Finite Group Databasebeta

← #15757894

Subgroups — #15757894

Order 3328.

782 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 111 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#13123877C211yesmaximal, characteristic, complemented
2#13114780C212yesmaximal, complemented
3C4×#88074C211yesmaximal, characteristic, complemented
4#13112722C212yesmaximal, complemented
5C13×#2388C211yesmaximal, characteristic
6C22×C13×#41C2211yescharacteristic
7C2×C4×D4×C13—21no—
8C2×C4×D4×C13C2211yescharacteristic
9C13×#395C411yescharacteristic, complemented
10C13×#394—21no—
11C13×#394C411yescharacteristic, complemented
12C13×#393C411yescharacteristic, complemented
13C13×#392—21no—
14C13×#392C411yescharacteristic, complemented
15C4×C13×C22⋊C4—21no—
16C4×C13×C22⋊C4C2211yescharacteristic
17C2×#88077—24no—
18C2×#88074—21no—
19C4×C22⋊Dic13C2212yes—
20#11718868—21no—
21#11718868C2212yes—
22#11718673—24no—
23#11718586—21no—
24#11718118—21no—
25C2×C13×#41—22no—
26C2×C13×#41—22no—
27C2×C13×#41—44no—
28C22×Q8×C13C2×C411yescharacteristic
29C22×D4×C13—21no—
30C22×D4×C13—21no—
31C22×D4×C13C2×C411yescharacteristic
32C22×D4×C13D411yescharacteristic
33C23×C4×C13—21no—
34C23×C4×C13D411yescharacteristic
35C23×C4×C13C2311yescharacteristic
36C4×D4×C13—22no—
37C4×D4×C13—22no—
38C4×D4×C13—44no—
39C2×C13×C4⋊C4—21no—
40C2×C13×C4⋊C4C2×C411yescharacteristic
41C2×C13×C22⋊C4—21no—
42C2×C13×C22⋊C4—21no—
43C2×C13×C22⋊C4—21no—
44C2×C13×C22⋊C4—21no—
45C2×C13×C22⋊C4—21no—
46C2×C13×C22⋊C4C2×C411yescharacteristic
47C2×C13×C22⋊C4D411yescharacteristic
48C2×C42×C13—21no—
49C2×C42×C13D411yescharacteristic
50C2×C22⋊Dic13—22no—
51C2×C4×Dic13—22no—
52C2×#6099—22no—
53C2×#6099D414yescomplemented
54C13×#95—21no—
55C13×#95—21no—
56C13×#95C2×C411yescharacteristic
57C13×#95C2×C411yescharacteristic
58#88077—48no—
59#88074—24no—
60#88071—22no—
61C22⋊#1055—22no—
62C22⋊#1055—22no—
63C22⋊#1055—22no—
64C4⋊#1055—22no—
65C4×#1055—22no—
66#9010—131nomaximal, Sylow
67C24×C13—21no—
68C24×C13C22⋊C411yescharacteristic
69C24×C13C2×D411yescharacteristic
70C13×#41—28no—
71C13×#41—44no—
72C13×#41—44no—
73C13×#41—44no—
74C2×Q8×C13—22no—
75C2×Q8×C13—42no—
76C2×D4×C13—22no—
77C2×D4×C13—22no—
78C2×D4×C13—22no—
79C2×D4×C13—41no—
80C2×D4×C13—42no—
81C2×D4×C13—42no—
82C2×D4×C13—42no—
83C2×D4×C13—42no—
84C2×D4×C13—42no—
85C2×D4×C13—42no—
86C2×D4×C13—42no—
87C22×C4×C13—24no—
88C22×C4×C13—21no—
89C22×C4×C13—21no—
90C22×C4×C13—21no—
91C22×C4×C13—21no—
92C22×C4×C13—21no—
93C22×C4×C13—42no—
94C22×C4×C13—42no—
95C22×C4×C13—42no—
96C22×C4×C13—41no—
97C22×C4×C13—41no—
98C22×C4×C13C22⋊C411yescharacteristic
99C22×C4×C13C22⋊C411yescharacteristic
100C22×C4×C13C22⋊C411yescharacteristic
101C22×C4×C13C22×C411yescharacteristic
102C22×C4×C13C2×D412yes—
103C22×C4×C13C2×D411yescharacteristic
104C22×C4×C13#4112yes—
105C22×Dic13—42no—
106C13×C4⋊C4—22no—
107C13×C4⋊C4—42no—
108C13×C22⋊C4—22no—
109C13×C22⋊C4—22no—
110C13×C22⋊C4—42no—
111C13×C22⋊C4—42no—
112C13×C22⋊C4—42no—
113C13×C22⋊C4—42no—
114C13×C22⋊C4—42no—
115C42×C13—21no—
116C42×C13—21no—
117C42×C13—42no—
118C22⋊Dic13—24no—
119C22⋊Dic13—42no—
120C4×Dic13—44no—
121#6099—44no—
122#6099—82no—
123C2×#1055—24no—
124#2975—131no—
125#2388D1311yescharacteristic
126#1789—132no—
127C4×#99—131no—
128#1297—132no—
129C23×C13—21no—
130C23×C13—21no—
131C23×C13—22no—
132C23×C13—21no—
133C23×C13—21no—
134C23×C13—21no—
135C23×C13—41no—
136C23×C13—41no—
137C23×C13—41no—
138C23×C13—41no—
139C23×C13—41no—
140C23×C13—41no—
141C23×C13—41no—
142C23×C13—41no—
143C23×C13—41no—
144C23×C13—41no—
145C23×C13C2×C22⋊C411yescharacteristic
146C23×C13#12112yes—
147Q8×C13—24no—
148Q8×C13—42no—
149D4×C13—24no—
150D4×C13—44no—
151D4×C13—42no—
152D4×C13—48no—
153D4×C13—48no—
154C2×C4×C13—21no—
155C2×C4×C13—21no—
156C2×C4×C13—24no—
157C2×C4×C13—22no—
158C2×C4×C13—22no—
159C2×C4×C13—22no—
160C2×C4×C13—21no—
161C2×C4×C13—21no—
162C2×C4×C13—41no—
163C2×C4×C13—42no—
164C2×C4×C13—44no—
165C2×C4×C13—41no—
166C2×C4×C13—41no—
167C2×C4×C13—41no—
168C2×C4×C13—41no—
169C2×C4×C13—42no—
170C2×C4×C13—44no—
171C2×C4×C13—41no—
172C2×C4×C13—41no—
173C2×C4×C13—41no—
174C2×C4×C13—41no—
175C2×C4×C13—41no—
176C2×C4×C13—81no—
177C2×C4×C13#10412yescomplemented
178C2×C4×C13C4×D412yes—
179C2×C4×C13#12011yescharacteristic
180C2×C4×C13#12311yescharacteristic
181C2×Dic13—44no—
182C2×Dic13—82no—
183C2×Dic13—82no—
184#1055—44no—
185C22×#41C2×D1311yescharacteristic
186C2×C4×D4—21no—
187C2×C4×D4C2×D1311yescharacteristic
188#436—264no—
189#431—261no—
190C2×#104—264no—
191C2×#99—261no—
192#407—132no—
193#407—261no—
194#395Dic1311yescharacteristic, complemented
195#394—21no—
196#394Dic1311yescharacteristic, complemented
197#393Dic1311yescharacteristic, complemented
198#392—21no—
199#392Dic1311yescharacteristic, complemented
200C4×C22⋊C4—21no—
201C4×C22⋊C4—132no—
202C4×C22⋊C4C2×D1311yescharacteristic
203#342—261no—
204C22×C13—21no—
205C22×C13—21no—
206C22×C13—22no—
207C22×C13—22no—
208C22×C13—22no—
209C22×C13—42no—
210C22×C13—42no—
211C22×C13—42no—
212C22×C13—41no—
213C22×C13—41no—
214C22×C13—42no—
215C22×C13—41no—
216C22×C13—81no—
217C22×C13—81no—
218C22×C13—81no—
219C22×C13—81no—
220C22×C13—81no—
221C22×C13—81no—
222C22×C13—81no—
223C22×C13#38611yescharacteristic
224C22×C13C2×#10411yescharacteristic
225C22×C13#42111yescharacteristic
226C4×C13—22no—
227C4×C13—21no—
228C4×C13—22no—
229C4×C13—42no—
230C4×C13—41no—
231C4×C13—42no—
232C4×C13—41no—
233C4×C13—81no—
234C4×C13B(4,𝔽2)12yes—
235Dic13—84no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—44no—
239C22×Q8C2×Dic1311yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2×Dic1311yescharacteristic
243C22×D4C22⋊D1311yescharacteristic
244C23×C4—21no—
245C23×C4C22⋊D1311yescharacteristic
246C23×C4C22×D1311yescharacteristic
247C2×M16—134no—
248C2×M16—262no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—44no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4C2×Dic1311yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—262no—
260C2×C22⋊C4C2×Dic1311yescharacteristic
261C2×C22⋊C4C22⋊D1311yescharacteristic
262C2×C42—21no—
263C2×C42—262no—
264C2×C42C22⋊D1311yescharacteristic
265C4⋊C8—262no—
266#104—528no—
267#99—264no—
268C22⋊C8—262no—
269C22⋊C8—262no—
270C22⋊C8—262no—
271C4×C8—262no—
272#95—21no—
273#95—21no—
274#95—262no—
275#95C2×Dic1311yescharacteristic
276#95C2×Dic1311yescharacteristic
277C2×C13—21no—
278C2×C13—21no—
279C2×C13—42no—
280C2×C13—81no—
281C2×C13—82no—
282C2×C13#187111yescharacteristic
283C2×C13#188411yescharacteristic
284C2×C13#188911yescharacteristic
285C24—21no—
286C24C22⋊Dic1311yescharacteristic
287C24C2×C22⋊D1311yescharacteristic
288#41—28no—
289#41—44no—
290#41—44no—
291#41—44no—
292C2×Q8—22no—
293C2×Q8—42no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—522no—
317C22×C4C22⋊Dic1311yescharacteristic
318C22×C4C22⋊Dic1311yescharacteristic
319C22×C4C22⋊Dic1311yescharacteristic
320C22×C4D4×D1312yes—
321C22×C4#612912yes—
322C22×C4C22×Dic1311yescharacteristic
323C22×C4C2×C22⋊D1311yescharacteristic
324M16—524no—
325M16—1042no—
326C2×C8—264no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—264no—
337C22⋊C4—522no—
338C42—21no—
339C42—21no—
340C42—42no—
341C42—524no—
342C13#901011yescharacteristic, complemented, Sylow
343C23—21no—
344C23—21no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23#8819312yes—
360C23C2×C22⋊Dic1311yescharacteristic
361Q8—24no—
362Q8—42no—
363D4—24no—
364D4—44no—
365D4—42no—
366D4—48no—
367D4—48no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—524no—
392C2×C4—1042no—
393C2×C4—1042no—
394C2×C4#8807712yescomplemented
395C2×C4D4×Dic1312yes—
396C2×C4#8818911yescharacteristic
397C2×C4#8819111yescharacteristic
398C8—524no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22—41no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#1171888111yescharacteristic
419C22C2×#8807711yescharacteristic
420C22#1171889711yescharacteristic
421C4—22no—
422C4—21no—
423C4—22no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—81no—
429C4—1044no—
430C4#1171869312yes—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—81no—
435C2—82no—
436C2#1311513911yescharacteristic
437C2#1311522011yescharacteristic
438C2#1311524511yescharacteristic
subgroupG/H
centerC2×C4#88191
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
Fitting, prime radicalC13×#2388C2
socleC22×C13#386
cosocleC23×C4×C13C23
hypercenter#2388D13
hypocenterC13#9010
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. #13123877C2semidirect
2. #13114780C2semidirect
3. C4×#88074C2semidirect
4. #13112722C2semidirect
9. C13×#395C4semidirect
11. C13×#394C4semidirect
12. C13×#393C4semidirect
14. C13×#392C4semidirect
53. C2×#6099D4semidirect
177. C2×C4×C13#104semidirect
194. #395Dic13semidirect
196. #394Dic13semidirect
197. #393Dic13semidirect
199. #392Dic13semidirect
342. C13#9010semidirect
394. C2×C4#88077semidirect

Maximal subgroups

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

subgroupindex
#131238772
#131147802
C4×#880742
#131127222
C13×#23882
#901013