Finite Group Databasebeta

← #15782075

Subgroups — #15782075

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718776C211yesmaximal, characteristic
2C2×#11718096C211yesmaximal, characteristic, complemented
3C2×#11718101C211yesmaximal, characteristic, complemented
4C2×#11718101C211yesmaximal, characteristic, complemented
5#13114865C211yesmaximal, characteristic, complemented
6#13112928C211yesmaximal, characteristic, complemented
7C2×C13×C22⋊C16C211yesmaximal, characteristic, complemented
8C23×C8×C13C2211yescharacteristic
9C2×#88184—21no—
10C22×D104C2211yescharacteristic
11C22×C13×C16C2211yescharacteristic
12C22×C13×C16C2211yescharacteristic
13#11718873—21no—
14#11718776—22no—
15#11718776—22no—
16C2×#88061—21no—
17C2×#88058C2211yescharacteristic
18C2×#88058C2211yescharacteristic
19C2×#88058C2211yescharacteristic
20C13×C22⋊C16—22no—
21#11718143—21no—
22#11718101—22no—
23#11718101—22no—
24#11718096—22no—
25C23×C4×C13D411yescharacteristic
26C22×C22⋊D13—41no—
27C22×D52—21no—
28C22×C8×C13—21no—
29C22×C8×C13—21no—
30C22×C8×C13—21no—
31C22×C8×C13—21no—
32C22×C8×C13—21no—
33C22×C8×C13—21no—
34C22×C8×C13D411yescharacteristic
35C22×C8×C13C2311yescharacteristic
36C2×C22⋊Dic13—41no—
37#88184—24no—
38#88184—42no—
39C2×#6103—41no—
40C2×C4⋊Dic13—21no—
41C2×C4⋊Dic13—21no—
42C2×C4⋊Dic13—21no—
43C2×D104—21no—
44C2×D104—21no—
45C2×D104—21no—
46C2×D104—21no—
47C2×D104—21no—
48C2×D104D412yes—
49C2×C13×C16—21no—
50C2×C13×C16—21no—
51C2×C13×C16—21no—
52C2×C13×C16—21no—
53C2×C13×C16—21no—
54C2×C13×C16D412yescomplemented
55#88071—41no—
56#88061—24no—
57#88058—21no—
58#88058—21no—
59#88058—21no—
60#88058—21no—
61#88058—21no—
62#88058D412yes—
63#13380—131nomaximal, Sylow
64C24×C13D811yescharacteristic
65C23×D13—41no—
66C22×C4×C13—21no—
67C22×C4×C13—21no—
68C22×C4×C13—21no—
69C22×C4×C13—21no—
70C22×C4×C13—21no—
71C22×C4×C13—21no—
72C22×C4×C13D811yescharacteristic
73C22×C4×C13C2×D411yescharacteristic
74C2×C22⋊D13—44no—
75C2×C22⋊D13—82no—
76C2×C22⋊D13—82no—
77C22×Dic13—41no—
78C22×Dic13—41no—
79C22×Dic13—41no—
80C2×D52—22no—
81C2×D52—41no—
82C2×D52—41no—
83C2×D52—41no—
84C2×D52—41no—
85C2×D52—41no—
86C2×C8×C13—21no—
87C2×C8×C13—21no—
88C2×C8×C13—21no—
89C2×C8×C13—41no—
90C2×C8×C13—41no—
91C2×C8×C13—41no—
92C2×C8×C13—41no—
93C2×C8×C13C2×D411yescharacteristic
94C2×C8×C13C2×D411yescharacteristic
95C2×C8×C13C2×D411yescharacteristic
96C2×C8×C13#4111yescharacteristic
97C2×C8×C13#4111yescharacteristic
98C2×C8×C13#4111yescharacteristic
99C22⋊Dic13—82no—
100#6103—44no—
101C4⋊Dic13—22no—
102C4⋊Dic13—41no—
103C4⋊Dic13—41no—
104C4⋊Dic13—41no—
105C4⋊Dic13—41no—
106C4⋊Dic13—41no—
107D104—22no—
108D104—22no—
109D104—41no—
110D104—41no—
111C13×C16—22no—
112C13×C16—41no—
113C2×#466—131no—
114C2×#366—131no—
115C2×#357—131no—
116C2×#357—131no—
117C2×C22⋊C16D1311yescharacteristic, complemented
118#1808—131no—
119#1380—131no—
120C23×C13—21no—
121C23×C13—21no—
122C23×C13—21no—
123C23×C13—21no—
124C23×C13—21no—
125C23×C13—21no—
126C23×C13D1612yes—
127C23×C13C2×D811yescharacteristic
128C22×D13—42no—
129C22×D13—81no—
130C22×D13—81no—
131C22×D13—81no—
132C22×D13—81no—
133C22×D13—81no—
134C22×D13—81no—
135C2×C4×C13—21no—
136C2×C4×C13—21no—
137C2×C4×C13—21no—
138C2×C4×C13—41no—
139C2×C4×C13—41no—
140C2×C4×C13—41no—
141C2×C4×C13—41no—
142C2×C4×C13#12011yescharacteristic
143C2×C4×C13#12111yescharacteristic
144C2×C4×C13#12311yescharacteristic
145C2×C4×C13#13511yescharacteristic
146C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C13#13711yescharacteristic
148C22⋊D13—88no—
149C2×Dic13—42no—
150C2×Dic13—81no—
151C2×Dic13—81no—
152C2×Dic13—81no—
153C2×Dic13—81no—
154C2×Dic13—81no—
155C2×Dic13—81no—
156C2×Dic13—81no—
157C2×Dic13—81no—
158D52—42no—
159D52—42no—
160D52—81no—
161D52—81no—
162C8×C13—41no—
163C8×C13#12111yescharacteristic
164C8×C13#12111yescharacteristic
165C8×C13#12311yescharacteristic
166C8×C13#12411yescharacteristic
167C22×D8—131no—
168C23×C8C2×D1311yescharacteristic
169C2×#121—261no—
170C22×C16C2×D1311yescharacteristic
171C22×C16C2×D1311yescharacteristic
172#466—262no—
173#466—262no—
174C2×#107—131no—
175C2×#107—131no—
176C2×#107—131no—
177C2×#102—261no—
178#380—261no—
179#366—262no—
180#357—262no—
181#357—262no—
182C22⋊C16—22no—
183#340—261no—
184C22×C13—22no—
185C22×C13—22no—
186C22×C13—22no—
187C22×C13—41no—
188C22×C13—41no—
189C22×C13—41no—
190C22×C13—41no—
191C22×C13#44711yescharacteristic
192C22×C13#46611yescharacteristic
193C22×C13#48011yescharacteristic
194C22×C13C2×D1611yescharacteristic
195C22×C13#50811yescharacteristic
196C22×C13#50911yescharacteristic
197C22×C13#51011yescharacteristic
198C2×D13—82no—
199C2×D13—82no—
200C2×D13—82no—
201C2×D13—161no—
202C2×D13—161no—
203C2×D13—161no—
204C2×D13—161no—
205C4×C13—41no—
206C4×C13#39411yescharacteristic
207C4×C13#45211yescharacteristic
208C4×C13#46811yescharacteristic
209C4×C13#48211yescharacteristic
210Dic13—82no—
211Dic13—161no—
212Dic13—161no—
213C22×D4—261no—
214C22×D4—521no—
215C23×C4D5211yescharacteristic
216C2×D8—132no—
217C2×D8—261no—
218C2×D8—261no—
219C2×D8—261no—
220C2×D8—261no—
221C2×D8—261no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8D5211yescharacteristic
229C22×C8C22×D1311yescharacteristic
230#121—264no—
231#121—522no—
232C2×C4⋊C4—261no—
233C2×C4⋊C4—261no—
234C2×C4⋊C4—261no—
235C2×C22⋊C4—521no—
236C2×C22⋊C4—521no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16C22⋊D1312yescomplemented
243#107—132no—
244#107—261no—
245#107—261no—
246#107—261no—
247#107—261no—
248#107—261no—
249#102—264no—
250#95—521no—
251C2×C13—42no—
252C2×C13#203311yescharacteristic
253C2×C13#218411yescharacteristic
254C2×C13#218911yescharacteristic
255C2×C13#221511yescharacteristic
256C2×C13#222011yescharacteristic
257C2×C13#223211yescharacteristic
258C2×C13#223411yescharacteristic
259D13—162no—
260C24—521no—
261C24D10411yescharacteristic
262C2×D4—262no—
263C2×D4—524no—
264C2×D4—521no—
265C2×D4—521no—
266C2×D4—521no—
267C2×D4—521no—
268C2×D4—521no—
269C2×D4—1042no—
270C2×D4—1042no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—521no—
278C22×C4—521no—
279C22×C4—521no—
280C22×C4D10411yescharacteristic
281C22×C4C2×D5211yescharacteristic
282D8—262no—
283D8—262no—
284D8—521no—
285D8—521no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8#612711yescharacteristic
294C2×C8D4×D1311yescharacteristic
295C2×C8D4×D1311yescharacteristic
296C2×C8#612911yescharacteristic
297C2×C8#612911yescharacteristic
298C2×C8C2×C22⋊D1311yescharacteristic
299C4⋊C4—262no—
300C4⋊C4—521no—
301C4⋊C4—521no—
302C4⋊C4—521no—
303C4⋊C4—521no—
304C4⋊C4—521no—
305C22⋊C4—524no—
306C22⋊C4—1042no—
307C16—22no—
308C16—41no—
309C13#1338011yescharacteristic, complemented, Sylow
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—522no—
317C23—1041no—
318C23—1041no—
319C23—1041no—
320C23—1041no—
321C23—1041no—
322C23—1041no—
323C23D20812yes—
324C23C2×D10411yescharacteristic
325D4—522no—
326D4—522no—
327D4—1048no—
328D4—1041no—
329D4—1041no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—522no—
338C2×C4—1041no—
339C2×C4—1041no—
340C2×C4—1041no—
341C2×C4—1041no—
342C2×C4—1041no—
343C2×C4—1041no—
344C2×C4—1041no—
345C2×C4—1041no—
346C2×C4#8813611yescharacteristic
347C2×C4#8814011yescharacteristic
348C2×C4#8815811yescharacteristic
349C2×C4#8816211yescharacteristic
350C2×C4#8816311yescharacteristic
351C2×C4#8818411yescharacteristic
352C8—41no—
353C8#8813711yescharacteristic
354C8#8813911yescharacteristic
355C8#8819211yescharacteristic
356C8#8819311yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—1042no—
365C22—1042no—
366C22—1042no—
367C22—2081no—
368C22—2081no—
369C22—2081no—
370C22—2081no—
371C22#1171839311yescharacteristic
372C22#1171839711yescharacteristic
373C22C2×D20811yescharacteristic
374C22#1171856511yescharacteristic
375C22#1171856911yescharacteristic
376C22#1171857011yescharacteristic
377C22#1171877611yescharacteristic
378C4—41no—
379C4—1042no—
380C4—2081no—
381C4—2081no—
382C4#1171839411yescharacteristic
383C4#1171839811yescharacteristic
384C4#1171862211yescharacteristic
385C4#1171879511yescharacteristic
386C2—42no—
387C2—2082no—
388C2#1311601911yescharacteristic
389C2#1311675011yescharacteristic
390C2#1311677811yescharacteristic
391C2#1311692211yescharacteristic
392C2#1311694911yescharacteristic
393C2#1311701711yescharacteristic
394C2#1311702811yescharacteristic
subgroupG/H
centerC23C2×D104
commutator, cosocleC22×C8×C13C23
FrattiniC22×C8C22×D13
FittingC2×C13×C22⋊C16C2
socleC23×C13C2×D8
prime radicalC2×#11718776C2
hypercenterC2×C22⋊C16D13
hypocenterC13#13380
LCSG ▷ C22×C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#11718096C2semidirect
3. C2×#11718101C2semidirect
4. C2×#11718101C2semidirect
5. #13114865C2semidirect
6. #13112928C2semidirect
7. C2×C13×C22⋊C16C2semidirect
54. C2×C13×C16D4semidirect
117. C2×C22⋊C16D13semidirect
242. C2×C16C22⋊D13semidirect
309. C13#13380semidirect

Maximal subgroups

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

subgroupindex
C2×#117187762
C2×#117180962
C2×#117181012
#131148652
#131129282
C2×C13×C22⋊C162
#1338013