Finite Group Databasebeta

← #16780486

Subgroups — #16780486

Order 3328.

530 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊#1057C211yesmaximal, characteristic
2C4×#88110C211yesmaximal, characteristic
3C2×Q8⋊#1057C212yesmaximal
4#13129672C212yesmaximal
5#13117311C211yesmaximal, characteristic
6C2×Q8×Dic13C2211yescharacteristic
7C2×#88110—21no—
8C2×#88110—21no—
9C2×C4⋊#1057C2212yes—
10C2×C4×#1057—21no—
11C2×C4×#1057C2212yes—
12C13×#391C411yescharacteristic
13#11718891C411yescharacteristic
14C4⋊C4×Dic13C2211yescharacteristic
15#11718634C411yescharacteristic
16C42×Dic13C2211yescharacteristic
17#11718589C411yescharacteristic
18#11718315—21no—
19#11718314—21no—
20#11718310—21no—
21#11718309—21no—
22Q8⋊#1057C418yescomplemented
23#11718290—21no—
24C22×Q8×C13C2×C411yescharacteristic
25C22×#1057—22no—
26C22×#1057—22no—
27C2×C13×C4⋊C4C2×C411yescharacteristic
28C2×C42×C13C2×C411yescharacteristic
29Q8×Dic13—22no—
30Q8×Dic13C2×C414yes—
31C2×C4⋊Dic13—21no—
32C2×C4⋊Dic13C2×C411yescharacteristic
33C2×C4⋊Dic13C2×C411yescharacteristic
34C2×#6101—21no—
35C2×#6101—21no—
36C2×C4×Dic13—21no—
37C2×C4×Dic13—21no—
38C2×C4×Dic13C2×C411yescharacteristic
39C2×C4×Dic13D411yescharacteristic
40C2×C4×Dic13D411yescharacteristic
41C2×C4×Dic13D411yescharacteristic
42C2×C4×Dic13D411yescharacteristic
43C2×C4×Dic13C2311yescharacteristic
44#88110—24no—
45#88110—24no—
46C4⋊#1057—22no—
47C4⋊#1057C2×C414yes—
48C4×#1057—22no—
49C4×#1057—42no—
50C4×#1057C2×C414yes—
51C13×#95—21no—
52C13×#95—21no—
53#88071—21no—
54#88071—21no—
55#88071—21no—
56#88071—21no—
57#88071—21no—
58#8180—131nomaximal, Sylow
59C2×Q8×C13—21no—
60C2×Q8×C13—22no—
61C2×Q8×C13—22no—
62C2×Q8×C13C4212yes—
63C22×C4×C13—21no—
64C22×C4×C13—21no—
65C22×C4×C13C22⋊C411yescharacteristic
66C22×C4×C13C22⋊C411yescharacteristic
67C22×C4×C13C22×C411yescharacteristic
68C22×Dic13—21no—
69C22×Dic13—21no—
70C22×Dic13C22⋊C411yescharacteristic
71C22×Dic13C22⋊C411yescharacteristic
72C22×Dic13C2×D411yescharacteristic
73C22×Dic13C2×D411yescharacteristic
74C2×#1057—24no—
75C2×#1057—24no—
76C2×#1057—24no—
77C2×#1057—24no—
78C2×#1057—42no—
79C2×#1057—42no—
80C13×C4⋊C4—22no—
81C42×C13—22no—
82C4⋊Dic13—21no—
83C4⋊Dic13—24no—
84C4⋊Dic13—22no—
85C4⋊Dic13C4212yes—
86#6101—42no—
87#6101—42no—
88C4×Dic13—22no—
89C4×Dic13—22no—
90C4×Dic13—22no—
91C4×Dic13—24no—
92C4×Dic13—21no—
93C4×Dic13—21no—
94C4×Dic13—22no—
95C4×Dic13—41no—
96C4×Dic13—41no—
97C4×Dic13C22×C412yes—
98C4×Dic13#4112yes—
99#2295—131no—
100C4×C4⋊C8—131no—
101C4×#97—131no—
102C2×Q8⋊C8—132no—
103#1279—132no—
104C23×C13C2×C22⋊C411yescharacteristic
105Q8×C13—24no—
106Q8×C13—42no—
107C2×C4×C13—21no—
108C2×C4×C13—21no—
109C2×C4×C13—22no—
110C2×C4×C13—21no—
111C2×C4×C13—21no—
112C2×C4×C13—21no—
113C2×C4×C13—41no—
114C2×C4×C13—41no—
115C2×C4×C13—41no—
116C2×C4×C13—41no—
117C2×C4×C13—41no—
118C2×C4×C13#10412yes—
119C2×C4×C13C2×C4211yescharacteristic
120C2×C4×C13#11711yescharacteristic
121C2×C4×C13#13112yes—
122C2×C4×C13#13112yes—
123C2×Dic13—21no—
124C2×Dic13—21no—
125C2×Dic13—21no—
126C2×Dic13—21no—
127C2×Dic13—22no—
128C2×Dic13—21no—
129C2×Dic13—21no—
130C2×Dic13—21no—
131C2×Dic13—41no—
132C2×Dic13—41no—
133C2×Dic13—41no—
134C2×Dic13—41no—
135C2×Dic13—41no—
136C2×Dic13#10412yes—
137C2×Dic13C4×D412yes—
138C2×Dic13C4×D412yes—
139C2×Dic13AGL(1,ℤ/8)11yescharacteristic
140C2×Dic13#13712yes—
141C2×Dic13#13711yescharacteristic
142#1057—28no—
143#1057—44no—
144C2×C4×Q8—131no—
145C2×C4⋊C8—132no—
146C2×#97—261no—
147C2×#97—261no—
148C2×C4×C8—132no—
149C2×C4×C8—261no—
150#391—131no—
151#391#26111yescharacteristic
152#389—131no—
153#389—131no—
154#387—261no—
155#387—261no—
156C4×C4⋊C4—131no—
157C4×C4⋊C4—261no—
158#376—261no—
159C43—131no—
160#336—261no—
161Q8⋊C8—138no—
162C22×C13C4×C22⋊C411yescharacteristic
163C22×C13#40811yescharacteristic
164C22×C13#40811yescharacteristic
165C22×C13#41711yescharacteristic
166C22×C13#41911yescharacteristic
167C22×C13C2×#10411yescharacteristic
168C22×C13C2×#10411yescharacteristic
169C4×C13—22no—
170C4×C13—41no—
171C4×C13—42no—
172C4×C13—81no—
173C4×C13#40512yes—
174C4×C13#43312yes—
175Dic13—22no—
176Dic13—22no—
177Dic13—41no—
178Dic13—42no—
179Dic13—81no—
180Dic13#44012yes—
181Dic13#44112yes—
182C22×Q8C2×#26111yescharacteristic
183C22×C8—262no—
184C22×C8—262no—
185C4×Q8—134no—
186C4×Q8—262no—
187C2×C4⋊C4—131no—
188C2×C4⋊C4—131no—
189C2×C4⋊C4—261no—
190C2×C4⋊C4—261no—
191C2×C4⋊C4—261no—
192C2×C4⋊C4C2×#26111yescharacteristic
193C2×C42—131no—
194C2×C42—131no—
195C2×C42—131no—
196C2×C42—131no—
197C2×C42—131no—
198C2×C42—131no—
199C2×C42—261no—
200C2×C42—261no—
201C2×C42C2×#26111yescharacteristic
202C4⋊C8—134no—
203C4⋊C8—262no—
204#97—264no—
205#97—264no—
206C4×C8—134no—
207C4×C8—262no—
208C4×C8—522no—
209#95—21no—
210#95—21no—
211#95—261no—
212#95—261no—
213#95—261no—
214#95—261no—
215#95—261no—
216C2×C13#148311yescharacteristic
217C2×C13#148311yescharacteristic
218C2×C13#148511yescharacteristic
219C2×C13#148511yescharacteristic
220C2×C13#175011yescharacteristic
221C2×C13C4×#10411yescharacteristic
222C2×C13#176311yescharacteristic
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8C4×#26112yes—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—131no—
230C22×C4—131no—
231C22×C4—131no—
232C22×C4—131no—
233C22×C4—261no—
234C22×C4—261no—
235C22×C4C22⋊#26111yescharacteristic
236C22×C4C22⋊#26111yescharacteristic
237C22×C4C22×#26111yescharacteristic
238C2×C8—264no—
239C2×C8—264no—
240C2×C8—264no—
241C2×C8—264no—
242C2×C8—522no—
243C2×C8—522no—
244C4⋊C4—22no—
245C4⋊C4—132no—
246C4⋊C4—264no—
247C4⋊C4—261no—
248C4⋊C4—262no—
249C4⋊C4—522no—
250C4⋊C4—522no—
251C42—22no—
252C42—132no—
253C42—132no—
254C42—261no—
255C42—262no—
256C42—261no—
257C42—262no—
258C42—264no—
259C42—262no—
260C42—262no—
261C42—521no—
262C42—521no—
263C13#818011yescharacteristic, complemented, Sylow
264C23C2×C22⋊#26111yescharacteristic
265Q8—24no—
266Q8—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—132no—
279C2×C4—132no—
280C2×C4—131no—
281C2×C4—131no—
282C2×C4—132no—
283C2×C4—132no—
284C2×C4—261no—
285C2×C4—261no—
286C2×C4—261no—
287C2×C4—261no—
288C2×C4—262no—
289C2×C4—261no—
290C2×C4—261no—
291C2×C4—261no—
292C2×C4—521no—
293C2×C4—521no—
294C2×C4—521no—
295C2×C4—521no—
296C2×C4—521no—
297C2×C4#8811812yes—
298C2×C4C2×C4×#26111yescharacteristic
299C2×C4#8823711yescharacteristic
300C2×C4#8823812yes—
301C2×C4#8823812yes—
302C8—268no—
303C8—524no—
304C22C4×C22⋊#26111yescharacteristic
305C22#1171914611yescharacteristic
306C22#1171914711yescharacteristic
307C22#1171915311yescharacteristic
308C22C2×#8811811yescharacteristic
309C22D13⋊#10411yescharacteristic
310C22#1171916411yescharacteristic
311C4—22no—
312C4—42no—
313C4—41no—
314C4—81no—
315C4—132no—
316C4—132no—
317C4—262no—
318C4—262no—
319C4—522no—
320C4—521no—
321C4—1041no—
322C4#1171906212yes—
323C4#1171906812yes—
324C2#1313015211yescharacteristic
325C2#1313015311yescharacteristic
326C2#1313015811yescharacteristic
327C2#1313015911yescharacteristic
328C2#1313056611yescharacteristic
329C2C4×#8811811yescharacteristic
330C2#1313060811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC2×C4×C13C2×C42
FrattiniC22×C4C22×#261
FittingC13×#391C4
socle, prime radicalC23×C13C2×C22⋊C4
cosocleC2×C4×Dic13C23
hypercenter#391#261
hypocenterC13#8180
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
22. Q8⋊#1057C4semidirect
263. C13#8180semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#10572
C4×#881102
C2×Q8⋊#10572
#131296722
#131173112
#818013