Finite Group Databasebeta

← #15865211

Subgroups — #15865211

Order 3328.

1178 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718763C211yesmaximal, characteristic, complemented
2C2×#11718771C211yesmaximal, characteristic
3C2×#11718771C211yesmaximal, characteristic
4#13117326C211yesmaximal, characteristic
5#13115442C218yesmaximal, complemented
6#13114532C211yesmaximal, characteristic, complemented
7#13114530C211yesmaximal, characteristic, complemented
8C22×C4×C8×C13C211yesmaximal, characteristic, complemented
9C22×#6127C2211yescharacteristic
10C23×C8×C13C2211yescharacteristic
11C23×C8×C13C2211yescharacteristic
12C22×C42×C13C2211yescharacteristic
13C2×#88180C2211yescharacteristic
14C22×#6099C2211yescharacteristic
15C22×#6099C2211yescharacteristic
16C2×C4×C8×C13—24no—
17C2×C4×C8×C13C2214yes—
18#11718874C414yes—
19#11718771—24no—
20#11718771—24no—
21C2×#88060C2214yes—
22C2×#88060C2214yes—
23#11718763—24no—
24C2×#88054C2214yes—
25#11718600C412yes—
26C4×#6103C2214yes—
27#11718590C412yes—
28#11718142C2214yes—
29#11718116C2214yes—
30C23×C4×C13C2×C412yes—
31C23×C4×C13C2311yescharacteristic
32C22×C22⋊D13C2×C412yes—
33C2×#6127—22no—
34C2×#6127—22no—
35C2×#6127—22no—
36C2×#6127—22no—
37C2×#6127—22no—
38C2×#6127—22no—
39C22×D52C2×C411yescharacteristic
40C22×C4×D13C2312yes—
41C22×Dic26C2×C411yescharacteristic
42C22×C8×C13—22no—
43C22×C8×C13—22no—
44C22×C8×C13—22no—
45C22×C8×C13—22no—
46C22×C8×C13D414yes—
47C22×C8×C13D414yes—
48C22×C8×C13C2312yes—
49C22×C8×C13C2312yes—
50C2×C42×C13—22no—
51C2×C42×C13—22no—
52C2×C42×C13C2×C412yes—
53C2×C42×C13C2312yes—
54C2×C22⋊Dic13C2×C412yes—
55C2×#6103C2×C418yes—
56#88180—22no—
57#88180—22no—
58C2×C4⋊Dic13C2×C412yes—
59C2×C4×Dic13C2312yes—
60C2×#6099—22no—
61C2×#6099—22no—
62C2×#6099—22no—
63C2×#6099—22no—
64C2×#6099—22no—
65C2×#6099—22no—
66C22×#1055C2312yes—
67C22×#1055C2312yes—
68C4×C8×C13—28no—
69#88071C2×C418yes—
70#88060—28no—
71#88060—28no—
72#88054—28no—
73#21046—131nomaximal, Sylow
74C24×C13C22×C411yescharacteristic
75C23×D13C22×C412yes—
76C22×C4×C13—24no—
77C22×C4×C13—28no—
78C22×C4×C13C22×C414yes—
79C22×C4×C13C22×C411yescharacteristic
80C22×C4×C13C2×D412yes—
81C22×C4×C13C2×D412yes—
82C22×C4×C13#4112yes—
83C22×C4×C13#4112yes—
84C22×C4×C13#4112yes—
85C22×C4×C13#4112yes—
86C22×C4×C13C2411yescharacteristic
87C2×C22⋊D13—24no—
88C2×C22⋊D13—24no—
89C2×C22⋊D13—24no—
90C22×Dic13C22×C412yes—
91C22×Dic13C22×C414yes—
92#6127—44no—
93#6127—44no—
94#6127—44no—
95#6127—44no—
96C2×D52—22no—
97C2×D52—22no—
98C2×D52—22no—
99C2×C4×D13—22no—
100C2×C4×D13—22no—
101C2×C4×D13—22no—
102C2×C4×D13—22no—
103C2×C4×D13—22no—
104C2×C4×D13—22no—
105C2×Dic26—22no—
106C2×Dic26—22no—
107C2×Dic26—22no—
108C2×C8×C13—28no—
109C2×C8×C13—28no—
110C2×C8×C13—22no—
111C2×C8×C13—22no—
112C2×C8×C13—22no—
113C2×C8×C13—22no—
114C2×C8×C13C2×D414yes—
115C2×C8×C13C2×D414yes—
116C42×C13—24no—
117C42×C13—22no—
118C42×C13—22no—
119C22⋊Dic13—24no—
120#6103—216no—
121C4⋊Dic13—24no—
122C4×Dic13—22no—
123C4×Dic13—22no—
124#6099—44no—
125#6099—44no—
126C2×#1055—22no—
127C2×#1055—22no—
128C2×#1055—22no—
129C2×#1055—22no—
130C2×#1055—22no—
131C2×#1055—22no—
132C2×#424—131no—
133C2×#408—131no—
134C2×#408—131no—
135C22×C4×C8D1311yescharacteristic, complemented
136#2297—131no—
137#1923—138no—
138#1729—131no—
139#1728—131no—
140C23×C13—24no—
141C23×C13#11712yes—
142C23×C13C4×D412yes—
143C23×C13C4×D412yes—
144C23×C13C23×C411yescharacteristic
145C22×D13—22no—
146C22×D13—22no—
147C22×D13—22no—
148C22×D13—28no—
149C2×C4×C13—216no—
150C2×C4×C13—24no—
151C2×C4×C13—24no—
152C2×C4×C13#11712yes—
153C2×C4×C13C4×D412yes—
154C2×C4×C13C4×D412yes—
155C2×C4×C13#12114yes—
156C2×C4×C13#12314yes—
157C2×C4×C13#12414yes—
158C2×C4×C13#12614yes—
159C2×C4×C13#13118yes—
160C2×C4×C13C22×D411yescharacteristic
161C2×C4×C13C22×D411yescharacteristic
162C2×C4×C13C2×#4111yescharacteristic
163C2×C4×C13C2×#4111yescharacteristic
164C2×C4×C13C2×#4111yescharacteristic
165C2×C4×C13C2×#4111yescharacteristic
166C22⋊D13—48no—
167C2×Dic13—22no—
168C2×Dic13—24no—
169C2×Dic13—22no—
170C2×Dic13—22no—
171C2×Dic13—28no—
172D52—44no—
173C4×D13—42no—
174C4×D13—42no—
175C4×D13—42no—
176C4×D13—42no—
177Dic26—44no—
178C8×C13—24no—
179C8×C13—24no—
180#1055—42no—
181#1055—42no—
182C22×#41—131no—
183C22×M16—131no—
184C22×M16—131no—
185C23×C8C2×D1311yescharacteristic
186C23×C8C2×D1311yescharacteristic
187C2×#117—131no—
188C22×C42C2×D1311yescharacteristic
189#424—264no—
190C2×C4⋊C8—134no—
191#408—264no—
192#408—264no—
193C2×C22⋊C8—134no—
194C2×C22⋊C8—134no—
195C2×C4×C8—24no—
196C2×C4×C8C2×D1314yes—
197#391—132no—
198#390—132no—
199#386—134no—
200C4×C22⋊C4—134no—
201#336—134no—
202#336—134no—
203C22×C13—24no—
204C22×C13—24no—
205C22×C13—24no—
206C22×C13#38814yes—
207C22×C13C2×#11711yescharacteristic
208C22×C13C2×C4×D411yescharacteristic
209C22×C13C2×C4×D411yescharacteristic
210C22×C13C2×#13114yes—
211C2×D13—44no—
212C2×D13—44no—
213C2×D13—44no—
214C2×D13—42no—
215C4×C13—28no—
216C4×C13#38814yes—
217C4×C13C2×#12111yescharacteristic
218C4×C13C2×#12311yescharacteristic
219C4×C13C2×#12411yescharacteristic
220C4×C13C2×#12611yescharacteristic
221Dic13—44no—
222Dic13—42no—
223C2×#41—262no—
224C2×#41—262no—
225C2×#41—262no—
226C2×#41—262no—
227C2×#41—262no—
228C2×#41—262no—
229C22×Q8—131no—
230C22×D4—132no—
231C22×D4—131no—
232C23×C4—132no—
233C23×C4C4×D1312yes—
234C23×C4C22×D1311yescharacteristic
235C2×M16—262no—
236C2×M16—262no—
237C2×M16—262no—
238C2×M16—262no—
239C2×M16—262no—
240C2×M16—262no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—132no—
246C22×C8—132no—
247C22×C8D5214yes—
248C22×C8C22⋊D1314yes—
249C22×C8C22×D1312yes—
250C22×C8C22×D1312yes—
251#117—262no—
252#117—262no—
253C2×C4⋊C4—132no—
254C2×C22⋊C4—132no—
255C2×C22⋊C4—138no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—132no—
259C2×C42C4×D1312yes—
260C2×C42C22×D1312yes—
261C4⋊C8—268no—
262C22⋊C8—268no—
263C22⋊C8—268no—
264C4×C8—28no—
265#95—138no—
266C2×C13—24no—
267C2×C13C2×#38811yescharacteristic
268C2×C13#292212yes—
269C2×C13#292912yes—
270C2×C13#293112yes—
271D13—82no—
272C24—132no—
273C24C2×C4×D1311yescharacteristic
274#41—524no—
275#41—524no—
276#41—524no—
277#41—524no—
278C2×Q8—262no—
279C2×Q8—262no—
280C2×Q8—262no—
281C2×D4—264no—
282C2×D4—264no—
283C2×D4—264no—
284C2×D4—262no—
285C2×D4—262no—
286C2×D4—262no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—132no—
290C22×C4—134no—
291C22×C4—262no—
292C22×C4—262no—
293C22×C4—262no—
294C22×C4—262no—
295C22×C4—262no—
296C22×C4—262no—
297C22×C4C2×C4×D1314yes—
298C22×C4C2×C4×D1311yescharacteristic
299C22×C4C2×D5212yes—
300C22×C4#612712yes—
301C22×C4#612712yes—
302C22×C4#612712yes—
303C22×C4#612712yes—
304C22×C4C2×C22⋊D1312yes—
305C22×C4C23×D1311yescharacteristic
306M16—524no—
307M16—524no—
308C2×C8—28no—
309C2×C8—28no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—262no—
315C2×C8—262no—
316C2×C8—262no—
317C2×C8—262no—
318C2×C8—262no—
319C2×C8—262no—
320C2×C8C2×D5214yes—
321C2×C8C2×C22⋊D1314yes—
322C4⋊C4—264no—
323C22⋊C4—264no—
324C22⋊C4—2616no—
325C42—24no—
326C42—22no—
327C42—22no—
328C42—262no—
329C42—262no—
330C13#2104611yescharacteristic, complemented, Sylow
331C23—24no—
332C23—262no—
333C23—262no—
334C23—268no—
335C23—262no—
336C23#8812612yes—
337C23C4×D5212yes—
338C23C4×C22⋊D1312yes—
339C23C22×C4×D1311yescharacteristic
340Q8—524no—
341D4—528no—
342D4—524no—
343C2×C4—216no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—262no—
347C2×C4—264no—
348C2×C4—262no—
349C2×C4—268no—
350C2×C4—262no—
351C2×C4—522no—
352C2×C4—522no—
353C2×C4—522no—
354C2×C4—522no—
355C2×C4#8812612yes—
356C2×C4C4×D5212yes—
357C2×C4#8812914yes—
358C2×C4#8813014yes—
359C2×C4#8815518yes—
360C2×C4C4×C22⋊D1312yes—
361C2×C4#8818314yes—
362C2×C4#8818414yes—
363C2×C4C22×D5211yescharacteristic
364C2×C4C2×#612711yescharacteristic
365C2×C4C2×#612711yescharacteristic
366C2×C4C2×#612711yescharacteristic
367C2×C4C2×#612711yescharacteristic
368C2×C4C22×C22⋊D1311yescharacteristic
369C8—24no—
370C8—24no—
371C8—522no—
372C8—522no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—522no—
377C22—524no—
378C22—524no—
379C22—524no—
380C22#1171860114yes—
381C22C2×#8812611yescharacteristic
382C22C2×C4×D5211yescharacteristic
383C22C2×#8815514yes—
384C22C2×C4×C22⋊D1311yescharacteristic
385C4—28no—
386C4—522no—
387C4—524no—
388C4#1171860114yes—
389C4C2×#8812911yescharacteristic
390C4C2×#8813011yescharacteristic
391C4C2×#8818311yescharacteristic
392C4C2×#8818411yescharacteristic
393C2—24no—
394C2—1042no—
395C2C2×#1171860111yescharacteristic
396C2#1312347812yes—
397C2#1312353012yes—
398C2#1312354912yes—
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC22×C4×C8×C13C2
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenterC22×C4×C8D13
hypocenterC13#21046
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C22×C4×C8

Complements

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

NH ≅ G/Ntype
1. C2×#11718763C2semidirect
5. #13115442C2semidirect
6. #13114532C2semidirect
7. #13114530C2semidirect
8. C22×C4×C8×C13C2semidirect
135. C22×C4×C8D13semidirect
330. C13#21046semidirect

Maximal subgroups

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

subgroupindex
C2×#117187632
C2×#117187712
#131173262
#131154422
#131145322
#131145302
C22×C4×C8×C132
#2104613