Finite Group Databasebeta

← #29654201

Subgroups — #29654201

Order 15616.

976 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 183 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#21456621C212yesmaximal
2#21451299C211yesmaximal, characteristic
3#21444861C214yesmaximal, complemented
4#21444759C214yesmaximal, complemented
5#21444339C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C2×C61×#544C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840095C2214yes—
10#16840075—22no—
11#16840010—24no—
12#16839912C2214yes—
13#16839901—22no—
14#16839726C2212yes—
15#16839701C2212yes—
16#16839679C2212yes—
17#16839652—22no—
18#16839641—21no—
19#16839641—21no—
20C4×C22⋊Dic61C2211yescharacteristic
21#16839499C2218yes—
22#16839488C2214yes—
23C61×#544—24no—
24C61×#544—22no—
25C61×#544C2214yes—
26C2×C4⋊Q8×C61—22no—
27C2×#125×C61—22no—
28C2×C22⋊Q8×C61—21no—
29C2×C22⋊Q8×C61C2212yes—
30C2×#117×C61—21no—
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic
33C22×C22⋊D61—21no—
34C2×#11758201—24no—
35C2×#11758194—22no—
36#14519751—22no—
37#14519751—22no—
38C2×C4⋊Dic61C2311yescharacteristic
39C2×C22⋊Dic61—21no—
40C2×#11758198—21no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×C4×Dic61—21no—
44#14519710—24no—
45#14519701—24no—
46#14519697—24no—
47C22⋊#92418—22no—
48C22⋊#92418C2314yes—
49#14519684—22no—
50#14519684—22no—
51#14519684C2312yes—
52C22×Q8×C61—21no—
53C22×Q8×C61D412yes—
54C23×C4×C61D411yescharacteristic
55C23×C4×C61D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C4⋊Q8×C61—28no—
58C4⋊Q8×C61—28no—
59#125×C61—28no—
60#125×C61—28no—
61C22⋊Q8×C61—24no—
62C22⋊Q8×C61—22no—
63C22⋊Q8×C61—22no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61D414yes—
66C22⋊Q8×C61C2314yes—
67#117×C61—24no—
68#117×C61—24no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61D412yes—
77C2×C4⋊C4×C61D412yes—
78C2×C4⋊C4×C61C2312yes—
79C2×C22⋊C4×C61—22no—
80C2×C42×C61—22no—
81C24×C61C2×D411yescharacteristic
82C23×D61—21no—
83C2×Q8×C61—24no—
84C2×Q8×C61—22no—
85C2×Q8×C61—82no—
86C2×Q8×C61—82no—
87C2×Q8×C61C2×D414yes—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—24no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—81no—
96C22×C4×C61—81no—
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D412yes—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2411yescharacteristic
103C2×C22⋊D61—44no—
104C2×C22⋊D61—41no—
105C2×C22⋊D61—41no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108C2×D244—22no—
109C2×D244—81no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61—28no—
114C4⋊C4×C61—24no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—24no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61C2×D414yes—
120C22⋊C4×C61—24no—
121C22⋊C4×C61—24no—
122C42×C61—28no—
123C22⋊Dic61—42no—
124#11758201—84no—
125#11758198—42no—
126#11758198—44no—
127#11758198—82no—
128C4⋊Dic61—22no—
129C4×Dic61—81no—
130#11758194—82no—
131C2×#92418—24no—
132C23×C61—22no—
133C23×C61—22no—
134C23×C61—41no—
135C23×C61#12011yescharacteristic
136C23×C61#12011yescharacteristic
137C23×C61C22×D411yescharacteristic
138C22×D61—42no—
139C22×D61—42no—
140C22×D61—41no—
141C22×D61—42no—
142Q8×C61—44no—
143Q8×C61—82no—
144Q8×C61—84no—
145C2×C4×C61—24no—
146C2×C4×C61—24no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—24no—
150C2×C4×C61—24no—
151C2×C4×C61—22no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—44no—
158C2×C4×C61—44no—
159C2×C4×C61—42no—
160C2×C4×C61—81no—
161C2×C4×C61#12012yes—
162C2×C4×C61#13714yes—
163C2×C4×C61C22×D412yes—
164C22⋊D61—82no—
165C22⋊D61—82no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171D244—44no—
172#92418—44no—
173#18326—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—44no—
182C22×C61#45412yes—
183C22×C61C2×#12011yescharacteristic
184C22×C61C2×#13712yes—
185C2×D61—84no—
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C4×C61—24no—
190C4×C61—22no—
191C4×C61—22no—
192C4×C61—44no—
193C4×C61—42no—
194C4×C61—44no—
195Dic61—82no—
196Dic61—81no—
197C2×#544D6111yescharacteristic, complemented
198#3193—612no—
199#2591—611no—
200#1617—614no—
201#1599—614no—
202#1508—612no—
203#1463—611no—
204C2×C61—22no—
205C2×C61—22no—
206C2×C61—41no—
207C2×C61#300411yescharacteristic
208C2×C61C2×#45411yescharacteristic
209C2×C61#302611yescharacteristic
210D61—161no—
211#544—24no—
212#544—22no—
213#544C2×D6114yes—
214C2×C4⋊Q8—22no—
215C2×#125—22no—
216C2×C22⋊Q8—21no—
217C2×C22⋊Q8C2×D6112yes—
218C2×#121—611no—
219C2×#117—21no—
220C22×C4⋊C4C2×D6111yescharacteristic
221#484—614no—
222#480—1222no—
223#468—1224no—
224#450—614no—
225#448—1222no—
226#407—612no—
227#400—612no—
228#394—612no—
229#388—1222no—
230#386—1221no—
231#386—1221no—
232C4×C22⋊C4—611no—
233#328—618no—
234#324—614no—
235C61#1832611yescharacteristic, complemented, Sylow
236C22×Q8—21no—
237C22×Q8C22⋊D6112yes—
238C22×D4—611no—
239C22×D4—1221no—
240C23×C4C22⋊D6111yescharacteristic
241C23×C4C22⋊D6111yescharacteristic
242C23×C4C22×D6111yescharacteristic
243C2×SD16—1224no—
244C2×M16—1222no—
245C4⋊Q8—28no—
246C4⋊Q8—28no—
247#125—28no—
248#125—28no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8C22⋊D6114yes—
254C22⋊Q8C22×D6114yes—
255#121—1222no—
256#121—1222no—
257#117—24no—
258#117—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—611no—
267C2×C4⋊C4C22⋊D6112yes—
268C2×C4⋊C4C22⋊D6112yes—
269C2×C4⋊C4C22×D6112yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—1222no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1222no—
275C2×C42—22no—
276C2×C42—1221no—
277#107—1224no—
278#103—1224no—
279#102—1224no—
280C22⋊C8—614no—
281C22⋊C8—1222no—
282#95—612no—
283#95—1222no—
284#95—1222no—
285C24—1221no—
286C24C2×C22⋊D6111yescharacteristic
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8C2×C22⋊D6114yes—
292C2×D4—1222no—
293C2×D4—4881no—
294C2×D4—2444no—
295C2×D4—2441no—
296C2×D4—2441no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—1222no—
307C22×C4—1221no—
308C22×C4C2×C22⋊D6112yes—
309C22×C4C2×C22⋊D6112yes—
310C22×C4C2×C22⋊D6112yes—
311C22×C4C2×C22⋊D6112yes—
312C22×C4C2×C22⋊D6111yescharacteristic
313C22×C4C23×D6111yescharacteristic
314SD16—4884no—
315M16—4882no—
316C2×C8—1224no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—1222no—
327C4⋊C4C2×C22⋊D6114yes—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—4882no—
331C22⋊C4—2442no—
332C22⋊C4—2444no—
333C22⋊C4—2442no—
334C42—28no—
335C42—4881no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—2442no—
340C23—2442no—
341C23—2441no—
342C23—2442no—
343C23#1451974611yescharacteristic
344C23#1451974611yescharacteristic
345C23C22×C22⋊D6111yescharacteristic
346Q8—44no—
347Q8—82no—
348Q8—84no—
349D4—4882no—
350D4—4882no—
351D4—2444no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4—2442no—
369C2×C4—2442no—
370C2×C4—2442no—
371C2×C4—2441no—
372C2×C4—2442no—
373C2×C4#1451974612yes—
374C2×C4#1451980814yes—
375C2×C4C22×C22⋊D6112yes—
376C8—2444no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—44no—
385C22—4884no—
386C22—4881no—
387C22—4881no—
388C22—4881no—
389C22#1683993512yes—
390C22C2×#1451974611yescharacteristic
391C22C2×#1451980812yes—
392C4—22no—
393C4—22no—
394C4—24no—
395C4—44no—
396C4—42no—
397C4—44no—
398C4—4882no—
399C4—4881no—
400C2—22no—
401C2—22no—
402C2—41no—
403C2—9761no—
404C2#2145505311yescharacteristic
405C2C2×#1683993511yescharacteristic
406C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#544C2
socleC22×C61C2×#120
prime radicalC2×#14519751C22
hypercenterC2×#544D61
hypocenterC61#18326
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #21444861C2semidirect
4. #21444759C2semidirect
5. #21444339C2semidirect
6. #21444149C2semidirect
7. C2×C61×#544C2semidirect
197. C2×#544D61semidirect
235. C61#18326semidirect

Maximal subgroups

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

subgroupindex
#214566212
#214512992
#214448612
#214447592
#214443392
#214441492
C2×C61×#5442
#1832661