132C2×C4×C13—41no—
133C2×C4×C13C4×D411yescharacteristic
134C2×Dic13—41no—
135C2×Dic13—41no—
136C2×Dic13#13511yescharacteristic
137C4×D13—42no—
138C4×D13—42no—
139C4×D13—81no—
140C4×D13—81no—
141C4×D13#12011yescharacteristic
142C4×D13C22⋊Q811yescharacteristic
143C4×D13AGL(1,ℤ/8)11yescharacteristic
144C4×D13#13711yescharacteristic
145Dic26—22no—
146Dic26—41no—
147Dic26—42no—
148Dic26—42no—
149#1057—41no—
150C8×C13C2×C4⋊C411yescharacteristic
151C8×C13C4×D411yescharacteristic
152#1055C22⋊Q811yescharacteristic
153#1055#12311yescharacteristic
154C22×Q16—131no—
155C2×C4×Q8—261no—
156C22×C16—131no—
157C22×C16—131no—
158C4×Q16—262no—
159C4×Q16—262no—
160C2×#107—131no—
161C2×#107—131no—
162C2×#107—131no—
163C2×#103—261no—
164C2×C4×C8—131no—
165#389—261no—
166#367—262no—
167#358—131no—
168#358—131no—
169#358—131no—
170#358—262no—
171#358#26111yescharacteristic, complemented
172#340—261no—
173C22×C13C4×D811yescharacteristic
174C2×D13#44711yescharacteristic
175C2×D13#47411yescharacteristic
176C2×D13C2×SD3211yescharacteristic
177C2×D13#50811yescharacteristic
178C2×D13#51011yescharacteristic
179C2×D13#51011yescharacteristic
180#261—22no—
181#261—41no—
182#261—82no—
183#261—161no—
184C4×C13—42no—
185C4×C13—81no—
186C4×C13#38511yescharacteristic
187C4×C13#44011yescharacteristic
188Dic13—42no—
189Dic13—81no—
190Dic13#45211yescharacteristic
191Dic13#47811yescharacteristic
192C22×Q8—261no—
193C2×Q16—132no—
194C2×Q16—132no—
195C2×Q16—131no—
196C2×Q16—261no—
197C2×Q16—261no—
198C2×Q16—261no—
199C2×Q16C2×#26111yescharacteristic
200C22×C8—131no—
201C22×C8—131no—
202C4×Q8—264no—
203C4×Q8—522no—
204C2×C4⋊C4—261no—
205C2×C4⋊C4—261no—
206C2×C4⋊C4—261no—
207C2×C4⋊C4—521no—
208C2×C4⋊C4—521no—
209C2×C42—131no—
210C2×C42—521no—
211C2×C16—131no—
212C2×C16—261no—
213C2×C16—261no—
214C2×C16—261no—
215C2×C16—261no—
216C2×C16—261no—
217C2×C16C2×#26111yescharacteristic
218#107—132no—
219#107—131no—
220#107—261no—
221#107—261no—
222#107—261no—
223#107—261no—
224#107C2×#26111yescharacteristic
225#103—264no—
226C4×C8—261no—
227C4×C8—261no—
228#95—521no—
229C2×C13#186211yescharacteristic
230C2×C13C4×SD3211yescharacteristic
231C2×C13#217611yescharacteristic
232D13#218511yescharacteristic
233D13#218911yescharacteristic
234D13#222511yescharacteristic
235D13#222911yescharacteristic
236C2×Q8—21no—
237C2×Q8—262no—
238C2×Q8—262no—
239C2×Q8—261no—
240C2×Q8—521no—
241C2×Q8—521no—
242C2×Q8—521no—
243C22×C4—131no—
244C22×C4—131no—
245C22×C4—131no—
246C22×C4—521no—
247C22×C4—521no—
248C22×C4—521no—
249C22×C4—521no—
250Q16—21no—
251Q16—132no—
252Q16—262no—
253Q16—262no—
254Q16—261no—
255Q16C4⋊#26112yescomplemented
256C2×C8—131no—
257C2×C8—131no—
258C2×C8—131no—
259C2×C8—131no—
260C2×C8—131no—
261C2×C8—261no—
262C2×C8—261no—
263C2×C8—261no—
264C2×C8C22×#26111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—262no—
267C4⋊C4—261no—
268C4⋊C4—521no—
269C4⋊C4—521no—
270C4⋊C4—521no—
271C4⋊C4—521no—
272C4⋊C4—524no—
273C4⋊C4—1042no—
274C42—261no—
275C42—261no—
276C42—524no—
277C16—21no—
278C16—261no—
279C16—521no—
280C13#1293811yescharacteristic, complemented, Sylow
281C23—131no—
282Q8—22no—
283Q8—41no—
284Q8—262no—
285Q8—522no—
286Q8—522no—
287Q8—521no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—131no—
291C2×C4—131no—
292C2×C4—131no—
293C2×C4—131no—
294C2×C4—131no—
295C2×C4—132no—
296C2×C4—261no—
297C2×C4—261no—
298C2×C4—261no—
299C2×C4—261no—
300C2×C4—261no—
301C2×C4—522no—
302C2×C4—522no—
303C2×C4—521no—
304C2×C4—522no—
305C2×C4—521no—
306C2×C4—1041no—
307C2×C4—1041no—
308C2×C4—1041no—
309C2×C4—1041no—
310C2×C4—1041no—
311C2×C4—1041no—
312C2×C4—1041no—
313C2×C4D4×#26111yescharacteristic
314C8—131no—
315C8—131no—
316C8—521no—
317C8C2×C4⋊#26111yescharacteristic
318C8D4×#26111yescharacteristic
319C22—131no—
320C22—131no—
321C22—131no—
322C22—131no—
323C22—131no—
324C22—131no—
325C22D8×#26111yescharacteristic
326C4—42no—
327C4—81no—
328C4—131no—
329C4—131no—
330C4—262no—
331C4—521no—
332C4—522no—
333C4—1042no—
334C4—1041no—
335C4—2081no—
336C4#1171907011yescharacteristic
337C4#1171910611yescharacteristic
338C2—131no—
339C2—131no—
340C2—131no—
341C2—131no—
342C2#1313079911yescharacteristic
343C2SD32×#26111yescharacteristic
344C2#1313123311yescharacteristic
subgroupG/H
centerC22D8×#261
commutatorC2×C8×C13C22×C4
FrattiniC2×C8C22×#261
FittingC13×#358C4
socleC22×C13C4×D8
prime radicalC22×D13C2×D8
cosocleC2×C8×D13C23
hypercenter#358#261
hypocenterC13#12938
LCSG ▷ C2×C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #358

Complements

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

NH ≅ G/Ntype
23. C13×#358C4semidirect
24. #11718156C4semidirect
25. #11718095C4semidirect
26. #11718084C4semidirect
92. C13×Q16C4⋊C4semidirect
101. Dic52C4⋊C4semidirect
171. #358#261semidirect
255. Q16C4⋊#261semidirect
280. C13#12938semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×#2612
C2×#117182222
C2×#117182802
#131309402
#131298932
(C2×#261)⋊C162
D13×#3582
#1293813