Finite Group Databasebeta

← #29653139

Subgroups — #29653139

Order 15616.

958 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 203 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#21455823C212yesmaximal
2#21450724C211yesmaximal, characteristic
3#21444827C214yesmaximal, complemented
4#21444759C214yesmaximal
5#21444383C212yesmaximal, complemented
6#21444143C211yesmaximal, characteristic, complemented
7C61×#3502C211yesmaximal, characteristic, complemented
8C2×#14519753—21no—
9C2×C4×D244C2211yescharacteristic
10#16840090C2214yes—
11#16840049—22no—
12#16839964—22no—
13#16839912C2214yes—
14C4×#11758201—24no—
15#16839815C2212yes—
16#16839692C2212yes—
17#16839679C2212yes—
18#16839653—22no—
19#16839628C2211yescharacteristic
20#16839619—21no—
21#16839499C2218yes—
22#16839488C2214yes—
23Q82×C61—22no—
24C61×#557—22no—
25C61×#557—24no—
26C61×#544—21no—
27C61×#544C2214yes—
28C61×#533—22no—
29C61×#533C2212yes—
30C61×#519—21no—
31C2×C4×Q8×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic
33C22×C4×D61—21no—
34C2×#11758201—24no—
35#14519753—44no—
36C4×D244—22no—
37C4×D244—22no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2311yescharacteristic
40C2×#11758196—22no—
41C2×#11758198—21no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44#14519708—24no—
45C4⋊#92418—22no—
46#14519701—24no—
47#14519697—24no—
48C22⋊#92418C2314yes—
49C4×#92418—22no—
50#14519684—22no—
51#14519684—21no—
52#14519684C2312yes—
53C22×Q8×C61D411yescharacteristic
54C4⋊Q8×C61—22no—
55C4⋊Q8×C61—22no—
56C4⋊Q8×C61—24no—
57C4⋊Q8×C61—28no—
58C4⋊Q8×C61D412yes—
59C4⋊Q8×C61D412yes—
60#125×C61—24no—
61#125×C61—24no—
62#125×C61—28no—
63#125×C61—22no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61C2314yes—
67C4×Q8×C61—24no—
68C4×Q8×C61—24no—
69C4×Q8×C61—24no—
70C4×Q8×C61—24no—
71C4×Q8×C61—22no—
72C4×Q8×C61—22no—
73C4×Q8×C61—22no—
74C4×Q8×C61D414yes—
75#117×C61—24no—
76#117×C61—22no—
77C2×C4⋊C4×C61D411yescharacteristic
78C2×C4⋊C4×C61C2312yes—
79C2×C42×C61D412yes—
80C2×C42×C61C2311yescharacteristic
81C23×D61—21no—
82C2×Q8×C61—24no—
83C2×Q8×C61—22no—
84C2×Q8×C61—22no—
85C2×Q8×C61—22no—
86C2×Q8×C61SD1614yescomplemented
87C2×Q8×C61C2×D414yes—
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2411yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—22no—
95C2×D244—22no—
96C2×D244—42no—
97C2×C4×D61—42no—
98C2×C4×D61—42no—
99C2×C4×D61—42no—
100C4⋊C4×C61—24no—
101C4⋊C4×C61—24no—
102C4⋊C4×C61—24no—
103C4⋊C4×C61—28no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—24no—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61C2×D414yes—
112C22⋊C4×C61—24no—
113C42×C61—28no—
114C42×C61—22no—
115C42×C61—22no—
116C42×C61—24no—
117C42×C61C2×D412yes—
118C42×C61C2×D412yes—
119#11758201—48no—
120#11758198—44no—
121#11758198—41no—
122#11758198—41no—
123#11758198—44no—
124C4⋊Dic61—22no—
125C4⋊Dic61—42no—
126#11758196—44no—
127C2×#92418—24no—
128C23×C61C22×D411yescharacteristic
129C22×D61—22no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136Q8×C61—28no—
137Q8×C61—44no—
138Q8×C61—44no—
139Q8×C61—84no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—24no—
143C2×C4×C61—24no—
144C2×C4×C61—22no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61#12011yescharacteristic
149C2×C4×C61#12011yescharacteristic
150C2×C4×C61#12012yes—
151C2×C4×C61C2×SD1614yes—
152C2×C4×C61C2×SD1612yes—
153C2×C4×C61#13712yes—
154C2×C4×C61C22×D412yes—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—42no—
160C2×Dic61—41no—
161D244—24no—
162D244—42no—
163C4×D61—84no—
164#92418—44no—
165#18220—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61C2×#12011yescharacteristic
169C22×C61C22×SD1611yescharacteristic
170C22×C61C2×#13711yescharacteristic
171C2×D61—42no—
172C2×D61—42no—
173C2×D61—42no—
174C2×D61—81no—
175C2×D61—81no—
176C2×D61—81no—
177C2×D61—81no—
178C4×C61—24no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—22no—
182C4×C61—44no—
183C4×C61—42no—
184C4×C61—44no—
185C4×C61#44814yes—
186Dic61—82no—
187Dic61—81no—
188#3502D6111yescharacteristic, complemented
189#3101—612no—
190#2536—611no—
191#1611—614no—
192#1599—614no—
193#1518—612no—
194#1461—611no—
195C2×C61—21no—
196C2×C61—21no—
197C2×C61C2×#44811yescharacteristic
198C2×C61#302011yescharacteristic
199C2×C61#302511yescharacteristic
200D61—82no—
201Q82—22no—
202#557—22no—
203#557—24no—
204#544—21no—
205#544C2×D6114yes—
206#533—22no—
207#533C2×D6112yes—
208C2×C22⋊Q8—1221no—
209#519—21no—
210C2×C4×Q8C2×D6111yescharacteristic
211C2×C4×D4—611no—
212#483—614no—
213#476—1222no—
214#461—1222no—
215#450—614no—
216C4×SD16—1224no—
217#431—612no—
218#398—612no—
219#394—612no—
220#388—1222no—
221#382—611no—
222#380—1221no—
223#328—618no—
224#324—614no—
225C61#1822011yescharacteristic, complemented, Sylow
226C22×Q8C22⋊D6111yescharacteristic
227C22×D4—611no—
228C23×C4—1221no—
229C2×SD16—1224no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—24no—
233C4⋊Q8—28no—
234C4⋊Q8C22⋊D6112yes—
235C4⋊Q8C22⋊D6112yes—
236#125—24no—
237#125—24no—
238#125—28no—
239#125—22no—
240C22⋊Q8—22no—
241C22⋊Q8—24no—
242C22⋊Q8—2444no—
243C22⋊Q8C22×D6114yes—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8C22⋊D6114yes—
252C4×D4—1222no—
253C4×D4—1222no—
254#117—24no—
255#117—22no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1222no—
259C2×C4⋊C4C22⋊D6111yescharacteristic
260C2×C4⋊C4C22×D6112yes—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1222no—
263C2×C22⋊C4—1222no—
264C2×C42C22⋊D6112yes—
265C2×C42C22×D6111yescharacteristic
266#106—1224no—
267C4⋊C8—1222no—
268#103—1224no—
269#102—1224no—
270C22⋊C8—614no—
271C4×C8—1222no—
272#95—612no—
273#95—1222no—
274#95—1221no—
275C24—1221no—
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8#1175820114yescomplemented
281C2×Q8C2×C22⋊D6114yes—
282C2×D4—1222no—
283C2×D4—2442no—
284C22×C4—1222no—
285C22×C4—1221no—
286C22×C4—2442no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4C2×C22⋊D6112yes—
290C22×C4C2×C22⋊D6112yes—
291C22×C4C2×C22⋊D6111yescharacteristic
292C22×C4C2×C22⋊D6111yescharacteristic
293C22×C4C23×D6111yescharacteristic
294SD16—2448no—
295C2×C8—1224no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—1222no—
308C4⋊C4—2444no—
309C4⋊C4—2442no—
310C4⋊C4C2×C22⋊D6114yes—
311C22⋊C4—24no—
312C22⋊C4—2444no—
313C22⋊C4—2441no—
314C22⋊C4—2441no—
315C22⋊C4—2444no—
316C42—28no—
317C42—22no—
318C42—22no—
319C42—24no—
320C42C2×C22⋊D6112yes—
321C42C2×C22⋊D6112yes—
322C23—1222no—
323C23—2441no—
324C23—2441no—
325C23—2441no—
326C23—2441no—
327C23—2441no—
328C23—2441no—
329C23C22×C22⋊D6111yescharacteristic
330Q8—28no—
331Q8—44no—
332Q8—44no—
333Q8—84no—
334D4—1224no—
335D4—2442no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—4884no—
345C2×C4—2442no—
346C2×C4—2442no—
347C2×C4—2441no—
348C2×C4—2442no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4#1451974611yescharacteristic
352C2×C4#1451974611yescharacteristic
353C2×C4#1451974612yes—
354C2×C4C2×#1175820114yes—
355C2×C4C2×#1175820112yes—
356C2×C4#1451980812yes—
357C2×C4C22×C22⋊D6112yes—
358C8—2444no—
359C22—21no—
360C22—21no—
361C22—4881no—
362C22—4881no—
363C22—4881no—
364C22—4881no—
365C22—2442no—
366C22—2442no—
367C22—2442no—
368C22C2×#1451974611yescharacteristic
369C22C22×#1175820111yescharacteristic
370C22C2×#1451980811yescharacteristic
371C4—21no—
372C4—21no—
373C4—22no—
374C4—24no—
375C4—44no—
376C4—42no—
377C4—44no—
378C4—4882no—
379C4—4881no—
380C4#1683990114yes—
381C2—21no—
382C2—21no—
383C2—4882no—
384C2C2×#1683990111yescharacteristic
385C2#2145518311yescharacteristic
386C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3502C2
socleC22×C61C2×#120
prime radicalC22×D244C23
hypercenter#3502D61
hypocenterC61#18220
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
3. #21444827C2semidirect
5. #21444383C2semidirect
6. #21444143C2semidirect
7. C61×#3502C2semidirect
86. C2×Q8×C61SD16semidirect
188. #3502D61semidirect
225. C61#18220semidirect
280. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214558232
#214507242
#214448272
#214447592
#214443832
#214441432
C61×#35022
#1822061