Finite Group Databasebeta

← #29654850

Subgroups — #29654850

Order 15616.

1262 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1262 conjugacy classes of subgroups — too many to draw. Only its 207 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#21456734C212yesmaximal, complemented
2#21450818C211yesmaximal, characteristic, complemented
3#21444754C218yesmaximal, complemented
4#21444314C212yesmaximal, complemented
5#21444143C211yesmaximal, characteristic, complemented
6C61×#3497C211yesmaximal, characteristic, complemented
7C2×#14519774C2211yescharacteristic
8C2×#14519766—21no—
9#16840138—22no—
10#16840110—22no—
11#16839957—24no—
12#16839895C2218yes—
13#16839815C2212yes—
14#16839679C2214yes—
15#16839664—22no—
16#16839663—21no—
17C4×C4⋊Dic61C2211yescharacteristic
18#16839496C2218yes—
19#16839488C2214yes—
20C2×2−1+4×C61—21no—
21C61×#550—22no—
22D4×Q8×C61—22no—
23C61×#548—24no—
24C61×#537—21no—
25C61×#537C2214yes—
26C61×#532—22no—
27C61×#532C2212yes—
28C2×C4×Q8×C61C2211yescharacteristic
29C22×D244—21no—
30C22×D244—21no—
31C22×D244C2312yes—
32C2×#11758199—28no—
33#14519774—22no—
34#14519774—22no—
35#14519766—44no—
36C2×C4⋊Dic61—21no—
37C2×#11758198—24no—
38C2×#11758198—24no—
39C2×C4×Dic61—21no—
40C4⋊#92418—22no—
41#14519697—28no—
42C22⋊#92418C2314yes—
43C4×#92418—22no—
44#14519684C2312yes—
452−1+4×C61—44no—
462−1+4×C61—44no—
47C2×#41×C61—24no—
48C2×#41×C61—24no—
49C2×#41×C61—22no—
50C22×Q8×C61—24no—
51C22×Q8×C61D411yescharacteristic
52C4⋊Q8×C61—22no—
53C4⋊Q8×C61D412yes—
54#127×C61—22no—
55#127×C61D412yes—
56#124×C61—24no—
57#124×C61—22no—
58C22⋊Q8×C61—24no—
59C22⋊Q8×C61—28no—
60#121×C61—24no—
61#121×C61—28no—
62#121×C61—22no—
63#121×C61—24no—
64#121×C61C2314yes—
65C4×Q8×C61—22no—
66C4×Q8×C61—22no—
67C4×D4×C61—24no—
68C4×D4×C61—24no—
69C4×D4×C61—22no—
70C4×D4×C61D414yes—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C4⋊C4×C61C2312yes—
73C2×C42×C61D412yes—
74C2×C42×C61C2311yescharacteristic
75C23×D61—22no—
76#41×C61—48no—
77#41×C61—416no—
78#41×C61—48no—
79#41×C61—44no—
80#41×C61—44no—
81C2×Q8×C61—24no—
82C2×Q8×C61—22no—
83C2×Q8×C61—22no—
84C2×Q8×C61—22no—
85C2×Q8×C61—416no—
86C2×Q8×C61D814yescomplemented
87C2×D4×C61—24no—
88C2×D4×C61—24no—
89C2×D4×C61—24no—
90C2×D4×C61—22no—
91C2×D4×C61—22no—
92C2×D4×C61C2×D414yes—
93C22×C4×C61—28no—
94C22×C4×C61—24no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—22no—
101C2×D244—24no—
102C2×D244—44no—
103C2×D244—41no—
104C2×D244—42no—
105C2×D244—41no—
106C2×D244—42no—
107C2×D244—42no—
108C2×D244—82no—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61C2×D414yes—
114C22⋊C4×C61—28no—
115C22⋊C4×C61—24no—
116C22⋊C4×C61—24no—
117C42×C61—22no—
118C42×C61—22no—
119C42×C61C2×D412yes—
120C42×C61C2×D412yes—
121#11758199—88no—
122#11758198—48no—
123#11758198—84no—
124C4⋊Dic61—42no—
125C4×Dic61—42no—
126C2×#92418—24no—
127C23×C61—24no—
128C23×C61C22×D411yescharacteristic
129C22×D61—44no—
130C22×D61—44no—
131C22×D61—42no—
132C22×D61—42no—
133C22×D61—42no—
134Q8×C61—28no—
135Q8×C61—48no—
136Q8×C61—44no—
137D4×C61—44no—
138D4×C61—48no—
139D4×C61—44no—
140D4×C61—44no—
141D4×C61—44no—
142D4×C61—44no—
143D4×C61—84no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—24no—
149C2×C4×C61—41no—
150C2×C4×C61—416no—
151C2×C4×C61—48no—
152C2×C4×C61—42no—
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61#12012yes—
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61C2×D814yes—
157C2×C4×C61C2×D812yes—
158C2×C4×C61AGL(1,ℤ/8)12yes—
159C2×C4×C61C22×D412yes—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163D244—48no—
164D244—82no—
165D244—84no—
166D244—82no—
167#92418—44no—
168#18401—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—44no—
172C22×C61—44no—
173C22×C61—44no—
174C22×C61C2×#12011yescharacteristic
175C22×C61C22×D811yescharacteristic
176C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
177C2×D61—84no—
178C2×D61—84no—
179C2×D61—82no—
180C2×D61—82no—
181C2×D61—82no—
182C4×C61—24no—
183C4×C61—21no—
184C4×C61—22no—
185C4×C61—44no—
186C4×C61—42no—
187C4×C61#44714yes—
188C4×C61#45412yes—
189Dic61—82no—
190#3497D6111yescharacteristic, complemented
191#3205—612no—
192#2548—611no—
193#1598—618no—
194#1502—612no—
195#1461—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61—84no—
199C2×C61C2×#44711yescharacteristic
200C2×C61C2×#45411yescharacteristic
201C2×C61#302711yescharacteristic
202D61—162no—
203C2×2−1+4—21no—
204#550—22no—
205D4×Q8—22no—
206#548—24no—
207#537—21no—
208#537C2×D6114yes—
209#532—22no—
210#532C2×D6112yes—
211C2×#127—611no—
212C2×#124—1221no—
213C2×C4×Q8C2×D6111yescharacteristic
214#493—1222no—
215#486—1222no—
216#459—1224no—
217#447—618no—
218#431—612no—
219#394—614no—
220#390—1221no—
221#390—1222no—
222C4×C4⋊C4—611no—
223#327—618no—
224#324—614no—
225C61#1840111yescharacteristic, complemented, Sylow
2262−1+4—44no—
2272−1+4—44no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—22no—
231C22×Q8—24no—
232C22×Q8C22⋊D6111yescharacteristic
233C22×D4—612no—
234C22×D4—1221no—
235C22×D4—1221no—
236C2×D8—1228no—
237C4⋊Q8—22no—
238C4⋊Q8C22⋊D6112yes—
239#127—22no—
240#127—1222no—
241#127—1222no—
242#127C22⋊D6112yes—
243#124—24no—
244#124—22no—
245#124—2444no—
246C22⋊Q8—24no—
247C22⋊Q8—28no—
248#121—24no—
249#121—28no—
250#121—22no—
251#121—24no—
252#121C22×D6114yes—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—22no—
258C4×D4C22⋊D6114yes—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4C22⋊D6111yescharacteristic
261C2×C4⋊C4C22×D6112yes—
262C2×C22⋊C4—1224no—
263C2×C22⋊C4—1224no—
264C2×C42—1221no—
265C2×C42C22⋊D6112yes—
266C2×C42C22×D6111yescharacteristic
267C4⋊C8—1222no—
268#102—1228no—
269C22⋊C8—614no—
270C4×C8—1222no—
271#95—612no—
272C24—1222no—
273#41—48no—
274#41—416no—
275#41—48no—
276#41—44no—
277#41—44no—
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—416no—
283C2×Q8#1175819914yescomplemented
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—1224no—
290C2×D4—4882no—
291C2×D4—2444no—
292C2×D4—2441no—
293C2×D4—2442no—
294C2×D4—2441no—
295C2×D4—2442no—
296C2×D4—2442no—
297C2×D4C2×C22⋊D6114yes—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—1222no—
301C22×C4C2×C22⋊D6111yescharacteristic
302C22×C4C2×C22⋊D6112yes—
303C22×C4C2×C22⋊D6112yes—
304C22×C4C2×C22⋊D6111yescharacteristic
305C22×C4C23×D6111yescharacteristic
306D8—4888no—
307C2×C8—1224no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—2442no—
313C4⋊C4C2×C22⋊D6114yes—
314C22⋊C4—28no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—4884no—
318C22⋊C4—2448no—
319C42—22no—
320C42—22no—
321C42—2442no—
322C42C2×C22⋊D6112yes—
323C42C2×C22⋊D6112yes—
324C23—24no—
325C23—2444no—
326C23—2444no—
327C23—2442no—
328C23—2442no—
329C23—2442no—
330C23C22×C22⋊D6111yescharacteristic
331Q8—28no—
332Q8—48no—
333Q8—44no—
334D4—44no—
335D4—48no—
336D4—44no—
337D4—44no—
338D4—44no—
339D4—44no—
340D4—84no—
341D4—4882no—
342D4—4884no—
343D4—4882no—
344D4—2448no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—41no—
351C2×C4—416no—
352C2×C4—48no—
353C2×C4—42no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4#1451974611yescharacteristic
358C2×C4#1451974612yes—
359C2×C4#1451974611yescharacteristic
360C2×C4C2×#1175819914yes—
361C2×C4C2×#1175819912yes—
362C2×C4#1451980312yes—
363C2×C4C22×C22⋊D6112yes—
364C8—2444no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—4884no—
371C22—4884no—
372C22—4882no—
373C22—4882no—
374C22—4882no—
375C22C2×#1451974611yescharacteristic
376C22C22×#1175819911yescharacteristic
377C22C2×#1451980311yescharacteristic
378C4—21no—
379C4—22no—
380C4—24no—
381C4—42no—
382C4—44no—
383C4—4882no—
384C4#1683989614yes—
385C4#1683993512yes—
386C2—21no—
387C2—21no—
388C2—84no—
389C2—9762no—
390C2C2×#1683989611yescharacteristic
391C2C2×#1683993511yescharacteristic
392C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3497C2
socleC22×C61C2×#120
hypercenter#3497D61
hypocenterC61#18401
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21456734C2semidirect
2. #21450818C2semidirect
3. #21444754C2semidirect
4. #21444314C2semidirect
5. #21444143C2semidirect
6. C61×#3497C2semidirect
86. C2×Q8×C61D8semidirect
190. #3497D61semidirect
225. C61#18401semidirect
283. C2×Q8#11758199semidirect

Maximal subgroups

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

subgroupindex
#214567342
#214508182
#214447542
#214443142
#214441432
C61×#34972
#1840161