Finite Group Databasebeta

← #29590784

Subgroups — #29590784

Order 15616.

434 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 91 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#21445807C211yesmaximal, characteristic
2#21444160C211yesmaximal, characteristic
3#21443843C212yesmaximal, complemented
4#21443819C212yesmaximal, complemented
5C61×#2395C211yesmaximal, characteristic
6#16839819—21no—
7C2×#14519692—21no—
8#16839622—21no—
9#16839529C2212yes—
10#16839486—21no—
11#16839486C2212yes—
12C2×C4×D4×C61C2211yescharacteristic
13C2×#117×C61C411yescharacteristic, complemented
14C61×#400C411yescharacteristic, complemented
15C61×#399C2211yescharacteristic
16C61×#398C411yescharacteristic, complemented
17C61×#397—21no—
18C61×#397C2211yescharacteristic
19C61×#394—21no—
20C4×C4⋊C4×C61—21no—
21C4×C22⋊C4×C61—21no—
22C4×C22⋊C4×C61C411yescharacteristic, complemented
23C2×C4⋊Dic61—41no—
24C2×C22⋊Dic61—22no—
25C4⋊#92418—41no—
26#14519692—42no—
27C22⋊#92418—22no—
28#14519688—41no—
29#14519684—41no—
30C22×D4×C61D411yescharacteristic
31C23×C4×C61C2×C411yescharacteristic
32C23×C4×C61D411yescharacteristic
33C4×D4×C61—41no—
34C4×D4×C61—41no—
35#117×C61—41no—
36#117×C61—41no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61C2×C411yescharacteristic
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61D411yescharacteristic
47C2×C22⋊C4×C61C2311yescharacteristic
48C2×C42×C61—21no—
49C2×C42×C61—21no—
50C2×C42×C61D411yescharacteristic
51#95×C61—21no—
52#95×C61—21no—
53#95×C61—21no—
54#95×C61—21no—
55#95×C61C2×C411yescharacteristic
56#95×C61C2×C411yescharacteristic
57C24×C61C22⋊C411yescharacteristic
58C24×C61C2×D411yescharacteristic
59C2×D4×C61—21no—
60C2×D4×C61—21no—
61C2×D4×C61—41no—
62C2×D4×C61—41no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×Dic61—42no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—42no—
81C4⋊C4×C61—42no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—42no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C42×C61—21no—
92C42×C61—21no—
93C42×C61—42no—
94C42×C61—41no—
95C42×C61—41no—
96C22⋊Dic61—42no—
97C22⋊Dic61—42no—
98C4⋊Dic61—82no—
99C2×#92418—42no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61C2×C22⋊C411yescharacteristic
110C23×C61AGL(1,ℤ/8)11yescharacteristic
111C23×C61#13711yescharacteristic
112D4×C61—42no—
113D4×C61—42no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—42no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—81no—
140C2×C4×C61#9911yescharacteristic
141C2×C4×C61#9911yescharacteristic
142C2×Dic61—82no—
143C2×Dic61—82no—
144C2×Dic61—82no—
145#92418—82no—
146#11822—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—41no—
154C22×C61—41no—
155C22×C61—41no—
156C22×C61—41no—
157C22×C61—41no—
158C22×C61—81no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61C2×#9911yescharacteristic
162C22×C61#41711yescharacteristic
163C22×C61#42111yescharacteristic
164C4×C61—21no—
165C4×C61—42no—
166C4×C61—41no—
167C4×C61—41no—
168C4×C61—41no—
169C4×C61—41no—
170C4×C61—81no—
171C4×C61—81no—
172C4×C61—81no—
173C4×C61#35212yes—
174Dic61—162no—
175#2395D6111yescharacteristic
176#1799—611no—
177#1466—611no—
178#1352—612no—
179#1344—612no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61—41no—
183C2×C61—41no—
184C2×C61—81no—
185C2×C61#151311yescharacteristic
186C2×C61C2×#35211yescharacteristic
187C2×C61#212611yescharacteristic
188C2×C4×D4C2×D6111yescharacteristic
189C2×#117Dic6111yescharacteristic, complemented
190#432—1221no—
191C2×#99—1221no—
192#400Dic6111yescharacteristic, complemented
193#399C2×D6111yescharacteristic
194#398Dic6111yescharacteristic, complemented
195#397—21no—
196#397C2×D6111yescharacteristic
197#394—21no—
198#380—1221no—
199C4×C4⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4Dic6111yescharacteristic, complemented
202#342—612no—
203#323—612no—
204#323—1221no—
205C61#1182211yescharacteristic, complemented, Sylow
206C22×D4C22⋊D6111yescharacteristic
207C23×C4C2×Dic6111yescharacteristic
208C23×C4C22⋊D6111yescharacteristic
209C4×D4—41no—
210C4×D4—41no—
211#117—41no—
212#117—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—2441no—
216C2×C4⋊C4C2×Dic6111yescharacteristic
217C2×C4⋊C4C2×Dic6111yescharacteristic
218C2×C4⋊C4C2×Dic6111yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—1222no—
224C2×C22⋊C4C22⋊D6111yescharacteristic
225C2×C22⋊C4C22×D6111yescharacteristic
226C2×C42—21no—
227C2×C42—21no—
228C2×C42C22⋊D6111yescharacteristic
229C4⋊C8—2441no—
230#99—2442no—
231C22⋊C8—1222no—
232#97—2441no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—2441no—
238#95C2×Dic6111yescharacteristic
239#95C2×Dic6111yescharacteristic
240C24C22⋊Dic6111yescharacteristic
241C24C2×C22⋊D6111yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—2442no—
257C22×C4C22⋊Dic6111yescharacteristic
258C22×C4C22⋊Dic6111yescharacteristic
259C22×C4C22⋊Dic6111yescharacteristic
260C22×C4C22×Dic6111yescharacteristic
261C22×C4C2×C22⋊D6111yescharacteristic
262C2×C8—2442no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—4882no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—2442no—
277C22⋊C4—2442no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—41no—
282C42—41no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊Dic6111yescharacteristic
293C23#1451980211yescharacteristic
294C23#1451980711yescharacteristic
295D4—42no—
296D4—42no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—4882no—
324C2×C4—4882no—
325C2×C4—4882no—
326C2×C4#1451969211yescharacteristic
327C2×C4#1451969211yescharacteristic
328C8—4882no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C2×#1451969211yescharacteristic
344C22#1683976311yescharacteristic
345C22#1683977911yescharacteristic
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—9762no—
356C4#1683955212yes—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—41no—
361C2—81no—
362C2#2144436211yescharacteristic
363C2C2×#1683955211yescharacteristic
364C2#2144748411yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2395C2
socleC22×C61C2×#99
prime radicalC22×D4×C61D4
cosocleC2×C22⋊C4×C61C23
hypercenter#2395D61
hypocenterC61#11822
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2395

Complements

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

NH ≅ G/Ntype
3. #21443843C2semidirect
4. #21443819C2semidirect
13. C2×#117×C61C4semidirect
14. C61×#400C4semidirect
16. C61×#398C4semidirect
22. C4×C22⋊C4×C61C4semidirect
189. C2×#117Dic61semidirect
192. #400Dic61semidirect
194. #398Dic61semidirect
201. C4×C22⋊C4Dic61semidirect
205. C61#11822semidirect

Maximal subgroups

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

subgroupindex
#214458072
#214441602
#214438432
#214438192
C61×#23952
#1182261