Finite Group Databasebeta

← #29577704

Subgroups — #29577704

Order 15616.

448 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 95 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#21454903C211yesmaximal, characteristic
2#21451028C211yesmaximal, characteristic
3#21444536C211yesmaximal, characteristic
4#21444162C211yesmaximal, characteristic
5#21443676C211yesmaximal, characteristic
6#21443674C211yesmaximal, characteristic
7C61×#1556C211yesmaximal, characteristic
8C2×C4×Dic122C2211yescharacteristic
9#16839891—22no—
10#16839890—22no—
11#16839819C2211yescharacteristic
12#16839819C2211yescharacteristic
13#16839816—21no—
14#16839812—21no—
15#16839681—21no—
16#16839681—21no—
17#16839667C412yes—
18#16839658—21no—
19#16839658—21no—
20#16839631C412yes—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839508—22no—
23#16839487—21no—
24#16839487—21no—
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#432C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C61×#332—22no—
29C22×Dic122C2×C411yescharacteristic
30C4×Dic122—21no—
31C4×Dic122—21no—
32C4×Dic122—21no—
33C4×Dic122—21no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×#11758196—22no—
37C2×#11758196—22no—
38C2×#11758196D411yescharacteristic
39C2×#11758196D411yescharacteristic
40C2×C4×Dic61C2×C412yes—
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43C4⋊#92418—21no—
44C4⋊#92418D412yes—
45#14519706—22no—
46#14519706—22no—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49#14519688—21no—
50#14519688—21no—
51#14519687—22no—
52C8×Dic61—22no—
53#14519684—21no—
54#14519684—22no—
55#14519684—22no—
56C22×Q8×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61C2×C412yes—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61—21no—
65C4⋊C8×C61D412yes—
66C22⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68#97×C61—21no—
69#97×C61—21no—
70C2×Q8×C61—41no—
71C2×Q8×C61—41no—
72C2×Q8×C61—82no—
73C22×C4×C61—22no—
74C22×C4×C61C22×C411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×Dic61—22no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C2×Dic122—21no—
83C2×Dic122—21no—
84C2×Dic122—22no—
85C2×Dic122—41no—
86C2×C8×C61—22no—
87C2×C8×C61—41no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C42×C61C2×D411yescharacteristic
93C42×C61C2×D411yescharacteristic
94C42×C61#4111yescharacteristic
95C42×C61#4111yescharacteristic
96C4⋊Dic61—24no—
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99#11758196—21no—
100#11758196—21no—
101#11758196—21no—
102#11758196—21no—
103#11758196—82no—
104#11758196—82no—
105C4×Dic61—21no—
106C4×Dic61—21no—
107C4×Dic61—21no—
108C4×Dic61—21no—
109C4×Dic61—42no—
110C2×#92418—22no—
111C2×#92418—41no—
112C23×C61C2×C22⋊C411yescharacteristic
113Q8×C61—82no—
114Q8×C61—82no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—22no—
118C2×C4×C61—41no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61C4×D411yescharacteristic
123C2×C4×C61C4×D411yescharacteristic
124C2×C4×C61#12011yescharacteristic
125C2×C4×C61#12111yescharacteristic
126C2×C4×C61#12111yescharacteristic
127C2×C4×C61#12311yescharacteristic
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—24no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61#10112yes—
141Dic122—42no—
142Dic122—42no—
143C8×C61—42no—
144C8×C61—81no—
145#92418—42no—
146#92418—81no—
147#9454—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61#38611yescharacteristic
151C22×C61C2×#10111yescharacteristic
152C22×C61#41311yescharacteristic
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—82no—
159C4×C61#45612yes—
160Dic61—22no—
161Dic61—44no—
162Dic61—41no—
163Dic61—41no—
164Dic61—41no—
165#2983—611no—
166#2567—611no—
167#1556—611no—
168#1556D6111yescharacteristic
169#1467—611no—
170#1285—611no—
171#1285—611no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61#188611yescharacteristic
175C2×C61#190111yescharacteristic
176C2×C61#191011yescharacteristic
177C2×C4⋊Q8C2×D6111yescharacteristic
178C2×C4×Q8—611no—
179#446—1222no—
180#446—1222no—
181#432—611no—
182#432—611no—
183#432—1221no—
184#432C2×D6111yescharacteristic
185#432C2×D6111yescharacteristic
186#431—1221no—
187#395—1221no—
188#395—1221no—
189#391—612no—
190#389—1221no—
191#389—1221no—
192#384—612no—
193C4×C4⋊C4—611no—
194#332—22no—
195#332—1222no—
196#324—1221no—
197#324—1221no—
198C61#945411yescharacteristic, complemented, Sylow
199C22×Q8—21no—
200C22×Q8—611no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4×Q8—1221no—
206C4×Q8—1221no—
207C4×Q8—1221no—
208C4×Q8—1221no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—611no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—1222no—
215C2×C4⋊C4—1222no—
216C2×C4⋊C4C4×D6112yes—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—612no—
220C2×C42C22×D6111yescharacteristic
221C4⋊C8—21no—
222C4⋊C8—612no—
223C4⋊C8—1222no—
224C4⋊C8—1222no—
225C4⋊C8—1221no—
226C4⋊C8C22⋊D6112yes—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—1221no—
230C22⋊C8—1221no—
231#97—21no—
232#97—21no—
233#97—1222no—
234#97—1221no—
235#97—1221no—
236C4×C8—1222no—
237#95—1222no—
238#95—1222no—
239#95—1221no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—82no—
243C2×Q8—1222no—
244C2×Q8—1221no—
245C2×Q8—1221no—
246C2×Q8—2441no—
247C22×C4—22no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—1222no—
253C22×C4C2×C4×D6111yescharacteristic
254C22×C4D4×D6111yescharacteristic
255C22×C4D4×D6111yescharacteristic
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—1222no—
259C2×C8—2441no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—1224no—
265C4⋊C4—1221no—
266C4⋊C4—1221no—
267C4⋊C4—1221no—
268C4⋊C4—1221no—
269C4⋊C4—1221no—
270C4⋊C4—1221no—
271C4⋊C4—4882no—
272C4⋊C4—4882no—
273C42—1221no—
274C42—1221no—
275C42—1221no—
276C42—1221no—
277C42—2442no—
278C42#1175822211yescharacteristic
279C42D4×D6111yescharacteristic
280C42#1175822411yescharacteristic
281C42C2×C22⋊D6111yescharacteristic
282C23C22⋊C4×D6111yescharacteristic
283Q8—82no—
284Q8—82no—
285Q8—2442no—
286Q8—2442no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—612no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1224no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4#1451973711yescharacteristic
308C2×C4C4×C22⋊D6111yescharacteristic
309C2×C4#1451974511yescharacteristic
310C2×C4#1451974811yescharacteristic
311C2×C4#1451975011yescharacteristic
312C2×C4#1451975911yescharacteristic
313C8—42no—
314C8—81no—
315C8—4881no—
316C8—2442no—
317C22—21no—
318C22—21no—
319C22#1683963811yescharacteristic
320C22#101×D6111yescharacteristic
321C22#1683974511yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—82no—
328C4—1222no—
329C4—2444no—
330C4—2441no—
331C4—2441no—
332C4—2441no—
333C4#1683994312yes—
334C2—21no—
335C2—21no—
336C2#2144620911yescharacteristic
337C2#2144628411yescharacteristic
338C2#2144634311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1556C2
socleC22×C61#386
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1556D61
hypocenterC61#9454
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
198. C61#9454semidirect

Maximal subgroups

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

subgroupindex
#214549032
#214510282
#214445362
#214441622
#214436762
#214436742
C61×#15562
#945461