Finite Group Databasebeta

← #16789688

Subgroups — #16789688

Order 3328.

936 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718304C214yesmaximal, complemented
2#13131063C211yesmaximal, characteristic, complemented
3C2×C22⋊#6092C211yesmaximal, characteristic, complemented
4C22×#88202C211yesmaximal, characteristic
5C22×C13×#41C411yescharacteristic
6C2×#88202—28no—
7C2×#88202—21no—
8C2×#88202—21no—
9C2×#88202C412yes—
10C2×#88202C2212yes—
11C22×#6099—21no—
12C22×#6099C411yescharacteristic
13C23×#1055—21no—
14C23×#1055C2211yescharacteristic
15C2×#88106C2212yescomplemented
16C22×#6092C2212yescomplemented
17#11718304—28no—
18C22⋊#6092—21no—
19C22⋊#6092—21no—
20C22⋊#6092—21no—
21C22⋊#6092—21no—
22C2×C13×#41—24no—
23C2×C13×#41—24no—
24C2×C13×#41—21no—
25C2×C13×#41—21no—
26C2×C13×#41C812yes—
27C2×C13×#41C2×C412yes—
28C22×Q8×C13C811yescharacteristic
29C22×D4×C13—21no—
30C22×D4×C13C811yescharacteristic
31C23×C4×C13—21no—
32C23×C4×C13C2×C411yescharacteristic
33#88202—28no—
34#88202—28no—
35#88202—24no—
36#88202—24no—
37#88202—24no—
38#88202D418yes—
39C2×#6099—28no—
40C2×#6099—21no—
41C2×#6099—24no—
42C2×#6099—22no—
43C2×#6099—22no—
44C2×#6099C2×C412yes—
45C22×#1055—28no—
46C22×#1055—21no—
47C22×#1055—22no—
48C22×#1055—22no—
49C22×#1055—24no—
50C22×#1055C2×C411yescharacteristic
51C22×#1055D411yescharacteristic
52C22×#1055D411yescharacteristic
53C22×#1055D411yescharacteristic
54C22×#1055D411yescharacteristic
55C22×#1055C2311yescharacteristic
56#88106—24no—
57C2×#6092—22no—
58C2×#6092—22no—
59C2×#6092—24no—
60#12683—131nomaximal, Sylow
61C24×C13—21no—
62C24×C13M1611yescharacteristic
63C13×#41—24no—
64C13×#41—28no—
65C13×#41—24no—
66C13×#41—24no—
67C13×#41—24no—
68C13×#41—22no—
69C13×#41—22no—
70C13×#41C22⋊C414yes—
71C13×#41M1614yes—
72C2×Q8×C13—24no—
73C2×Q8×C13—21no—
74C2×Q8×C13C2×C812yes—
75C2×D4×C13—28no—
76C2×D4×C13—22no—
77C2×D4×C13—22no—
78C2×D4×C13—24no—
79C2×D4×C13—21no—
80C2×D4×C13C2×C812yes—
81C22×C4×C13—24no—
82C22×C4×C13—24no—
83C22×C4×C13—21no—
84C22×C4×C13—22no—
85C22×C4×C13—22no—
86C22×C4×C13—22no—
87C22×C4×C13—22no—
88C22×C4×C13C22⋊C411yescharacteristic
89C22×C4×C13C22⋊C411yescharacteristic
90C22×C4×C13C2×C811yescharacteristic
91C22×C4×C13C2×C811yescharacteristic
92C22×C4×C13M1611yescharacteristic
93C22×C4×C13C22×C411yescharacteristic
94#6099—24no—
95#6099—28no—
96#6099—22no—
97#6099—28no—
98#6099C22⋊C414yes—
99C2×#1055—24no—
100C2×#1055—28no—
101C2×#1055—22no—
102C2×#1055—22no—
103C2×#1055—22no—
104C2×#1055—28no—
105C2×#1055—22no—
106C2×#1055—22no—
107C2×#1055—22no—
108C2×#1055C22⋊C411yescharacteristic
109C2×#1055C22⋊C411yescharacteristic
110C2×#1055C2×D411yescharacteristic
111C2×#1055C2×D411yescharacteristic
112C2×#1055C2×D414yes—
113#6092—42no—
114#6092—42no—
115C22×#131—131no—
116C2×#350—134no—
117#2112—131no—
118C2×C22⋊C16—131no—
119C23×C13—24no—
120C23×C13—22no—
121C23×C13—22no—
122C23×C13—22no—
123C23×C13—41no—
124C23×C13—42no—
125C23×C13—42no—
126C23×C13C22⋊C811yescharacteristic
127C23×C13C22⋊C811yescharacteristic
128C23×C13C2×M1611yescharacteristic
129Q8×C13—24no—
130Q8×C13—22no—
131Q8×C13C22⋊C814yes—
132D4×C13—28no—
133D4×C13—24no—
134D4×C13—24no—
135D4×C13—24no—
136D4×C13—22no—
137D4×C13C22⋊C814yes—
138C2×C4×C13—24no—
139C2×C4×C13—28no—
140C2×C4×C13—22no—
141C2×C4×C13—22no—
142C2×C4×C13—21no—
143C2×C4×C13—24no—
144C2×C4×C13—21no—
145C2×C4×C13—22no—
146C2×C4×C13—22no—
147C2×C4×C13—21no—
148C2×C4×C13—24no—
149C2×C4×C13—21no—
150C2×C4×C13C22⋊C811yescharacteristic
151C2×C4×C13C22⋊C811yescharacteristic
152C2×C4×C13C2×C22⋊C412yes—
153C2×C4×C13C2×C22⋊C411yescharacteristic
154C2×C4×C13C22×C811yescharacteristic
155C2×C4×C13C2×M1612yes—
156#1055—21no—
157#1055—24no—
158#1055—21no—
159#1055#12012yes—
160#1055#12012yes—
161C22×#41#26111yescharacteristic
162C2×#131—132no—
163C2×#131—132no—
164C2×#131—261no—
165C2×#131—261no—
166C2×#131—268no—
167C22×M16—131no—
168C22×M16—261no—
169C23×C8—131no—
170C23×C8—261no—
171C2×M32—132no—
172C22×C16—132no—
173#350—268no—
174C22⋊C16—261no—
175C22⋊C16—261no—
176C22⋊C16—261no—
177C22⋊C16—261no—
178C22×C13—21no—
179C22×C13—21no—
180C22×C13—21no—
181C22×C13—21no—
182C22×C13—21no—
183C22×C13—21no—
184C22×C13—21no—
185C22×C13—24no—
186C22×C13—44no—
187C22×C13—48no—
188C22×C13—42no—
189C22×C13—42no—
190C22×C13#35012yescomplemented
191C22×C13C2×C22⋊C811yescharacteristic
192C22×C13#40712yes—
193C4×C13—21no—
194C4×C13—24no—
195C4×C13—21no—
196C4×C13#37911yescharacteristic
197C4×C13C2×C22⋊C812yes—
198C4×C13#40711yescharacteristic
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41#105712yes—
204C2×#41C2×#26112yes—
205C22×Q8#105711yescharacteristic
206C22×D4—21no—
207C22×D4#105711yescharacteristic
208C23×C4—21no—
209C23×C4C2×#26111yescharacteristic
210#131—138no—
211#131—264no—
212#131—264no—
213#131—268no—
214#131—268no—
215#131—264no—
216C2×M16—132no—
217C2×M16—262no—
218C2×M16—262no—
219C2×M16—264no—
220C2×M16—268no—
221C2×M16—261no—
222C22×C8—131no—
223C22×C8—131no—
224C22×C8—131no—
225C22×C8—131no—
226C22×C8—131no—
227C22×C8—131no—
228C22×C8—264no—
229C22×C8—262no—
230C22×C8—262no—
231C22×C8—268no—
232C22×C8—261no—
233M32—264no—
234C2×C16—262no—
235C2×C16—262no—
236C2×C16—264no—
237C2×C13—22no—
238C2×C13—22no—
239C2×C13—41no—
240C2×C13—44no—
241C2×C13#177911yescharacteristic
242C2×C13C2×#35011yescharacteristic
243C2×C13#211711yescharacteristic
244C24—21no—
245C24#612211yescharacteristic
246#41—24no—
247#41—28no—
248#41—24no—
249#41—24no—
250#41—24no—
251#41—22no—
252#41—22no—
253#41#612214yes—
254#41C22⋊#26114yes—
255C2×Q8—24no—
256C2×Q8—21no—
257C2×Q8C2×#105712yes—
258C2×D4—28no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—21no—
263C2×D4C2×#105712yes—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4C2×#105711yescharacteristic
272C22×C4C2×#105711yescharacteristic
273C22×C4#612211yescharacteristic
274C22×C4C22⋊#26111yescharacteristic
275C22×C4C22⋊#26111yescharacteristic
276C22×C4C22×#26111yescharacteristic
277M16—134no—
278M16—268no—
279M16—268no—
280M16—264no—
281M16—262no—
282C2×C8—134no—
283C2×C8—131no—
284C2×C8—131no—
285C2×C8—131no—
286C2×C8—131no—
287C2×C8—262no—
288C2×C8—262no—
289C2×C8—262no—
290C2×C8—262no—
291C2×C8—268no—
292C2×C8—264no—
293C2×C8—268no—
294C2×C8—262no—
295C2×C8—262no—
296C16—522no—
297C16—522no—
298C13#1268311yescharacteristic, complemented, Sylow
299C23—24no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—41no—
304C23—42no—
305C23—42no—
306C23C22⋊#105711yescharacteristic
307C23C22⋊#105711yescharacteristic
308C23C2×#612211yescharacteristic
309Q8—24no—
310Q8—22no—
311Q8C22⋊#105714yes—
312D4—28no—
313D4—24no—
314D4—24no—
315D4—24no—
316D4—22no—
317D4C22⋊#105714yes—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4C22⋊#105711yescharacteristic
331C2×C4C22⋊#105711yescharacteristic
332C2×C4C22×#105711yescharacteristic
333C2×C4C2×#612212yes—
334C2×C4C2×C22⋊#26112yes—
335C2×C4C2×C22⋊#26111yescharacteristic
336C8—132no—
337C8—132no—
338C8—261no—
339C8—261no—
340C8—264no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—44no—
350C22—48no—
351C22—42no—
352C22—42no—
353C22#1171830412yescomplemented
354C22C2×C22⋊#105711yescharacteristic
355C22#1171916912yes—
356C4—21no—
357C4—24no—
358C4—21no—
359C4C2×C22⋊#105712yes—
360C4#1171916911yescharacteristic
361C4#1171917111yescharacteristic
362C2—22no—
363C2—22no—
364C2—41no—
365C2—44no—
366C2#1313065811yescharacteristic
367C2C2×#1171830411yescharacteristic
368C2#1313107811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#261
commutatorC2×C4×C13C22×C8
FrattiniC22×C4C22×#261
Fitting, prime radicalC22×C13×#41C4
socleC22×C13C2×C22⋊C8
cosocleC22×#1055C23
hypercenterC22×#41#261
hypocenterC13#12683
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#11718304C2semidirect
2. #13131063C2semidirect
3. C2×C22⋊#6092C2semidirect
15. C2×#88106C22semidirect
16. C22×#6092C22semidirect
190. C22×C13#350semidirect
298. C13#12683semidirect
353. C22#11718304semidirect

Maximal subgroups

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

subgroupindex
C2×#117183042
#131310632
C2×C22⋊#60922
C22×#882022
#1268313