Finite Group Databasebeta

← #16795255

Subgroups — #16795255

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131751C212yesmaximal, complemented
2#13130432C212yesmaximal, complemented
3#13130429C212yesmaximal, complemented
4SD16×#1057C214yesmaximal, complemented
5C2×#11718253C211yesmaximal, characteristic, complemented
6#13130131C212yesmaximal, complemented
7#13130039C211yesmaximal, characteristic, complemented
8#135×Dic13C211yesmaximal, characteristic
9C2×C13×#135C411yescharacteristic
10#41×Dic13C2212yes—
11Q8×#1057C2214yes—
12D4×#1057—24no—
13D4×#1057C2214yes—
14#11719119—22no—
15#11719119C2211yescharacteristic
16#11719119C2211yescharacteristic
17C2×C4⋊#1057—22no—
18C2×C4⋊#1057C2212yes—
19#11718894C412yes—
20Q16×Dic13C411yescharacteristic
21Q16×Dic13C2211yescharacteristic
22SD16×Dic13C412yes—
23SD16×Dic13C2212yes—
24D8×Dic13C411yescharacteristic
25D8×Dic13C2211yescharacteristic
26#11718767C411yescharacteristic
27C2×C8×Dic13C2211yescharacteristic
28Q8⋊#1057C2214yes—
29D4⋊#1057C2214yes—
30#11718253C2212yes—
31#11718253C2212yes—
32#11718251C2212yes—
33C8×#1057C2212yes—
34C2×C13×#41C2×C412yes—
35C22×#1057—22no—
36C22×#1057—24no—
37C13×#135C818yescomplemented
38C2×C13×Q16C2×C411yescharacteristic
39C2×C13×SD16C2×C412yes—
40C2×C13×D8C2×C411yescharacteristic
41C22×C8×C13C2×C411yescharacteristic
42Q8×Dic13C2×C412yes—
43Q8×Dic13C2312yes—
44D4×Dic13—22no—
45D4×Dic13—22no—
46D4×Dic13C2×C412yes—
47D4×Dic13C2312yes—
48#88180—22no—
49#88180C2×C412yes—
50C2×C4×Dic13—22no—
51C2×C4×Dic13C2311yescharacteristic
52C22×#1055C2×C411yescharacteristic
53C22⋊#1057—21no—
54C22⋊#1057—24no—
55C22⋊#1057D412yes—
56C4⋊#1057—24no—
57C4⋊#1057—22no—
58C4⋊#1057—21no—
59C4⋊#1057D412yes—
60C4⋊#1057C2314yes—
61C4×#1057—24no—
62C4×#1057C2312yes—
63C4×#1057C2312yes—
64#88075C2×C412yes—
65#88075C2×C412yes—
66#88072C2×C412yes—
67#88072C2×C412yes—
68#88058C2×C411yescharacteristic
69#88058C2×C411yescharacteristic
70#88057C2×C412yes—
71C8×Dic13C2×C411yescharacteristic
72C8×Dic13C2×C411yescharacteristic
73C8×Dic13C2311yescharacteristic
74C8×Dic13C2311yescharacteristic
75#17858—131nomaximal, Sylow
76C13×#41—24no—
77C13×#41C2×C818yescomplemented
78C2×Q8×C13C22×C412yes—
79C2×D4×C13—22no—
80C2×D4×C13C22×C412yes—
81C22×C4×C13—22no—
82C22×C4×C13C22×C411yescharacteristic
83C22×Dic13—22no—
84C22×Dic13—22no—
85C22×Dic13C2×D411yescharacteristic
86C22×Dic13#4111yescharacteristic
87C2×#1057—24no—
88C2×#1057—22no—
89C2×#1057—84no—
90C2×#1057—82no—
91C2×#1057C2×D414yes—
92C13×Q16C2×C814yescomplemented
93C13×SD16C2×C818yescomplemented
94C13×D8C2×C814yescomplemented
95C2×C8×C13C2×C814yescomplemented
96C2×C8×C13C22×C411yescharacteristic
97C2×C8×C13C22×C411yescharacteristic
98C22⋊Dic13—22no—
99C22⋊Dic13—22no—
100C22⋊Dic13—22no—
101C4⋊Dic13—22no—
102C4⋊Dic13—22no—
103C4⋊Dic13C22×C412yes—
104C4⋊Dic13C22×C412yes—
105C4×Dic13—22no—
106C4×Dic13—22no—
107C4×Dic13—22no—
108C4×Dic13C22×C411yescharacteristic
109C4×Dic13C2×D411yescharacteristic
110C4×Dic13#4111yescharacteristic
111C4×Dic13C2411yescharacteristic
112C2×#1055—22no—
113C2×#1055C22×C411yescharacteristic
114C2×#1055C22×C411yescharacteristic
115#2988—132no—
116C4×#135—131no—
117#1580—132no—
118#1579—132no—
119C8×SD16—134no—
120C2×#334—131no—
121#1477—132no—
122#1450—131no—
123C23×C13—22no—
124C23×C13C4×D411yescharacteristic
125Q8×C13—22no—
126Q8×C13C22×C814yes—
127D4×C13—28no—
128D4×C13—22no—
129D4×C13C22×C814yes—
130C2×C4×C13—22no—
131C2×C4×C13—28no—
132C2×C4×C13C4×D411yescharacteristic
133C2×C4×C13C22×C812yes—
134C2×C4×C13#13112yes—
135C2×C4×C13C23×C411yescharacteristic
136C2×Dic13—22no—
137C2×Dic13—22no—
138C2×Dic13—22no—
139C2×Dic13—41no—
140C2×Dic13—42no—
141C2×Dic13—42no—
142C2×Dic13—82no—
143C2×Dic13C4×D411yescharacteristic
144C2×Dic13C4×D411yescharacteristic
145C2×Dic13C22×D411yescharacteristic
146C2×Dic13C2×#4111yescharacteristic
147#1057—24no—
148#1057—42no—
149#1057—44no—
150C8×C13C22×C812yes—
151C8×C13C22×C812yes—
152#1055—21no—
153#1055—21no—
154C2×#135#26111yescharacteristic
155C4×#41—132no—
156C8×Q8—134no—
157C4×Q16—131no—
158C4×Q16—131no—
159C4×SD16—132no—
160C4×SD16—132no—
161C4×D8—131no—
162C4×D8—131no—
163C8×D4—134no—
164C8×D4—264no—
165#431—131no—
166#431—131no—
167#431—262no—
168#427—131no—
169C2×C4⋊C8—132no—
170C2×C4⋊C8—262no—
171#416—132no—
172C2×C4×C8—131no—
173#334—132no—
174#334—132no—
175Q8⋊C8—134no—
176D4⋊C8—134no—
177#322—132no—
178C82—132no—
179C22×C13—21no—
180C22×C13—21no—
181C22×C13—24no—
182C22×C13—42no—
183C22×C13—42no—
184C22×C13C8×D412yes—
185C22×C13C2×C4×D411yescharacteristic
186C4×C13—24no—
187C4×C13C8×D412yes—
188C4×C13C23×C811yescharacteristic
189C4×C13C2×#13111yescharacteristic
190Dic13—21no—
191Dic13—21no—
192Dic13—21no—
193Dic13—42no—
194Dic13—42no—
195Dic13#57712yes—
196C2×#41C2×#26112yes—
197#135#105718yescomplemented
198C2×Q16C2×#26111yescharacteristic
199C2×SD16C2×#26112yes—
200C2×D8C2×#26111yescharacteristic
201C22×C8—131no—
202C22×C8—264no—
203C22×C8—262no—
204C22×C8C2×#26111yescharacteristic
205C4×Q8—132no—
206C4×Q8—132no—
207C4×D4—132no—
208C4×D4—132no—
209C4×D4—262no—
210C4×D4—262no—
211#117—132no—
212#117—262no—
213C2×C42—131no—
214C2×C42—262no—
215#107—131no—
216#107—131no—
217#106—132no—
218C4⋊C8—132no—
219C4⋊C8—134no—
220C4⋊C8—262no—
221C4⋊C8—261no—
222C4⋊C8—264no—
223#103—132no—
224#103—132no—
225#102—132no—
226#102—132no—
227C22⋊C8—132no—
228C22⋊C8—264no—
229C22⋊C8—261no—
230C4×C8—132no—
231C4×C8—131no—
232C4×C8—131no—
233C4×C8—131no—
234C4×C8—131no—
235C4×C8—132no—
236C4×C8—264no—
237C2×C13—22no—
238C2×C13—44no—
239C2×C13C2×C8×D411yescharacteristic
240C2×C13#295211yescharacteristic
241C2×C13#295711yescharacteristic
242#41—24no—
243#41C2×#105718yescomplemented
244C2×Q8C22×#26112yes—
245C2×D4—22no—
246C2×D4C22×#26112yes—
247C22×C4—22no—
248C22×C4—131no—
249C22×C4—131no—
250C22×C4—262no—
251C22×C4—262no—
252C22×C4C22×#26111yescharacteristic
253Q16C2×#105714yescomplemented
254SD16C2×#105718yescomplemented
255D8C2×#105714yescomplemented
256C2×C8—134no—
257C2×C8—131no—
258C2×C8—131no—
259C2×C8—262no—
260C2×C8—262no—
261C2×C8—264no—
262C2×C8—1044no—
263C2×C8—1042no—
264C2×C8C2×#105714yescomplemented
265C2×C8C22×#26111yescharacteristic
266C2×C8C22×#26111yescharacteristic
267C4⋊C4—132no—
268C4⋊C4—132no—
269C4⋊C4—262no—
270C4⋊C4—262no—
271C22⋊C4—262no—
272C22⋊C4—262no—
273C22⋊C4—262no—
274C42—131no—
275C42—131no—
276C42—131no—
277C42—131no—
278C42—262no—
279C42—262no—
280C42—262no—
281C13#1785811yescharacteristic, complemented, Sylow
282C23—22no—
283C23D4×#26111yescharacteristic
284Q8—22no—
285Q8C22×#105714yes—
286D4—28no—
287D4—22no—
288D4C22×#105714yes—
289C2×C4—22no—
290C2×C4—28no—
291C2×C4—131no—
292C2×C4—131no—
293C2×C4—131no—
294C2×C4—131no—
295C2×C4—262no—
296C2×C4—262no—
297C2×C4—262no—
298C2×C4—522no—
299C2×C4—522no—
300C2×C4—521no—
301C2×C4—1042no—
302C2×C4#8823812yes—
303C2×C4D4×#26111yescharacteristic
304C2×C4C22×#105712yes—
305C2×C4C23×#26111yescharacteristic
306C8—264no—
307C8—261no—
308C8—261no—
309C8—524no—
310C8—522no—
311C8C22×#105712yes—
312C8C22×#105712yes—
313C22—21no—
314C22—21no—
315C22—24no—
316C22—42no—
317C22—42no—
318C22D4×#105712yes—
319C22C2×D4×#26111yescharacteristic
320C4—24no—
321C4—132no—
322C4—261no—
323C4—261no—
324C4—261no—
325C4—522no—
326C4—522no—
327C4D4×#105712yes—
328C4C2×#8823811yescharacteristic
329C4C23×#105711yescharacteristic
330C2—22no—
331C2—44no—
332C2C2×D4×#105711yescharacteristic
333C2#1313159911yescharacteristic
334C2#1313161511yescharacteristic
subgroupG/H
centerC22C2×D4×#261
commutatorC4×C13C23×C8
FrattiniC2×C4C23×#261
Fitting, prime radicalC2×C13×#135C4
socleC22×C13C2×C4×D4
cosocleC4×Dic13C24
hypercenterC2×#135#261
hypocenterC13#17858
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #13131751C2semidirect
2. #13130432C2semidirect
3. #13130429C2semidirect
4. SD16×#1057C2semidirect
5. C2×#11718253C2semidirect
6. #13130131C2semidirect
7. #13130039C2semidirect
37. C13×#135C8semidirect
77. C13×#41C2×C8semidirect
92. C13×Q16C2×C8semidirect
93. C13×SD16C2×C8semidirect
94. C13×D8C2×C8semidirect
95. C2×C8×C13C2×C8semidirect
197. #135#1057semidirect
243. #41C2×#1057semidirect
253. Q16C2×#1057semidirect
254. SD16C2×#1057semidirect
255. D8C2×#1057semidirect
264. C2×C8C2×#1057semidirect
281. C13#17858semidirect

Maximal subgroups

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

subgroupindex
#131317512
#131304322
#131304292
SD16×#10572
C2×#117182532
#131301312
#131300392
#135×Dic132
#1785813