Finite Group Databasebeta

← #15772435

Subgroups — #15772435

Order 3328.

700 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 91 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#13113163C211yesmaximal, characteristic, complemented
2C2×#11718117C211yesmaximal, characteristic, complemented
3#13112784C212yesmaximal, complemented
4#13112764C212yesmaximal, complemented
5C13×#2405C211yesmaximal, characteristic
6C23×D4×C13C411yescharacteristic
7C2×C13×#123—21no—
8C22×C13×C22⋊C4C2211yescharacteristic
9C13×#394—21no—
10C13×#394—21no—
11C13×#386—21no—
12C13×#385C411yescharacteristic
13C13×#385C2211yescharacteristic
14C13×#379C411yescharacteristic
15C13×#379C411yescharacteristic
16C13×#379C2211yescharacteristic
17#11718868—21no—
18C2×C22⋊#1055—21no—
19#11718117—21no—
20#11718117—21no—
21#11718117C2212yes—
22#11718117C2212yes—
23C25×C13C2×C411yescharacteristic
24C25×C13C2×C411yescharacteristic
25C22×D4×C13—21no—
26C22×D4×C13—21no—
27C22×D4×C13—44no—
28C22×D4×C13—41no—
29C22×D4×C13—41no—
30C23×C4×C13C2×C411yescharacteristic
31C23×C4×C13D411yescharacteristic
32C23×C4×C13D411yescharacteristic
33C13×#123—44no—
34C2×C13×C4⋊C4—21no—
35C2×C13×C4⋊C4—21no—
36C2×C13×C22⋊C4—21no—
37C2×C13×C22⋊C4—21no—
38C2×C13×C22⋊C4—21no—
39C2×C13×C22⋊C4—21no—
40C2×C13×C22⋊C4—21no—
41C2×C13×C22⋊C4—21no—
42C2×C13×C22⋊C4—41no—
43C2×C13×C22⋊C4—41no—
44C2×C13×C22⋊C4C2×C411yescharacteristic
45C2×C13×C22⋊C4C2×C411yescharacteristic
46C2×C13×C22⋊C4C2×C411yescharacteristic
47C2×C13×C22⋊C4D411yescharacteristic
48C2×C13×C22⋊C4D411yescharacteristic
49C2×C13×C22⋊C4C2311yescharacteristic
50C2×#6099—41no—
51C22×#1055—41no—
52C13×#95—21no—
53C13×#95—21no—
54C22⋊#1055—22no—
55C22⋊#1055—22no—
56C22⋊#1055—41no—
57C22⋊#1055—41no—
58#11541—131nomaximal, Sylow
59C24×C13—21no—
60C24×C13—21no—
61C24×C13—21no—
62C24×C13—22no—
63C24×C13—22no—
64C24×C13—41no—
65C24×C13—42no—
66C24×C13—41no—
67C24×C13—42no—
68C24×C13—41no—
69C24×C13—81no—
70C24×C13—81no—
71C24×C13C22⋊C411yescharacteristic
72C24×C13C22⋊C411yescharacteristic
73C24×C13C22×C411yescharacteristic
74C2×D4×C13—24no—
75C2×D4×C13—42no—
76C2×D4×C13—42no—
77C2×D4×C13—42no—
78C2×D4×C13—42no—
79C2×D4×C13—44no—
80C2×D4×C13—44no—
81C2×D4×C13—44no—
82C2×D4×C13—41no—
83C2×D4×C13—41no—
84C2×D4×C13—44no—
85C22×C4×C13—21no—
86C22×C4×C13—21no—
87C22×C4×C13—21no—
88C22×C4×C13—21no—
89C22×C4×C13—21no—
90C22×C4×C13—21no—
91C22×C4×C13—21no—
92C22×C4×C13—42no—
93C22×C4×C13—41no—
94C22×C4×C13—42no—
95C22×C4×C13—81no—
96C22×C4×C13C22⋊C411yescharacteristic
97C22×C4×C13C22⋊C411yescharacteristic
98C22×C4×C13C2×D411yescharacteristic
99C22×C4×C13C2×D411yescharacteristic
100C13×C4⋊C4—42no—
101C13×C4⋊C4—42no—
102C13×C22⋊C4—22no—
103C13×C22⋊C4—41no—
104C13×C22⋊C4—41no—
105C13×C22⋊C4—41no—
106C13×C22⋊C4—41no—
107C13×C22⋊C4—42no—
108C13×C22⋊C4—42no—
109C13×C22⋊C4—42no—
110C13×C22⋊C4—41no—
111C13×C22⋊C4—81no—
112C13×C22⋊C4—81no—
113C13×C22⋊C4—81no—
114#6099—82no—
115C2×#1055—42no—
116C2×#1055—42no—
117C2×#1055—82no—
118#2405D1311yescharacteristic
119#1462—131no—
120C2×#323—131no—
121#1326—132no—
122#1316—132no—
123C23×C13—22no—
124C23×C13—21no—
125C23×C13—21no—
126C23×C13—21no—
127C23×C13—21no—
128C23×C13—22no—
129C23×C13—21no—
130C23×C13—21no—
131C23×C13—21no—
132C23×C13—41no—
133C23×C13—42no—
134C23×C13—42no—
135C23×C13—41no—
136C23×C13—41no—
137C23×C13—41no—
138C23×C13—41no—
139C23×C13—41no—
140C23×C13—42no—
141C23×C13—41no—
142C23×C13—41no—
143C23×C13—42no—
144C23×C13—42no—
145C23×C13—41no—
146C23×C13—41no—
147C23×C13—41no—
148C23×C13—42no—
149C23×C13—81no—
150C23×C13—81no—
151C23×C13—82no—
152C23×C13—84no—
153C23×C13—84no—
154C23×C13—84no—
155C23×C13—84no—
156C23×C13—82no—
157C23×C13#9911yescharacteristic
158C23×C13#9911yescharacteristic
159C23×C13C2×C22⋊C411yescharacteristic
160C23×C13#13112yes—
161D4×C13—44no—
162D4×C13—48no—
163D4×C13—44no—
164C2×C4×C13—22no—
165C2×C4×C13—21no—
166C2×C4×C13—21no—
167C2×C4×C13—21no—
168C2×C4×C13—21no—
169C2×C4×C13—42no—
170C2×C4×C13—41no—
171C2×C4×C13—42no—
172C2×C4×C13—42no—
173C2×C4×C13—41no—
174C2×C4×C13—41no—
175C2×C4×C13—41no—
176C2×C4×C13—41no—
177C2×C4×C13—41no—
178C2×C4×C13—42no—
179C2×C4×C13—41no—
180C2×C4×C13—41no—
181C2×C4×C13—41no—
182C2×C4×C13—41no—
183C2×C4×C13—41no—
184C2×C4×C13—41no—
185C2×C4×C13—81no—
186C2×C4×C13—81no—
187#1055—82no—
188#1055—82no—
189C23×D4Dic1311yescharacteristic
190C2×#123—21no—
191C22×C22⋊C4C2×D1311yescharacteristic
192#407—261no—
193C2×C22⋊C8—261no—
194#394—21no—
195#394—21no—
196#386—21no—
197#385Dic1311yescharacteristic
198#385C2×D1311yescharacteristic
199#379Dic1311yescharacteristic
200#379Dic1311yescharacteristic
201#379C2×D1311yescharacteristic
202#323—132no—
203#323—132no—
204#323—261no—
205#323—261no—
206C22×C13—21no—
207C22×C13—21no—
208C22×C13—21no—
209C22×C13—21no—
210C22×C13—21no—
211C22×C13—21no—
212C22×C13—22no—
213C22×C13—21no—
214C22×C13—42no—
215C22×C13—42no—
216C22×C13—42no—
217C22×C13—42no—
218C22×C13—42no—
219C22×C13—42no—
220C22×C13—41no—
221C22×C13—41no—
222C22×C13—42no—
223C22×C13—41no—
224C22×C13—41no—
225C22×C13—41no—
226C22×C13—84no—
227C22×C13—81no—
228C22×C13—81no—
229C22×C13—81no—
230C22×C13—81no—
231C22×C13—84no—
232C22×C13—84no—
233C22×C13—84no—
234C22×C13—81no—
235C22×C13—81no—
236C22×C13AΣL(1,𝔽16)12yes—
237C22×C13#40811yescharacteristic
238C22×C13C2×#9911yescharacteristic
239C22×C13#41311yescharacteristic
240C4×C13—41no—
241C4×C13—42no—
242C4×C13—41no—
243C4×C13—41no—
244C4×C13—41no—
245C4×C13—81no—
246C4×C13—81no—
247C25C2×Dic1311yescharacteristic
248C25C2×Dic1311yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—44no—
252C22×D4—41no—
253C22×D4—41no—
254C23×C4C2×Dic1311yescharacteristic
255C23×C4C22⋊D1311yescharacteristic
256C23×C4C22⋊D1311yescharacteristic
257C2×M16—521no—
258C22×C8—521no—
259#123—44no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4C2×Dic1311yescharacteristic
271C2×C22⋊C4C2×Dic1311yescharacteristic
272C2×C22⋊C4C2×Dic1311yescharacteristic
273C2×C22⋊C4C22⋊D1311yescharacteristic
274C2×C22⋊C4C22⋊D1311yescharacteristic
275C2×C22⋊C4C22×D1311yescharacteristic
276C22⋊C8—262no—
277C22⋊C8—262no—
278C22⋊C8—521no—
279C22⋊C8—521no—
280#95—21no—
281#95—21no—
282C2×C13—21no—
283C2×C13—21no—
284C2×C13—22no—
285C2×C13—41no—
286C2×C13—42no—
287C2×C13—82no—
288C2×C13—81no—
289C2×C13#146811yescharacteristic
290C2×C13C2×AΣL(1,𝔽16)11yescharacteristic
291C2×C13#212111yescharacteristic
292C24—21no—
293C24—21no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—42no—
301C24—41no—
302C24—81no—
303C24—81no—
304C24C22⋊Dic1311yescharacteristic
305C24C22⋊Dic1311yescharacteristic
306C24C22×Dic1311yescharacteristic
307C2×D4—24no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—44no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—81no—
329C22×C4C22⋊Dic1311yescharacteristic
330C22×C4C22⋊Dic1311yescharacteristic
331C22×C4C2×C22⋊D1311yescharacteristic
332C22×C4C2×C22⋊D1311yescharacteristic
333M16—1042no—
334C2×C8—522no—
335C2×C8—522no—
336C2×C8—1042no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C22⋊C4—22no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—41no—
348C22⋊C4—81no—
349C22⋊C4—81no—
350C22⋊C4—81no—
351C13#1154111yescharacteristic, complemented, Sylow
352C23—21no—
353C23—21no—
354C23—22no—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—81no—
379C23—81no—
380C23—82no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—84no—
385C23—82no—
386C23#8807411yescharacteristic
387C23#8807411yescharacteristic
388C23#8820212yes—
389C23C2×C22⋊Dic1311yescharacteristic
390D4—44no—
391D4—48no—
392D4—44no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—81no—
415C2×C4—81no—
416C8—1042no—
417C8—1042no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—84no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—84no—
444C22—84no—
445C22—84no—
446C22—81no—
447C22—81no—
448C22#1171812812yes—
449C22C2×#8807411yescharacteristic
450C22#1171889511yescharacteristic
451C22#1171889811yescharacteristic
452C4—41no—
453C4—42no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—81no—
458C4—81no—
459C2—21no—
460C2—21no—
461C2—22no—
462C2—41no—
463C2—42no—
464C2—82no—
465C2—81no—
466C2#1311318411yescharacteristic
467C2C2×#1171812811yescharacteristic
468C2#1311648211yescharacteristic
subgroupG/H
centerC22C2×#88074
commutatorC24×C13C22×C4
FrattiniC2×C22⋊C4C22×D13
FittingC13×#2405C2
socleC22×C13C2×#99
prime radicalC23×D4×C13C4
cosocleC2×C13×C22⋊C4C23
hypercenter#2405D13
hypocenterC13#11541
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2405

Complements

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

NH ≅ G/Ntype
1. #13113163C2semidirect
2. C2×#11718117C2semidirect
3. #13112784C2semidirect
4. #13112764C2semidirect
351. C13#11541semidirect

Maximal subgroups

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

subgroupindex
#131131632
C2×#117181172
#131127842
#131127642
C13×#24052
#1154113