Finite Group Databasebeta

← #30599734

Subgroups — #30599734

Order 15616.

454 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 92 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#21462560C211yesmaximal, characteristic
2#21461934C211yesmaximal, characteristic, complemented
3#21461377C211yesmaximal, characteristic, complemented
4#21461145C211yesmaximal, characteristic
5#21460711C211yesmaximal, characteristic, complemented
6#21460651C211yesmaximal, characteristic, complemented
7D4⋊C8×D61C211yesmaximal, characteristic
8C2×D61⋊M16—21no—
9#16840797—21no—
10#16840794—22no—
11#16840790—21no—
12#16840781C2211yescharacteristic
13C2×#14519847—21no—
14#16840766C2211yescharacteristic
15#16840766C2211yescharacteristic
16#16840733—21no—
17C4×#11758213C2211yescharacteristic
18#16840638—21no—
19#16840625—22no—
20#16840610—21no—
21D4⋊#92420—21no—
22C4×D4×D61C2211yescharacteristic
23C4⋊C8×D61C2211yescharacteristic
24C4×C8×D61C2211yescharacteristic
25#16839491C411yescharacteristic
26D4⋊#92418C411yescharacteristic
27#16839489C411yescharacteristic
28C61×D4⋊C8C411yescharacteristic
29C2×#11758213—21no—
30C2×#11758213—21no—
31C2×#11758213D411yescharacteristic
32C2×#11758213D411yescharacteristic
33D61⋊M16—42no—
34D61⋊M16—42no—
35C2×#11758217—22no—
36C2×D61⋊C8—22no—
37#14519847—42no—
38C4⋊#92420—21no—
39C4⋊#92420—21no—
40C4⋊#92420—21no—
41C4⋊#92420—21no—
42C4⋊#92420D412yes—
43#14519834—22no—
44#14519834—21no—
45C22⋊#92420—22no—
46#14519832—21no—
47C4×#92420—21no—
48C4×#92420—21no—
49C2×D4×D61Q811yescharacteristic
50C22×C4×D61—21no—
51C2×C8×D61—21no—
52C2×C8×D61—21no—
53C4×D244C2×C411yescharacteristic
54D4×Dic61D411yescharacteristic
55#14519739D411yescharacteristic
56C4×C22⋊D61—21no—
57#14519737—21no—
58C4⋊C4×D61Q811yescharacteristic
59C22⋊C4×D61—21no—
60C42×D61C2311yescharacteristic
61C4⋊#92418C2×C411yescharacteristic
62#14519706—21no—
63C4×#92418C2×C411yescharacteristic
64C8×Dic61—21no—
65C4×D4×C61C2×C411yescharacteristic
66C4⋊C8×C61C2×C411yescharacteristic
67C4×C8×C61C2×C411yescharacteristic
68C23×D61—21no—
69C2×D4×C61C4⋊C411yescharacteristic
70C22×C4×C61—21no—
71C2×C22⋊D61—21no—
72C22×Dic61—21no—
73D4×D61—22no—
74D4×D61—41no—
75C2×D244C4⋊C411yescharacteristic
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—41no—
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C2×C4×D61—41no—
83C2×C4×D61C2×D411yescharacteristic
84C2×C4×D61C2×Q811yescharacteristic
85C2×C4×D61#4111yescharacteristic
86#11758217—44no—
87C2×#92420—21no—
88C2×#92420—21no—
89C2×#92420—22no—
90C2×#92420—41no—
91C2×#92420—41no—
92#11758213—21no—
93#11758213—21no—
94#11758213—21no—
95#11758213—41no—
96#11758213—41no—
97#11758213—41no—
98#11758213—41no—
99#11758213D812yes—
100D61⋊C8—44no—
101C2×C8×C61—21no—
102C2×C8×C61—21no—
103C4⋊C4×C61C4⋊C411yescharacteristic
104C22⋊C4×C61—21no—
105C42×C61C22×C411yescharacteristic
106C22⋊Dic61—21no—
107#11758198—21no—
108#11758198—21no—
109C4⋊Dic61C4⋊C411yescharacteristic
110#11758196—21no—
111C4×Dic61C2×D411yescharacteristic
112C4×Dic61C2×D411yescharacteristic
113C4×Dic61#4111yescharacteristic
114C2×#92418—21no—
115C2×#92418—21no—
116C8×D61—42no—
117C8×D61—41no—
118C8×D61—41no—
119C23×C61—21no—
120C22×D61—21no—
121C22×D61—42no—
122C22×D61—41no—
123C22×D61—41no—
124C22×D61—81no—
125C22×D61C22⋊Q811yescharacteristic
126D4×C61—21no—
127D4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61C2×C4⋊C411yescharacteristic
132C2×C4×C61C4×D411yescharacteristic
133C2×C4×C61C4×D411yescharacteristic
134C22⋊D61—42no—
135C2×Dic61—21no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61#12011yescharacteristic
139C2×Dic61C22⋊Q811yescharacteristic
140C2×Dic61#12311yescharacteristic
141D244—21no—
142D244#10812yescomplemented
143C4×D61—21no—
144C4×D61—21no—
145C4×D61—41no—
146C4×D61—41no—
147C4×D61—41no—
148C4×D61—41no—
149C4×D61—41no—
150C4×D61—41no—
151C4×D61—41no—
152C4×D61C2×D811yescharacteristic
153C4×D61#13511yescharacteristic
154C4×D61AGL(1,ℤ/8)11yescharacteristic
155C4×D61#13711yescharacteristic
156#92420—22no—
157#92420—42no—
158#92420—41no—
159#92420—81no—
160#92420—81no—
161C8×C61—22no—
162C8×C61—41no—
163#92418—41no—
164#92418—41no—
165#8906—611nomaximal, Sylow
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61#38511yescharacteristic
170C2×D61—21no—
171C2×D61—21no—
172C2×D61—22no—
173C2×D61—41no—
174C2×D61—41no—
175C2×D61—82no—
176C2×D61—82no—
177C2×D61#47411yescharacteristic
178C2×D61#47811yescharacteristic
179C4×C61—21no—
180C4×C61—41no—
181C4×C61C2×#10811yescharacteristic
182C4×C61#43011yescharacteristic
183C4×C61C4×D811yescharacteristic
184C4×C61#44011yescharacteristic
185Dic61—21no—
186Dic61—21no—
187Dic61—41no—
188Dic61#44711yescharacteristic
189Dic61#45211yescharacteristic
190D4×M16—611no—
191#1950—611no—
192#1567—611no—
193#1486—611no—
194C2×D4⋊C8—611no—
195#1301—611no—
196#1278—611no—
197C2×C61—81no—
198C2×C61#185511yescharacteristic
199C2×C61#186111yescharacteristic
200C2×C61#186211yescharacteristic
201D61—21no—
202D61—21no—
203D61—42no—
204C22×M16—1221no—
205C2×C4×D4—611no—
206#436—1221no—
207#435—1222no—
208C8×D4—1221no—
209#431—611no—
210C2×#108—1221no—
211#423—611no—
212#423—611no—
213C2×C4⋊C8—611no—
214#407—1221no—
215C4×M16—611no—
216C2×C4×C8—611no—
217#341—1221no—
218#335—1222no—
219D4⋊C8—611no—
220D4⋊C8—611no—
221D4⋊C8—611no—
222D4⋊C8—1221no—
223D4⋊C8—1221no—
224D4⋊C8#692311yescharacteristic
225C61#890611yescharacteristic, complemented, Sylow
226C22×D4—611no—
227C23×C4—1221no—
228C2×M16—611no—
229C2×M16—611no—
230C2×M16—1221no—
231C2×M16—1222no—
232C2×M16—1221no—
233C2×M16—2442no—
234C2×M16—2442no—
235C22×C8—1221no—
236C22×C8—1221no—
237C22×C8—1222no—
238C4×D4—611no—
239C4×D4—611no—
240C4×D4—611no—
241C4×D4—1221no—
242C4×D4—1221no—
243C4×D4C2×#692311yescharacteristic
244C2×C4⋊C4—611no—
245C2×C22⋊C4—1221no—
246C2×C42—611no—
247#108—2442no—
248C4⋊C8—612no—
249C4⋊C8—611no—
250C4⋊C8—1221no—
251C4⋊C8—1221no—
252C4⋊C8—1221no—
253C4⋊C8—1221no—
254C4⋊C8—1221no—
255C4⋊C8C2×#692311yescharacteristic
256C22⋊C8—1222no—
257C22⋊C8—1222no—
258C22⋊C8—1221no—
259#97—1221no—
260C4×C8—611no—
261C4×C8—1221no—
262C4×C8—1221no—
263C4×C8—1221no—
264C4×C8C2×#692311yescharacteristic
265C24—1221no—
266C2×D4—611no—
267C2×D4—1221no—
268C2×D4—1222no—
269C2×D4—2441no—
270C2×D4C4⋊#692311yescharacteristic
271C22×C4—21no—
272C22×C4—611no—
273C22×C4—611no—
274C22×C4—611no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—2441no—
280C22×C4—2441no—
281C22×C4—2441no—
282C22×C4—2441no—
283M16—612no—
284M16—1221no—
285M16—1221no—
286M16—1221no—
287M16—2444no—
288M16—2441no—
289M16—2441no—
290M16—2441no—
291M16—2441no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—1221no—
295C2×C8—1221no—
296C2×C8—1222no—
297C2×C8—1221no—
298C2×C8—1221no—
299C2×C8—2442no—
300C2×C8—2441no—
301C2×C8—2441no—
302C2×C8—2444no—
303C2×C8—2441no—
304C2×C8—2441no—
305C4⋊C4—611no—
306C4⋊C4—1221no—
307C4⋊C4C4⋊#692311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—1221no—
310C22⋊C4—1221no—
311C22⋊C4—1221no—
312C42—611no—
313C42—611no—
314C42—611no—
315C42C22×#692311yescharacteristic
316C23—21no—
317C23—611no—
318C23—1221no—
319C23—4881no—
320C23—2441no—
321C23—2441no—
322C23—2442no—
323D4—21no—
324D4—21no—
325D4—612no—
326D4—1221no—
327D4—2442no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—611no—
332C2×C4—611no—
333C2×C4—611no—
334C2×C4—611no—
335C2×C4—611no—
336C2×C4—611no—
337C2×C4—611no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4C2×C4⋊#692311yescharacteristic
351C2×C4D4×#692311yescharacteristic
352C2×C4D4×#692311yescharacteristic
353C8—22no—
354C8—41no—
355C8—1222no—
356C8—4881no—
357C8—4881no—
358C8—2441no—
359C8—2442no—
360C8—2441no—
361C8—2441no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—611no—
366C22—611no—
367C22—1221no—
368C22—1221no—
369C22—1222no—
370C22—4882no—
371C22—4882no—
372C22—2441no—
373C22—2441no—
374C22#1684070911yescharacteristic
375C4—21no—
376C4—41no—
377C4—611no—
378C4—611no—
379C4—1221no—
380C4—1221no—
381C4—2441no—
382C4D61⋊#10811yescharacteristic
383C4#1684077711yescharacteristic
384C4D8×#692311yescharacteristic
385C4#1684080111yescharacteristic
386C2—81no—
387C2—1221no—
388C2—1221no—
389C2—2442no—
390C2#2146177311yescharacteristic
391C2#2146178011yescharacteristic
392C2#2146178211yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×D4⋊C8C4
socleC22×C61#385
prime radicalC2×D4×D61Q8
cosocleC42×D61C23
hypercenterD4⋊C8#6923
hypocenterC61#8906
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #21461934C2semidirect
3. #21461377C2semidirect
5. #21460711C2semidirect
6. #21460651C2semidirect
142. D244#108semidirect
225. C61#8906semidirect

Maximal subgroups

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

subgroupindex
#214625602
#214619342
#214613772
#214611452
#214607112
#214606512
D4⋊C8×D612
#890661