Finite Group Databasebeta

← #15755503

Subgroups — #15755503

Order 3328.

794 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13115305C212yesmaximal, complemented
2#13115134C211yesmaximal, characteristic
3#13113200C212yesmaximal, complemented
4#13113174C211yesmaximal, characteristic
5C13×#3501C211yesmaximal, characteristic
6C13×#554—24no—
7C13×#554—22no—
8D4×Q8×C13—22no—
9C13×#544—21no—
10C13×#543—21no—
11C13×#543C412yes—
12C13×#533—21no—
13C2×C13×C22⋊Q8—21no—
14C2×C13×C22⋊Q8C2212yes—
15C13×#518C412yes—
16C2×C4×D4×C13C2211yescharacteristic
17C22×#6099C2211yescharacteristic
18C2×#88076—22no—
19#11718868C2212yes—
20C2×C22⋊#1055—21no—
21C2×C22⋊#1055C2211yescharacteristic
22#11718122—22no—
23#11718120—21no—
24#11718120—24no—
25#11718117—21no—
26#11718116—21no—
27C22×Q8×C13—21no—
28C22×Q8×C13D412yes—
29C22×D4×C13C2×C411yescharacteristic
30C23×C4×C13D411yescharacteristic
31C23×C4×C13C2311yescharacteristic
32C13×C4⋊Q8—21no—
33C13×C4⋊Q8—21no—
34C13×C4⋊Q8—24no—
35C13×#125—22no—
36C13×#125—24no—
37C13×C22⋊Q8—22no—
38C13×C22⋊Q8—22no—
39C13×C22⋊Q8—24no—
40C13×C22⋊Q8—24no—
41C13×C22⋊Q8—22no—
42C13×C22⋊Q8—22no—
43C13×C22⋊Q8—24no—
44C13×C22⋊Q8—24no—
45C13×C22⋊Q8—22no—
46C13×C22⋊Q8—22no—
47C13×C22⋊Q8—22no—
48C13×C22⋊Q8—22no—
49C4×Q8×C13—22no—
50C4×Q8×C13—22no—
51C4×D4×C13—24no—
52C4×D4×C13—24no—
53C4×D4×C13—22no—
54C4×D4×C13—22no—
55C13×#117—22no—
56C2×C13×C4⋊C4—21no—
57C2×C13×C4⋊C4—22no—
58C2×C13×C4⋊C4D411yescharacteristic
59C2×C13×C4⋊C4D414yes—
60C2×C13×C4⋊C4D412yes—
61C2×C13×C22⋊C4—22no—
62C2×C13×C22⋊C4C2×C411yescharacteristic
63C2×C13×C22⋊C4C2×C414yes—
64C2×C13×C22⋊C4D411yescharacteristic
65C2×C42×C13D411yescharacteristic
66C2×#6099—22no—
67C2×#6099—22no—
68C2×#6099—42no—
69C22×#1055—21no—
70C22×#1055—21no—
71#88076—44no—
72C22⋊#1055—22no—
73C22⋊#1055—22no—
74C22⋊#1055—22no—
75C22⋊#1055—42no—
76#8537—131nomaximal, Sylow
77C24×C13C22⋊C411yescharacteristic
78C24×C13C22×C411yescharacteristic
79C2×Q8×C13—22no—
80C2×Q8×C13—22no—
81C2×Q8×C13—22no—
82C2×Q8×C13—22no—
83C2×Q8×C13—42no—
84C2×Q8×C13—42no—
85C2×Q8×C13—81no—
86C2×Q8×C13—81no—
87C2×D4×C13—22no—
88C2×D4×C13—22no—
89C2×D4×C13—22no—
90C22×C4×C13—22no—
91C22×C4×C13—22no—
92C22×C4×C13—21no—
93C22×C4×C13—21no—
94C22×C4×C13—41no—
95C22×C4×C13—41no—
96C22×C4×C13—41no—
97C22×C4×C13—41no—
98C22×C4×C13C22⋊C411yescharacteristic
99C22×C4×C13C22⋊C411yescharacteristic
100C22×C4×C13C22⋊C414yes—
101C22×C4×C13C22⋊C414yes—
102C22×C4×C13C22⋊C411yescharacteristic
103C22×C4×C13C2×D411yescharacteristic
104C22×C4×C13C2×D412yes—
105C22×C4×C13C2×D411yescharacteristic
106C22×C4×C13C2×D412yes—
107C13×C4⋊C4—22no—
108C13×C4⋊C4—24no—
109C13×C4⋊C4—22no—
110C13×C4⋊C4—22no—
111C13×C4⋊C4—24no—
112C13×C4⋊C4—22no—
113C13×C4⋊C4—24no—
114C13×C4⋊C4—22no—
115C13×C4⋊C4—22no—
116C13×C4⋊C4—22no—
117C13×C4⋊C4—24no—
118C13×C22⋊C4—24no—
119C13×C22⋊C4—24no—
120C13×C22⋊C4—22no—
121C13×C22⋊C4—24no—
122C13×C22⋊C4—22no—
123C42×C13—24no—
124C42×C13—21no—
125C42×C13—21no—
126#6099—24no—
127#6099—42no—
128#6099—42no—
129#6099—42no—
130C2×#1055—22no—
131C2×#1055—41no—
132C2×#1055—41no—
133C2×#1055—41no—
134C2×#1055—41no—
135C2×#1055—41no—
136C2×#1055—42no—
137#3501D1311yescharacteristic
138#1901—132no—
139#1870—131no—
140#1472—132no—
141#1465—131no—
142C23×C13—21no—
143C23×C13—21no—
144C23×C13—21no—
145C23×C13—21no—
146C23×C13—41no—
147C23×C13—41no—
148C23×C13—41no—
149C23×C13—41no—
150C23×C13#10112yescomplemented
151C23×C13C2×C22⋊C412yes—
152C23×C13C2×C22⋊C411yescharacteristic
153Q8×C13—42no—
154Q8×C13—42no—
155Q8×C13—82no—
156Q8×C13—82no—
157Q8×C13—82no—
158D4×C13—28no—
159C2×C4×C13—21no—
160C2×C4×C13—21no—
161C2×C4×C13—21no—
162C2×C4×C13—22no—
163C2×C4×C13—21no—
164C2×C4×C13—22no—
165C2×C4×C13—21no—
166C2×C4×C13—24no—
167C2×C4×C13—24no—
168C2×C4×C13—21no—
169C2×C4×C13—42no—
170C2×C4×C13—41no—
171C2×C4×C13—41no—
172C2×C4×C13—41no—
173C2×C4×C13—41no—
174C2×C4×C13—44no—
175C2×C4×C13—41no—
176C2×C4×C13—44no—
177C2×C4×C13—81no—
178C2×C4×C13—82no—
179C2×C4×C13—82no—
180C2×C4×C13#10112yescomplemented
181C2×C4×C13#12011yescharacteristic
182C2×C4×C13#12012yes—
183C2×C4×C13#12011yescharacteristic
184#1055—42no—
185#1055—81no—
186#1055—82no—
187C22×M16—131no—
188#554—24no—
189#554—22no—
190D4×Q8—22no—
191#544—21no—
192#543—21no—
193#543Dic1312yes—
194#533—21no—
195C2×C22⋊Q8—21no—
196C2×C22⋊Q8C2×D1312yes—
197#518Dic1312yes—
198C2×C4×D4C2×D1311yescharacteristic
199C2×#101—262no—
200#407—132no—
201C2×C22⋊C8—131no—
202C2×C22⋊C8—261no—
203#343—262no—
204#336—261no—
205#324—261no—
206#324—264no—
207#323—261no—
208C22×C13—21no—
209C22×C13—21no—
210C22×C13—22no—
211C22×C13—22no—
212C22×C13—22no—
213C22×C13—22no—
214C22×C13—42no—
215C22×C13—41no—
216C22×C13—41no—
217C22×C13—81no—
218C22×C13—81no—
219C22×C13—81no—
220C22×C13#37911yescharacteristic
221C22×C13C2×#10111yescharacteristic
222C22×C13C2×#10111yescharacteristic
223C4×C13—22no—
224C4×C13—21no—
225C4×C13—22no—
226C4×C13—41no—
227C4×C13—44no—
228C4×C13—44no—
229C4×C13—41no—
230C4×C13B(4,𝔽2)11yescharacteristic
231C4×C13#45811yescharacteristic
232C22×Q8—21no—
233C22×Q8C22⋊D1312yes—
234C22×D4C2×Dic1311yescharacteristic
235C23×C4C22⋊D1311yescharacteristic
236C23×C4C22×D1311yescharacteristic
237C2×M16—262no—
238C2×M16—262no—
239C2×M16—522no—
240C22×C8—261no—
241C22×C8—261no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—24no—
245#125—22no—
246#125—24no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—22no—
264C4×D4—22no—
265#117—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C22⋊D1311yescharacteristic
269C2×C4⋊C4C22⋊D1314yes—
270C2×C4⋊C4C22⋊D1312yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4C2×Dic1311yescharacteristic
273C2×C22⋊C4C2×Dic1314yes—
274C2×C22⋊C4C22⋊D1311yescharacteristic
275C2×C42C22⋊D1311yescharacteristic
276#101—524no—
277C22⋊C8—262no—
278C22⋊C8—262no—
279C22⋊C8—262no—
280C22⋊C8—522no—
281C2×C13—21no—
282C2×C13—21no—
283C2×C13—42no—
284C2×C13—42no—
285C2×C13#178411yescharacteristic
286C2×C13#179011yescharacteristic
287C2×C13#179111yescharacteristic
288C24C22⋊Dic1311yescharacteristic
289C24C22×Dic1311yescharacteristic
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—81no—
297C2×Q8—81no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4C22⋊Dic1311yescharacteristic
310C22×C4C22⋊Dic1311yescharacteristic
311C22×C4C22⋊Dic1314yes—
312C22×C4C22⋊Dic1314yes—
313C22×C4C22⋊Dic1311yescharacteristic
314C22×C4C2×C22⋊D1311yescharacteristic
315C22×C4C2×C22⋊D1312yes—
316C22×C4C2×C22⋊D1311yescharacteristic
317C22×C4C2×C22⋊D1312yes—
318M16—264no—
319M16—522no—
320M16—522no—
321M16—522no—
322C2×C8—262no—
323C2×C8—521no—
324C2×C8—521no—
325C2×C8—521no—
326C2×C8—521no—
327C2×C8—521no—
328C2×C8—522no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—22no—
343C22⋊C4—24no—
344C22⋊C4—22no—
345C42—24no—
346C42—21no—
347C42—21no—
348C13#853711yescharacteristic, complemented, Sylow
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23#8807612yescomplemented
358C23C2×C22⋊Dic1312yes—
359C23C2×C22⋊Dic1311yescharacteristic
360Q8—42no—
361Q8—42no—
362Q8—82no—
363Q8—82no—
364Q8—82no—
365D4—28no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4#8807612yescomplemented
388C2×C4#8820711yescharacteristic
389C2×C4#8820712yes—
390C2×C4#8820711yescharacteristic
391C8—522no—
392C8—1041no—
393C8—1042no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—42no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22C2×#8807611yescharacteristic
407C22C2×#8807611yescharacteristic
408C22#1171890811yescharacteristic
409C4—22no—
410C4—21no—
411C4—22no—
412C4—41no—
413C4—44no—
414C4—44no—
415C4—41no—
416C4#1171890411yescharacteristic
417C4#1171890511yescharacteristic
418C2—21no—
419C2—21no—
420C2—42no—
421C2—42no—
422C2#1311476011yescharacteristic
423C2#1311478411yescharacteristic
424C2#1311478711yescharacteristic
subgroupG/H
centerC22#11718908
commutatorC24×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#3501C2
socleC22×C13#379
prime radicalC22×D4×C13C2×C4
cosocleC23×C4×C13C23
hypercenter#3501D13
hypocenterC13#8537
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #13115305C2semidirect
3. #13113200C2semidirect
150. C23×C13#101semidirect
180. C2×C4×C13#101semidirect
348. C13#8537semidirect
357. C23#88076semidirect
387. C2×C4#88076semidirect

Maximal subgroups

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

subgroupindex
#131153052
#131151342
#131132002
#131131742
C13×#35012
#853713