Finite Group Databasebeta

← #15760919

Subgroups — #15760919

Order 3328.

628 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 107 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#13123865C211yesmaximal, characteristic, complemented
2#13113174C211yesmaximal, characteristic, complemented
3#13113163C212yesmaximal, complemented
4#13112742C212yesmaximal, complemented
5C13×#2534C211yesmaximal, characteristic
6C2×C13×#123C411yescharacteristic
7C2×C13×#120C411yescharacteristic
8C2×C4×D4×C13C2211yescharacteristic
9C22×C13×C22⋊C4C2211yescharacteristic
10C13×#396—21no—
11C13×#394—21no—
12C13×#388—21no—
13C13×#386—21no—
14C13×#381C411yescharacteristic
15C13×#380C2211yescharacteristic
16C13×#379C411yescharacteristic
17#11718868C2212yes—
18#11718868C2212yes—
19C2×C22⋊#1055—21no—
20#11718672—24no—
21#11718587—21no—
22#11718120—21no—
23#11718117—21no—
24#11718117—24no—
25C25×C13C2×C411yescharacteristic
26C22×D4×C13—21no—
27C22×D4×C13C2×C411yescharacteristic
28C23×C4×C13C2×C411yescharacteristic
29C23×C4×C13D411yescharacteristic
30C23×C4×C13C2311yescharacteristic
31C13×#123—42no—
32C13×#120—42no—
33C4×D4×C13—22no—
34C4×D4×C13—22no—
35C2×C13×C4⋊C4—21no—
36C2×C13×C4⋊C4—21no—
37C2×C13×C4⋊C4D411yescharacteristic
38C2×C13×C22⋊C4—21no—
39C2×C13×C22⋊C4—21no—
40C2×C13×C22⋊C4—21no—
41C2×C13×C22⋊C4—21no—
42C2×C13×C22⋊C4—21no—
43C2×C13×C22⋊C4—22no—
44C2×C13×C22⋊C4—22no—
45C2×C13×C22⋊C4—21no—
46C2×C13×C22⋊C4C2×C411yescharacteristic
47C2×C13×C22⋊C4C2×C411yescharacteristic
48C2×C13×C22⋊C4C2×C411yescharacteristic
49C2×C13×C22⋊C4D411yescharacteristic
50C2×C42×C13D411yescharacteristic
51C2×#6099—22no—
52C2×#6099—22no—
53C22×#1055—22no—
54C13×#95—21no—
55C13×#95—21no—
56C13×#95—21no—
57C22⋊#1055—22no—
58C22⋊#1055—22no—
59C22⋊#1055—22no—
60C22⋊#1055—22no—
61C22⋊#1055D414yescomplemented
62C4⋊#1055—22no—
63#88043—22no—
64#9515—131nomaximal, Sylow
65C24×C13—21no—
66C24×C13—22no—
67C24×C13—42no—
68C24×C13—41no—
69C24×C13—41no—
70C24×C13—81no—
71C24×C13C22⋊C411yescharacteristic
72C24×C13C22×C411yescharacteristic
73C2×D4×C13—22no—
74C2×D4×C13—22no—
75C2×D4×C13—41no—
76C2×D4×C13—42no—
77C2×D4×C13—42no—
78C2×D4×C13—81no—
79C22×C4×C13—21no—
80C22×C4×C13—21no—
81C22×C4×C13—21no—
82C22×C4×C13—21no—
83C22×C4×C13—21no—
84C22×C4×C13—22no—
85C22×C4×C13—41no—
86C22×C4×C13—41no—
87C22×C4×C13—81no—
88C22×C4×C13—81no—
89C22×C4×C13C22⋊C411yescharacteristic
90C22×C4×C13C22⋊C411yescharacteristic
91C22×C4×C13C22⋊C411yescharacteristic
92C22×C4×C13C2×D412yes—
93C22×C4×C13C2×D411yescharacteristic
94C22×C4×C13C2×D411yescharacteristic
95C22×C4×C13#4112yes—
96C13×C4⋊C4—22no—
97C13×C4⋊C4—42no—
98C13×C4⋊C4—81no—
99C13×C22⋊C4—22no—
100C13×C22⋊C4—22no—
101C13×C22⋊C4—42no—
102C13×C22⋊C4—42no—
103C13×C22⋊C4—42no—
104C13×C22⋊C4—41no—
105C13×C22⋊C4—42no—
106C13×C22⋊C4—41no—
107C13×C22⋊C4—41no—
108C13×C22⋊C4—41no—
109C13×C22⋊C4—81no—
110C42×C13—21no—
111C42×C13—21no—
112#6099—82no—
113#6099—82no—
114C2×#1055—24no—
115C2×#1055—42no—
116C2×#1055—82no—
117#2973—131no—
118#2534D1311yescharacteristic
119#1465—131no—
120#1462—132no—
121#1305—132no—
122C23×C13—21no—
123C23×C13—21no—
124C23×C13—21no—
125C23×C13—21no—
126C23×C13—22no—
127C23×C13—24no—
128C23×C13—41no—
129C23×C13—41no—
130C23×C13—41no—
131C23×C13—41no—
132C23×C13—41no—
133C23×C13—41no—
134C23×C13—41no—
135C23×C13—42no—
136C23×C13—41no—
137C23×C13—41no—
138C23×C13—41no—
139C23×C13—82no—
140C23×C13—82no—
141C23×C13—84no—
142C23×C13—84no—
143C23×C13#10012yes—
144C23×C13C2×C22⋊C411yescharacteristic
145C23×C13C4×D412yes—
146D4×C13—44no—
147D4×C13—84no—
148C2×C4×C13—21no—
149C2×C4×C13—21no—
150C2×C4×C13—21no—
151C2×C4×C13—21no—
152C2×C4×C13—21no—
153C2×C4×C13—42no—
154C2×C4×C13—42no—
155C2×C4×C13—41no—
156C2×C4×C13—41no—
157C2×C4×C13—42no—
158C2×C4×C13—41no—
159C2×C4×C13—41no—
160C2×C4×C13—41no—
161C2×C4×C13—41no—
162C2×C4×C13—41no—
163C2×C4×C13—41no—
164C2×C4×C13—41no—
165C2×C4×C13—41no—
166C2×C4×C13—41no—
167C2×C4×C13—41no—
168C2×C4×C13—41no—
169C2×C4×C13—81no—
170C2×C4×C13—82no—
171C2×C4×C13#12011yescharacteristic
172C2×C4×C13#12112yes—
173C2×C4×C13#12311yescharacteristic
174#1055—44no—
175#1055—82no—
176C2×#123Dic1311yescharacteristic
177C2×#120Dic1311yescharacteristic
178C2×C4×D4C2×D1311yescharacteristic
179C22×C22⋊C4C2×D1311yescharacteristic
180#435—264no—
181#432—261no—
182#407—132no—
183#407—132no—
184C2×C22⋊C8—261no—
185#396—21no—
186#394—21no—
187#388—21no—
188#386—21no—
189#381Dic1311yescharacteristic
190#380C2×D1311yescharacteristic
191#379Dic1311yescharacteristic
192#324—261no—
193#323—264no—
194#323—261no—
195C22×C13—21no—
196C22×C13—21no—
197C22×C13—22no—
198C22×C13—22no—
199C22×C13—22no—
200C22×C13—22no—
201C22×C13—42no—
202C22×C13—42no—
203C22×C13—42no—
204C22×C13—42no—
205C22×C13—42no—
206C22×C13—41no—
207C22×C13—41no—
208C22×C13—42no—
209C22×C13—81no—
210C22×C13—81no—
211C22×C13—81no—
212C22×C13—84no—
213C22×C13—84no—
214C22×C13—81no—
215C22×C13—81no—
216C22×C13#38611yescharacteristic
217C22×C13C2×#10011yescharacteristic
218C22×C13#41311yescharacteristic
219C4×C13—21no—
220C4×C13—22no—
221C4×C13—21no—
222C4×C13—41no—
223C4×C13—41no—
224C4×C13—41no—
225C4×C13—81no—
226C4×C13—81no—
227C4×C13—81no—
228C25C2×Dic1311yescharacteristic
229C22×D4—21no—
230C22×D4C2×Dic1311yescharacteristic
231C23×C4C2×Dic1311yescharacteristic
232C23×C4C22⋊D1311yescharacteristic
233C23×C4C22×D1311yescharacteristic
234C2×M16—262no—
235C2×M16—262no—
236C22×C8—262no—
237#123—42no—
238#120—42no—
239C4×D4—22no—
240C4×D4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C22⋊D1311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4C2×Dic1311yescharacteristic
253C2×C22⋊C4C2×Dic1311yescharacteristic
254C2×C22⋊C4C2×Dic1311yescharacteristic
255C2×C22⋊C4C22⋊D1311yescharacteristic
256C2×C42C22⋊D1311yescharacteristic
257C4⋊C8—262no—
258C22⋊C8—134no—
259C22⋊C8—262no—
260C22⋊C8—262no—
261C22⋊C8—262no—
262C22⋊C8—262no—
263#97—262no—
264#95—21no—
265#95—21no—
266#95—21no—
267C2×C13—21no—
268C2×C13—21no—
269C2×C13—42no—
270C2×C13—42no—
271C2×C13—82no—
272C2×C13#189111yescharacteristic
273C2×C13#190011yescharacteristic
274C2×C13#191311yescharacteristic
275C24—21no—
276C24—22no—
277C24—42no—
278C24—41no—
279C24—41no—
280C24—81no—
281C24C22⋊Dic1311yescharacteristic
282C24C22×Dic1311yescharacteristic
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—81no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4C22⋊Dic1311yescharacteristic
300C22×C4C22⋊Dic1311yescharacteristic
301C22×C4C22⋊Dic1311yescharacteristic
302C22×C4D4×D1312yes—
303C22×C4#612912yes—
304C22×C4C2×C22⋊D1311yescharacteristic
305C22×C4C2×C22⋊D1311yescharacteristic
306M16—1042no—
307M16—1042no—
308C2×C8—264no—
309C2×C8—522no—
310C2×C8—1042no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—81no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—81no—
325C42—21no—
326C42—21no—
327C13#951511yescharacteristic, complemented, Sylow
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—22no—
333C23—24no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—82no—
346C23—82no—
347C23—84no—
348C23—84no—
349C23#8807312yes—
350C23D4×Dic1312yes—
351C23C2×C22⋊Dic1311yescharacteristic
352D4—44no—
353D4—84no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4#8818911yescharacteristic
378C2×C4#8819111yescharacteristic
379C2×C4#8819312yes—
380C8—524no—
381C8—1042no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—84no—
400C22—84no—
401C22—81no—
402C22—81no—
403C22C2×#8807311yescharacteristic
404C22#1171888111yescharacteristic
405C22#1171889811yescharacteristic
406C4—21no—
407C4—22no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C4—81no—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—42no—
419C2—82no—
420C2#1311525511yescharacteristic
421C2#1311530011yescharacteristic
422C2#1311537911yescharacteristic
subgroupG/H
centerC22#11718881
commutatorC24×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2534C2
socleC22×C13#386
prime radicalC2×C13×#120C4
cosocleC23×C4×C13C23
hypercenter#2534D13
hypocenterC13#9515
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. #13123865C2semidirect
2. #13113174C2semidirect
3. #13113163C2semidirect
4. #13112742C2semidirect
61. C22⋊#1055D4semidirect
327. C13#9515semidirect

Maximal subgroups

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

subgroupindex
#131238652
#131131742
#131131632
#131127422
C13×#25342
#951513