Finite Group Databasebeta

← #30595807

Subgroups — #30595807

Order 15616.

556 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 90 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#21461709C211yesmaximal, characteristic, complemented
2#21461482C211yesmaximal, characteristic, complemented
3#21460637C212yesmaximal, complemented
4#21460636C212yesmaximal, complemented
5#21454151C211yesmaximal, characteristic
6C22×#11758217—21no—
7C2×C22⋊#92420—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519761—21no—
13C2×#14519751C411yescharacteristic
14C22×C4×Dic61C2211yescharacteristic
15#16839663C411yescharacteristic
16#16839663C2211yescharacteristic
17#16839661C411yescharacteristic
18#16839661C2211yescharacteristic
19#16839654—21no—
20#16839654—21no—
21#16839639—21no—
22C61×#380C411yescharacteristic
23C2×#11758217—24no—
24C2×#11758217—42no—
25C2×#11758217—42no—
26C22×#92420—22no—
27C22×#92420—22no—
28C22⋊#92420—42no—
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C23×Dic61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33#14519761—44no—
34#14519751—42no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×C4×Dic61—42no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C2×C4×Dic61C2311yescharacteristic
48#14519684—21no—
49#14519684—21no—
50C23×C4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61—21no—
54#95×C61—21no—
55C24×C61C22⋊C411yescharacteristic
56C23×D61—21no—
57C22×C4×C61—22no—
58C22×C4×C61—22no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—41no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C2×C22⋊D61—42no—
65C2×C22⋊D61—42no—
66C2×C22⋊D61—42no—
67C22×Dic61—22no—
68C22×Dic61—22no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—22no—
72C22×Dic61—22no—
73C22×Dic61—21no—
74C22×Dic61—41no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C2×D244—22no—
80C2×D244—41no—
81C2×D244—41no—
82#11758217—48no—
83C2×#92420—24no—
84C2×#92420—42no—
85C2×#92420—42no—
86C2×#92420—42no—
87C2×#92420—42no—
88C2×#92420—42no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—41no—
91C22⋊C4×C61—42no—
92#11758198—42no—
93#11758198—42no—
94#11758198—42no—
95#11758198—41no—
96#11758198—42no—
97#11758198—41no—
98C4⋊Dic61—42no—
99C4⋊Dic61—42no—
100C4⋊Dic61—41no—
101C4×Dic61—41no—
102C4×Dic61—41no—
103C4×Dic61—41no—
104C4×Dic61—41no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61#9912yes—
112C23×C61C2×C22⋊C411yescharacteristic
113C22×D61—41no—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61—41no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61—41no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—44no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61#13112yes—
133C22⋊D61—84no—
134C2×Dic61—24no—
135C2×Dic61—24no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—44no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61#10412yes—
150C2×Dic61AGL(1,ℤ/8)11yescharacteristic
151C2×Dic61AGL(1,ℤ/8)12yes—
152C2×Dic61#13711yescharacteristic
153D244—82no—
154#92420—44no—
155#92420—82no—
156#7694—611nomaximal, Sylow
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—22no—
160C22×C61—41no—
161C22×C61—41no—
162C22×C61—41no—
163C22×C61—41no—
164C22×C61#40811yescharacteristic
165C22×C61C2×#9911yescharacteristic
166C22×C61#41311yescharacteristic
167C22×C61#41711yescharacteristic
168C22×C61#41811yescharacteristic
169C22×C61C2×#10411yescharacteristic
170C22×C61#42111yescharacteristic
171C2×D61—81no—
172C2×D61—81no—
173C2×D61—81no—
174C2×D61—81no—
175C2×D61—81no—
176C2×D61—81no—
177C2×D61—81no—
178C4×C61—22no—
179C4×C61—41no—
180C4×C61—81no—
181C4×C61—81no—
182Dic61—21no—
183Dic61—22no—
184Dic61—21no—
185Dic61—22no—
186Dic61—41no—
187Dic61—81no—
188#2854—611no—
189#1817—611no—
190#1728—611no—
191#1271—612no—
192#1270—612no—
193C2×C61—42no—
194C2×C61#146811yescharacteristic
195C2×C61#148411yescharacteristic
196C2×C61#149511yescharacteristic
197C2×C61#150411yescharacteristic
198C2×C61#151011yescharacteristic
199C2×C61#152911yescharacteristic
200C2×C61#155711yescharacteristic
201D61—161no—
202C22×M16—1221no—
203C2×#123—1221no—
204C2×#121—611no—
205C22×C42—611no—
206C2×C22⋊C8—1221no—
207#390—611no—
208#390—611no—
209#389—611no—
210#389—611no—
211#388—1221no—
212#388—1221no—
213#386—1221no—
214#380#692311yescharacteristic
215#336—612no—
216#336—612no—
217#336—1221no—
218#336—1221no—
219C61#769411yescharacteristic, complemented, Sylow
220C22×D4—611no—
221C22×D4—1221no—
222C23×C4—611no—
223C23×C4—611no—
224C23×C4C2×#692311yescharacteristic
225C2×M16—1224no—
226C2×M16—2442no—
227C2×M16—2442no—
228C22×C8—1222no—
229C22×C8—1222no—
230#123—2444no—
231#121—2442no—
232C2×C4⋊C4—611no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4C2×#692311yescharacteristic
236C2×C4⋊C4C2×#692311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—1221no—
239C2×C22⋊C4—1221no—
240C2×C22⋊C4—1221no—
241C2×C22⋊C4—1221no—
242C2×C22⋊C4—1221no—
243C2×C42—611no—
244C2×C42—611no—
245C2×C42—611no—
246C2×C42—611no—
247C2×C42—2442no—
248C22⋊C8—2442no—
249#95—21no—
250#95—1221no—
251#95—1221no—
252C24—1221no—
253C24C22⋊#692311yescharacteristic
254C2×D4—1222no—
255C2×D4—2441no—
256C2×D4—2441no—
257C2×D4—2442no—
258C2×D4—2442no—
259C2×D4—2442no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4—611no—
268C22×C4—611no—
269C22×C4—1222no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4—1222no—
273C22×C4—1222no—
274C22×C4—1222no—
275C22×C4—1221no—
276C22×C4—2441no—
277C22×C4C22⋊#692311yescharacteristic
278C22×C4C22×#692311yescharacteristic
279M16—2448no—
280C2×C8—1224no—
281C2×C8—2442no—
282C2×C8—2442no—
283C2×C8—2442no—
284C2×C8—2442no—
285C2×C8—2442no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—2441no—
289C4⋊C4—2442no—
290C4⋊C4—2442no—
291C22⋊C4—42no—
292C22⋊C4—2442no—
293C22⋊C4—2441no—
294C22⋊C4—2441no—
295C22⋊C4—2442no—
296C22⋊C4—2442no—
297C22⋊C4—2442no—
298C42—2441no—
299C42—2441no—
300C42—2441no—
301C42—2441no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—2441no—
309C23—2441no—
310C23—2441no—
311C23—2441no—
312C23—2441no—
313C23—2441no—
314C23—2441no—
315C23#1451983612yes—
316C23C2×C22⋊#692311yescharacteristic
317D4—4882no—
318D4—4884no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—44no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—611no—
332C2×C4—611no—
333C2×C4—612no—
334C2×C4—612no—
335C2×C4—1224no—
336C2×C4—1224no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—2444no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4#1451986412yes—
351C8—4882no—
352C8—2444no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—4881no—
361C22—4881no—
362C22—4881no—
363C22—4881no—
364C22—4881no—
365C22—4881no—
366C22—4881no—
367C22#1684073511yescharacteristic
368C22C2×#1451983611yescharacteristic
369C22#1684074911yescharacteristic
370C22#1684075311yescharacteristic
371C22#1684075411yescharacteristic
372C22D61⋊#10411yescharacteristic
373C22#1684076111yescharacteristic
374C4—22no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4—1221no—
379C4—1222no—
380C4—1221no—
381C4—1222no—
382C4—4881no—
383C4—2441no—
384C2—42no—
385C2—9761no—
386C2#2146108611yescharacteristic
387C2#2146113811yescharacteristic
388C2#2146117611yescharacteristic
389C2#2146120711yescharacteristic
390C2#2146122611yescharacteristic
391C2#2146128711yescharacteristic
392C2#2146135111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C4
cosocleC2×C4×Dic61C23
hypercenter#380#6923
hypocenterC61#7694
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
1. #21461709C2semidirect
2. #21461482C2semidirect
3. #21460637C2semidirect
4. #21460636C2semidirect
219. C61#7694semidirect

Maximal subgroups

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

subgroupindex
#214617092
#214614822
#214606372
#214606362
#214541512
#769461