Finite Group Databasebeta

← #30599731

Subgroups — #30599731

Order 15616.

392 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 88 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#21462519C211yesmaximal, characteristic
2#21461692C211yesmaximal, characteristic
3#21461389C211yesmaximal, characteristic, complemented
4#21461375C211yesmaximal, characteristic, complemented
5#21460716C211yesmaximal, characteristic, complemented
6#21460654C211yesmaximal, characteristic, complemented
7#21444200C211yesmaximal, characteristic
8#16840787—22no—
9#16840784—21no—
10#16840781—21no—
11#16840766C2211yescharacteristic
12#16840766C2211yescharacteristic
13D61⋊(C4⋊C8)C2211yescharacteristic
14C2×#14519842—21no—
15#16840737—21no—
16#16840727—21no—
17C4×C4⋊#6923C2211yescharacteristic
18#16840632—21no—
19#16840626—21no—
20#16840625—21no—
21#16840624—22no—
22C4×#11758226C2211yescharacteristic
23#16839814C2211yescharacteristic
24C4×C8×D61C2211yescharacteristic
25Q8⋊#92418C411yescharacteristic, complemented
26#16839491C411yescharacteristic, complemented
27#16839489C411yescharacteristic, complemented
28C61×Q8⋊C8C411yescharacteristic, complemented
29C2×#11758213—21no—
30C2×#11758213—21no—
31C2×D61⋊C8—21no—
32C2×C4⋊#6923—21no—
33C2×C4×#6923—21no—
34#14519844—21no—
35C4⋊#92420—21no—
36C4⋊#92420—21no—
37C4⋊#92420—21no—
38C4⋊#92420D411yescharacteristic
39C4⋊#92420D411yescharacteristic
40C4⋊#92420D412yes—
41#14519842—42no—
42#14519834—22no—
43#14519834—22no—
44#14519832—22no—
45C4×#92420—22no—
46(C2×#6923)⋊C4—21no—
47C2×#11758226D411yescharacteristic
48C2×C8×D61—21no—
49Q8×Dic61Q811yescharacteristic
50C4×D244—21no—
51C4×D244C2×C411yescharacteristic
52#14519739—21no—
53#14519739Q811yescharacteristic
54#14519735—21no—
55#14519735D411yescharacteristic
56C42×D61—21no—
57C42×D61C2311yescharacteristic
58C4⋊#92418C2×C411yescharacteristic
59#14519689—21no—
60C4×#92418C2×C411yescharacteristic
61C8×Dic61—21no—
62C8×Dic61—21no—
63C4×Q8×C61C2×C411yescharacteristic
64C4⋊C8×C61C2×C411yescharacteristic
65C4×C8×C61C2×C411yescharacteristic
66C22×#6923—41no—
67C2×Q8×C61C4⋊C411yescharacteristic
68#11758226—21no—
69#11758226—21no—
70#11758226—21no—
71#11758226—21no—
72C2×D244—21no—
73C2×D244C4⋊C411yescharacteristic
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61C2×D411yescharacteristic
78C2×C4×D61#4111yescharacteristic
79C2×C4×D61#4111yescharacteristic
80C2×#92420—21no—
81C2×#92420—22no—
82C2×#92420—21no—
83C2×#92420—41no—
84C4⋊#6923—22no—
85C4⋊#6923—41no—
86C4×#6923—41no—
87C4×#6923—41no—
88#11758213—41no—
89#11758213—41no—
90#11758213—41no—
91#11758213—41no—
92D61⋊C8—41no—
93D61⋊C8—41no—
94C2×C8×C61—21no—
95C2×C8×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61C4⋊C411yescharacteristic
98C42×C61—21no—
99C42×C61C22×C411yescharacteristic
100#11758198—21no—
101#11758198—21no—
102#11758198—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61C4⋊C411yescharacteristic
105C4×Dic61—21no—
106C4×Dic61—21no—
107C4×Dic61C2×D411yescharacteristic
108C4×Dic61C2×D411yescharacteristic
109C4×Dic61C2×Q811yescharacteristic
110C2×#92418—21no—
111C2×#92418—21no—
112C8×D61—42no—
113C22×D61—21no—
114C22×D61#12311yescharacteristic
115C2×#6923—42no—
116C2×#6923—81no—
117C2×#6923—81no—
118Q8×C61—21no—
119Q8×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61C2×C4⋊C411yescharacteristic
123C2×C4×C61C4×D411yescharacteristic
124C2×C4×C61C4×D411yescharacteristic
125C2×Dic61—21no—
126C2×Dic61#12011yescharacteristic
127C2×Dic61C22⋊Q811yescharacteristic
128C2×Dic61C22⋊Q811yescharacteristic
129D244—21no—
130D244—21no—
131D244—41no—
132D244—41no—
133C4×D61—21no—
134C4×D61—21no—
135C4×D61—41no—
136C4×D61—41no—
137C4×D61—41no—
138C4×D61—41no—
139C4×D61—41no—
140C4×D61—81no—
141C4×D61#13511yescharacteristic
142C4×D61#13511yescharacteristic
143C4×D61AGL(1,ℤ/8)11yescharacteristic
144C4×D61#13711yescharacteristic
145#92420—41no—
146#92420—41no—
147#92420—42no—
148#92420—81no—
149C8×C61—41no—
150C8×C61—41no—
151#92418—22no—
152#92418—41no—
153#8905—611nomaximal, Sylow
154C22×C61#38511yescharacteristic
155C2×D61—21no—
156C2×D61—21no—
157C2×D61—41no—
158C2×D61—41no—
159C2×D61—41no—
160C2×D61#48211yescharacteristic
161C2×D61#48511yescharacteristic
162#6923—82no—
163C4×C61—21no—
164C4×C61—41no—
165C4×C61—41no—
166C4×C61#42711yescharacteristic
167C4×C61#42811yescharacteristic
168C4×C61#44311yescharacteristic
169C4×C61#44411yescharacteristic
170Dic61—21no—
171Dic61—21no—
172Dic61—41no—
173Dic61#44911yescharacteristic
174Dic61#45211yescharacteristic
175#2925—611no—
176#1806—611no—
177#1571—611no—
178#1566—611no—
179#1476—611no—
180#1303—611no—
181#1279—611no—
182C2×C61#185411yescharacteristic
183C2×C61#186111yescharacteristic
184C2×C61#186611yescharacteristic
185D61—21no—
186D61—21no—
187D61—81no—
188C4×#41—611no—
189#433—1222no—
190#432—1221no—
191#431—611no—
192#431—1221no—
193#423—611no—
194#423—611no—
195C2×C4⋊C8—611no—
196C2×#104—1221no—
197#408—1221no—
198#405—1221no—
199C2×C4×C8—611no—
200C4×C4⋊C4—611no—
201#339—1221no—
202#335—1221no—
203#335—1221no—
204#334—1222no—
205Q8⋊C8—611no—
206Q8⋊C8#692311yescharacteristic, complemented
207D4⋊C8—611no—
208D4⋊C8—611no—
209C61#890511yescharacteristic, complemented, Sylow
210C2×#41—611no—
211C2×M16—1221no—
212C2×M16—1221no—
213C22×C8—1221no—
214C22×C8—1221no—
215C4×Q8—611no—
216C4×Q8C2×#692311yescharacteristic
217C4×D4—611no—
218C4×D4—611no—
219C4×D4—1221no—
220C4×D4—1221no—
221#117—611no—
222#117—1221no—
223C2×C4⋊C4—1221no—
224C2×C42—611no—
225C2×C42—1221no—
226C2×C42—1221no—
227C4⋊C8—612no—
228C4⋊C8—611no—
229C4⋊C8—611no—
230C4⋊C8—611no—
231C4⋊C8—1221no—
232C4⋊C8—1221no—
233C4⋊C8—1221no—
234C4⋊C8—1221no—
235C4⋊C8C2×#692311yescharacteristic
236#104—2442no—
237C22⋊C8—1221no—
238C22⋊C8—1222no—
239C22⋊C8—1222no—
240#97—1222no—
241C4×C8—611no—
242C4×C8—1221no—
243C4×C8—1221no—
244C4×C8—1222no—
245C4×C8C2×#692311yescharacteristic
246#95—1221no—
247#41—1221no—
248#41—1221no—
249#41—1221no—
250#41—1221no—
251C2×Q8C4⋊#692311yescharacteristic
252C2×D4—611no—
253C2×D4—1221no—
254C22×C4—611no—
255C22×C4—611no—
256C22×C4—611no—
257C22×C4—1221no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—2441no—
261M16—2441no—
262M16—2441no—
263M16—2441no—
264M16—2441no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—1221no—
268C2×C8—1221no—
269C2×C8—1222no—
270C2×C8—1221no—
271C2×C8—1221no—
272C2×C8—2442no—
273C2×C8—2441no—
274C2×C8—2441no—
275C2×C8—2441no—
276C4⋊C4—21no—
277C4⋊C4—611no—
278C4⋊C4—1221no—
279C4⋊C4—1222no—
280C4⋊C4—2441no—
281C4⋊C4C4⋊#692311yescharacteristic
282C22⋊C4—1221no—
283C22⋊C4—1221no—
284C22⋊C4—1221no—
285C42—21no—
286C42—611no—
287C42—611no—
288C42—611no—
289C42—1221no—
290C42—1221no—
291C42—2441no—
292C42—2441no—
293C42C22×#692311yescharacteristic
294C23—611no—
295C23—1221no—
296Q8—21no—
297Q8—21no—
298D4—1221no—
299D4—1221no—
300D4—2441no—
301D4—2441no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—611no—
305C2×C4—611no—
306C2×C4—611no—
307C2×C4—611no—
308C2×C4—611no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—1221no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—4881no—
315C2×C4—4881no—
316C2×C4—4881no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4—2442no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4C2×C4⋊#692311yescharacteristic
324C2×C4D4×#692311yescharacteristic
325C2×C4D4×#692311yescharacteristic
326C8—41no—
327C8—41no—
328C8—1222no—
329C8—4881no—
330C8—2441no—
331C8—2441no—
332C8—2442no—
333C8—2441no—
334C22—611no—
335C22—611no—
336C22—1221no—
337C22—1221no—
338C22—2441no—
339C22—2441no—
340C22—2441no—
341C22#1684070911yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—611no—
346C4—611no—
347C4—1221no—
348C4—1221no—
349C4—4882no—
350C4—2441no—
351C4#1684077211yescharacteristic
352C4#1684077311yescharacteristic
353C4#1684080611yescharacteristic
354C4#1684080711yescharacteristic
355C2—1221no—
356C2—1221no—
357C2—4881no—
358C2#2146177011yescharacteristic
359C2#2146178111yescharacteristic
360C2#2146178711yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×Q8⋊C8C4
socleC22×C61#385
prime radicalC2×#11758226D4
cosocleC42×D61C23
hypercenterQ8⋊C8#6923
hypocenterC61#8905
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #21461389C2semidirect
4. #21461375C2semidirect
5. #21460716C2semidirect
6. #21460654C2semidirect
25. Q8⋊#92418C4semidirect
26. #16839491C4semidirect
27. #16839489C4semidirect
28. C61×Q8⋊C8C4semidirect
206. Q8⋊C8#6923semidirect
209. C61#8905semidirect

Maximal subgroups

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

subgroupindex
#214625192
#214616922
#214613892
#214613752
#214607162
#214606542
#214442002
#890561