Finite Group Databasebeta

← #29597366

Subgroups — #29597366

Order 15616.

700 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 103 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#21448931C211yesmaximal, characteristic, complemented
2C2×#16839559C211yesmaximal, characteristic, complemented
3#21445778C212yesmaximal, complemented
4#21443945C212yesmaximal, complemented
5C2×C61×#454C211yesmaximal, characteristic
6C2×#14519704—22no—
7#16839763—22no—
8C2×#14519696C2212yes—
9#16839669C2212yes—
10#16839655—22no—
11#16839642—21no—
12C22⋊C4×Dic61—22no—
13#16839608—21no—
14#16839559—22no—
15C2×2−1+4×C61C2211yescharacteristic
16C2×AGL(1,ℤ/8)×C61—21no—
17C2×#124×C61C2211yescharacteristic
18C61×#454—22no—
19C61×#454C414yescomplemented
20C2×#104×C61—21no—
21C2×#101×C61C2211yescharacteristic
22C23×Dic61—21no—
23C2×#11758194—22no—
24C2×C4⋊Dic61D412yes—
25C2×C22⋊Dic61—22no—
26C2×C22⋊Dic61—21no—
27C2×C4×Dic61—22no—
28C2×C4×Dic61D412yes—
29#14519704—44no—
30#14519701—42no—
31#14519698—42no—
32#14519696—22no—
33#14519696—22no—
34#14519684—21no—
35#14519684—21no—
36#14519684—22no—
37#14519684—22no—
382−1+4×C61—21no—
392−1+4×C61—22no—
402−1+4×C61—42no—
412−1+4×C61C2×C412yescomplemented
42C2×#41×C61—21no—
43C2×#41×C61—22no—
44C2×#41×C61—41no—
45C22×Q8×C61—41no—
46C22×Q8×C61C2311yescharacteristic
47C22×D4×C61D411yescharacteristic
48AGL(1,ℤ/8)×C61—24no—
49AGL(1,ℤ/8)×C61—42no—
50C2×SD16×C61—41no—
51C2×D8×C61—41no—
52C2×M16×C61—21no—
53#124×C61—22no—
54#124×C61C2×C412yes—
55#124×C61D412yes—
56C2×C22⋊C4×C61—22no—
57C2×C42×C61D411yescharacteristic
58#104×C61—24no—
59#101×C61—21no—
60#101×C61C2×C412yescomplemented
61C24×C61—21no—
62#41×C61—22no—
63#41×C61—24no—
64#41×C61—22no—
65#41×C61—24no—
66#41×C61—42no—
67#41×C61—41no—
68#41×C61—42no—
69#41×C61—42no—
70#41×C61—44no—
71#41×C61—42no—
72#41×C61—41no—
73C2×Q8×C61—21no—
74C2×Q8×C61—22no—
75C2×Q8×C61—41no—
76C2×Q8×C61—41no—
77C2×Q8×C61—42no—
78C2×Q8×C61—42no—
79C2×Q8×C61—44no—
80C2×Q8×C61C22×C411yescharacteristic
81C2×Q8×C61C2×D411yescharacteristic
82C2×Q8×C61#4112yes—
83C2×D4×C61—21no—
84C2×D4×C61—22no—
85C2×D4×C61—21no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C2×D4×C61C22⋊C412yes—
90C22×C4×C61—21no—
91C22×C4×C61—42no—
92C22×C4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D412yes—
95C22×Dic61—22no—
96C22×Dic61—22no—
97C22×Dic61—44no—
98C22×Dic61—81no—
99SD16×C61—44no—
100D8×C61—44no—
101M16×C61—22no—
102M16×C61—41no—
103C2×C8×C61—41no—
104C22⋊C4×C61—24no—
105C22⋊C4×C61—24no—
106C42×C61—21no—
107C42×C61C22⋊C412yes—
108C22⋊Dic61—44no—
109C22⋊Dic61—81no—
110C4⋊Dic61—42no—
111C4×Dic61—44no—
112C4×Dic61—42no—
113#11758194—42no—
114#11758194—42no—
115C2×#92418—42no—
116C23×C61—22no—
117C23×C61—42no—
118C23×C61—42no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61#12011yescharacteristic
123Q8×C61—22no—
124Q8×C61—24no—
125Q8×C61—41no—
126Q8×C61—41no—
127Q8×C61—42no—
128Q8×C61—42no—
129Q8×C61—41no—
130D4×C61—22no—
131D4×C61—24no—
132D4×C61—42no—
133D4×C61—42no—
134D4×C61—41no—
135D4×C61—42no—
136D4×C61—42no—
137D4×C61—81no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—22no—
143C2×C4×C61—44no—
144C2×C4×C61—42no—
145C2×C4×C61—44no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C4×D412yes—
151C2×C4×C61#12112yes—
152C2×C4×C61#12311yescharacteristic
153C2×Dic61—24no—
154C2×Dic61—42no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159C2×Dic61—84no—
160C8×C61—42no—
161#92418—82no—
162#13057—611nomaximal, Sylow
163C22×C61—22no—
164C22×C61—44no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—81no—
168C22×C61—81no—
169C22×C61—82no—
170C22×C61—82no—
171C22×C61#38611yescharacteristic
172C22×C61B(4,𝔽2)11yescharacteristic
173C22×C61B(4,𝔽2)11yescharacteristic
174C4×C61—22no—
175C4×C61—21no—
176C4×C61—22no—
177C4×C61—21no—
178C4×C61—42no—
179C4×C61—42no—
180Dic61—44no—
181Dic61—82no—
182C2×#454D6111yescharacteristic
183#2372—611no—
184C2×#355—611no—
185#1793—612no—
186#1394—612no—
187C2×C61—21no—
188C2×C61—21no—
189C2×C61—82no—
190C2×C61—82no—
191C2×C61#188911yescharacteristic
192C2×C61#220412yes—
193C2×2−1+4C2×D6111yescharacteristic
194C2×AGL(1,ℤ/8)—21no—
195C2×#124C2×D6111yescharacteristic
196#454—22no—
197#454Dic6114yescomplemented
198C2×#104—21no—
199C2×#104—1222no—
200#417—1222no—
201C2×#101—612no—
202C2×#101C2×D6111yescharacteristic
203#391—612no—
204#388—1222no—
205#386—1221no—
206C4×C22⋊C4—1222no—
207#376—1221no—
208#355—1222no—
209C61#1305711yescharacteristic, complemented, Sylow
2102−1+4—22no—
2112−1+4—21no—
2122−1+4—42no—
2132−1+4C2×Dic6112yescomplemented
214C2×#41—22no—
215C2×#41—21no—
216C2×#41—41no—
217C22×Q8—41no—
218C22×Q8C22×D6111yescharacteristic
219C22×D4C22⋊D6111yescharacteristic
220C23×C4—1221no—
221AGL(1,ℤ/8)—24no—
222AGL(1,ℤ/8)—42no—
223C2×SD16—41no—
224C2×D8—41no—
225C2×M16—21no—
226C2×M16—1222no—
227#124—22no—
228#124C2×Dic6112yes—
229#124C22⋊D6112yes—
230C2×C4⋊C4—612no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—1221no—
233C2×C22⋊C4—1222no—
234C2×C42—612no—
235C2×C42—1222no—
236C2×C42C22⋊D6111yescharacteristic
237#104—24no—
238#104—2444no—
239#103—2442no—
240#102—2442no—
241#101—21no—
242#101—1222no—
243#101—1222no—
244#101C2×Dic6112yescomplemented
245#95—1221no—
246#95—1221no—
247#95—1222no—
248#95—1222no—
249C24—21no—
250#41—24no—
251#41—22no—
252#41—22no—
253#41—24no—
254#41—42no—
255#41—41no—
256#41—42no—
257#41—42no—
258#41—44no—
259#41—42no—
260#41—41no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—44no—
268C2×Q8#1175822412yes—
269C2×Q8C22×Dic6111yescharacteristic
270C2×Q8C2×C22⋊D6111yescharacteristic
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4C22⋊Dic6112yes—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—4881no—
284C22×C4—2444no—
285C22×C4D4×D6112yes—
286C22×C4C2×C22⋊D6111yescharacteristic
287SD16—44no—
288D8—44no—
289M16—22no—
290M16—41no—
291M16—2442no—
292M16—2442no—
293C2×C8—41no—
294C2×C8—2442no—
295C4⋊C4—2442no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—4881no—
299C22⋊C4—2444no—
300C42—21no—
301C42—2442no—
302C42—2444no—
303C42C22⋊Dic6112yes—
304C23—22no—
305C23—42no—
306C23—42no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23#1451974511yescharacteristic
311Q8—22no—
312Q8—24no—
313Q8—42no—
314Q8—41no—
315Q8—41no—
316Q8—42no—
317Q8—41no—
318D4—22no—
319D4—24no—
320D4—42no—
321D4—42no—
322D4—41no—
323D4—42no—
324D4—42no—
325D4—81no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—1224no—
338C2×C4—4884no—
339C2×C4—2442no—
340C2×C4—2442no—
341C2×C4—2442no—
342C2×C4—2442no—
343C2×C4—2442no—
344C2×C4C2×C22⋊Dic6111yescharacteristic
345C2×C4D4×Dic6112yes—
346C2×C4#1451974912yes—
347C2×C4#1451975811yescharacteristic
348C8—42no—
349C8—4882no—
350C22—22no—
351C22—44no—
352C22—42no—
353C22—42no—
354C22—81no—
355C22—81no—
356C22—82no—
357C22—82no—
358C22#1683964211yescharacteristic
359C22#1683994811yescharacteristic
360C22#1683994811yescharacteristic
361C4—21no—
362C4—22no—
363C4—22no—
364C4—21no—
365C4—42no—
366C4—42no—
367C4—4882no—
368C4—2444no—
369C2—21no—
370C2—21no—
371C2—82no—
372C2—82no—
373C2#2144622811yescharacteristic
374C2#2144784312yes—
subgroupG/H
centerC22#16839948
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
Fitting, prime radicalC2×C61×#454C2
socleC22×C61B(4,𝔽2)
cosocleC22×Q8×C61C23
hypercenterC2×#454D61
hypocenterC61#13057
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#454

Complements

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

NH ≅ G/Ntype
1. #21448931C2semidirect
2. C2×#16839559C2semidirect
3. #21445778C2semidirect
4. #21443945C2semidirect
19. C61×#454C4semidirect
41. 2−1+4×C61C2×C4semidirect
60. #101×C61C2×C4semidirect
197. #454Dic61semidirect
209. C61#13057semidirect
213. 2−1+4C2×Dic61semidirect
244. #101C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214489312
C2×#168395592
#214457782
#214439452
C2×C61×#4542
#1305761