Finite Group Databasebeta

← #29577833

Subgroups — #29577833

Order 15616.

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

normal subgroups only

This group has 838 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#21448417C211yesmaximal, characteristic, complemented
2#21445763C212yesmaximal, complemented
3#21444146C211yesmaximal, characteristic, complemented
4#21443671C212yesmaximal, complemented
5C61×#2404C211yesmaximal, characteristic
6C2×#14519704—24no—
7#16839726C2212yes—
8#16839655—24no—
9#16839622—21no—
10#16839618—21no—
11C4×C22⋊Dic61C2212yes—
12#16839486—22no—
13C22×#41×C61C2211yescharacteristic
14C2×#120×C61—21no—
15C22×C22⋊C4×C61C2211yescharacteristic
16C61×#393C412yescomplemented
17C61×#392C412yescomplemented
18C61×#386—21no—
19C61×#386—22no—
20C61×#381C2211yescharacteristic
21C2×#11758194—22no—
22C2×C4⋊Dic61—22no—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26C2×C4×Dic61D414yescomplemented
27#14519704—48no—
28C22⋊#92418—22no—
29C22⋊#92418—22no—
30#14519684—22no—
31#14519684—22no—
32C25×C61D411yescharacteristic
33C2×#41×C61—22no—
34C2×#41×C61—22no—
35C2×#41×C61—44no—
36C22×Q8×C61C2×C411yescharacteristic
37C22×D4×C61—21no—
38C22×D4×C61—22no—
39C22×D4×C61C2×C411yescharacteristic
40C23×C4×C61—21no—
41C23×C4×C61D411yescharacteristic
42C23×C4×C61C2311yescharacteristic
43#120×C61—44no—
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61—21no—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61C2×C412yes—
50C2×C22⋊C4×C61D412yes—
51#95×C61—22no—
52#95×C61—21no—
53#95×C61C2×C412yes—
54C24×C61—21no—
55C24×C61—21no—
56C24×C61—21no—
57C24×C61—42no—
58C24×C61—42no—
59C24×C61—81no—
60C24×C61C22⋊C412yes—
61C24×C61C2×D411yescharacteristic
62#41×C61—28no—
63#41×C61—44no—
64#41×C61—44no—
65#41×C61—44no—
66C2×Q8×C61—22no—
67C2×Q8×C61—42no—
68C2×D4×C61—24no—
69C2×D4×C61—22no—
70C2×D4×C61—42no—
71C2×D4×C61—42no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61—44no—
75C2×D4×C61—44no—
76C2×D4×C61—82no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—22no—
81C22×C4×C61—42no—
82C22×C4×C61—42no—
83C22×C4×C61—42no—
84C22×C4×C61—42no—
85C22×C4×C61—81no—
86C22×C4×C61C22⋊C412yes—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61#4112yes—
91C22×Dic61—24no—
92C22⋊C4×C61—24no—
93C22⋊C4×C61—24no—
94C22⋊C4×C61—44no—
95C22⋊C4×C61—42no—
96C22⋊C4×C61—82no—
97C22⋊Dic61—44no—
98C22⋊Dic61—82no—
99C22⋊Dic61—82no—
100C4⋊Dic61—82no—
101C4×Dic61—44no—
102#11758194—44no—
103C2×#92418—42no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—22no—
107C23×C61—22no—
108C23×C61—22no—
109C23×C61—22no—
110C23×C61—41no—
111C23×C61—42no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—41no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61—41no—
119C23×C61—42no—
120C23×C61—42no—
121C23×C61—81no—
122C23×C61—81no—
123C23×C61—88no—
124C23×C61C2×C22⋊C411yescharacteristic
125C23×C61#12011yescharacteristic
126C23×C61#12311yescharacteristic
127Q8×C61—24no—
128Q8×C61—42no—
129D4×C61—24no—
130D4×C61—42no—
131D4×C61—44no—
132D4×C61—44no—
133D4×C61—88no—
134C2×C4×C61—24no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—48no—
141C2×C4×C61—42no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—81no—
149C2×C4×C61C4×D412yes—
150C2×C4×C61#12112yes—
151C2×Dic61—44no—
152C2×Dic61—44no—
153C2×Dic61—44no—
154#92418—82no—
155#9479—611nomaximal, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—44no—
164C22×C61—44no—
165C22×C61—44no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—82no—
171C22×C61—82no—
172C22×C61—84no—
173C22×C61—84no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61#38611yescharacteristic
177C22×C61#42112yes—
178C22×C61B(4,𝔽2)12yes—
179C4×C61—22no—
180C4×C61—22no—
181C4×C61—42no—
182C4×C61—42no—
183C4×C61—82no—
184Dic61—84no—
185#2404D6111yescharacteristic
186#2312—611no—
187#1789—612no—
188#1462—611no—
189#1284—612no—
190C2×C61—22no—
191C2×C61—22no—
192C2×C61—41no—
193C2×C61—82no—
194C2×C61—82no—
195C2×C61#188911yescharacteristic
196C2×C61#189711yescharacteristic
197C2×C61#191111yescharacteristic
198C22×#41C2×D6111yescharacteristic
199C2×#120—21no—
200C22×C22⋊C4C2×D6111yescharacteristic
201C2×#104—1224no—
202#407—612no—
203#393Dic6112yescomplemented
204#392Dic6112yescomplemented
205#388—1224no—
206#386—21no—
207#386—22no—
208#381C2×D6111yescharacteristic
209#380—1221no—
210#379—1221no—
211C4×C22⋊C4—612no—
212#323—1222no—
213C61#947911yescharacteristic, complemented, Sylow
214C25C22⋊D6111yescharacteristic
215C2×#41—22no—
216C2×#41—22no—
217C2×#41—44no—
218C22×Q8C2×Dic6111yescharacteristic
219C22×D4—21no—
220C22×D4—22no—
221C22×D4C2×Dic6111yescharacteristic
222C23×C4—21no—
223C23×C4C22⋊D6111yescharacteristic
224C23×C4C22×D6111yescharacteristic
225C2×M16—1222no—
226#120—44no—
227C2×C4⋊C4—1222no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—1222no—
234C2×C22⋊C4—1222no—
235C2×C22⋊C4—1222no—
236C2×C22⋊C4C2×Dic6112yes—
237C2×C22⋊C4C22⋊D6112yes—
238C2×C42—614no—
239#104—2448no—
240C22⋊C8—1222no—
241C22⋊C8—1222no—
242#95—22no—
243#95—21no—
244#95—1222no—
245#95—1222no—
246#95C2×Dic6112yes—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—42no—
251C24—42no—
252C24—81no—
253C24C22⋊Dic6112yes—
254C24C2×C22⋊D6111yescharacteristic
255#41—28no—
256#41—44no—
257#41—44no—
258#41—44no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—1224no—
280C22×C4C22⋊Dic6112yes—
281C22×C4D4×D6112yes—
282C22×C4#1175822412yes—
283C22×C4C22×Dic6111yescharacteristic
284C22×C4C2×C22⋊D6111yescharacteristic
285M16—2444no—
286C2×C8—2442no—
287C4⋊C4—4882no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—82no—
293C22⋊C4—4882no—
294C22⋊C4—4882no—
295C22⋊C4—2444no—
296C42—2444no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—41no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—41no—
314C23—81no—
315C23—81no—
316C23—88no—
317C23C2×C22⋊Dic6111yescharacteristic
318C23#1451974511yescharacteristic
319C23#1451975811yescharacteristic
320Q8—24no—
321Q8—42no—
322D4—24no—
323D4—42no—
324D4—44no—
325D4—44no—
326D4—88no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—48no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—2444no—
343C2×C4—2444no—
344C2×C4—2444no—
345C2×C4D4×Dic6112yes—
346C2×C4#1451974912yes—
347C8—4882no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—84no—
365C22—84no—
366C22—81no—
367C22—81no—
368C22#1683964211yescharacteristic
369C22#1683977912yes—
370C22#1683994812yes—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—4884no—
377C2—22no—
378C2—22no—
379C2—41no—
380C2—82no—
381C2—82no—
382C2#2144622811yescharacteristic
383C2#2144626811yescharacteristic
384C2#2144635211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#2404C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenter#2404D61
hypocenterC61#9479
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. #21448417C2semidirect
2. #21445763C2semidirect
3. #21444146C2semidirect
4. #21443671C2semidirect
16. C61×#393C4semidirect
17. C61×#392C4semidirect
26. C2×C4×Dic61D4semidirect
203. #393Dic61semidirect
204. #392Dic61semidirect
213. C61#9479semidirect

Maximal subgroups

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

subgroupindex
#214484172
#214457632
#214441462
#214436712
C61×#24042
#947961