Finite Group Databasebeta

← #29712494

Subgroups — #29712494

Order 15616.

1466 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 191 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#21451535C212yesmaximal, complemented
2#21448399C211yesmaximal, characteristic, complemented
3#21446982C212yesmaximal, complemented
4#21446943C212yesmaximal, complemented
5#21446278C212yesmaximal, complemented
6#21446227C214yesmaximal, complemented
7#21445896C211yesmaximal, characteristic, complemented
8C61×#3491C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C2×#14519745C2214yes—
11C2×#14519736C2211yescharacteristic
12C2×#14519692C2214yes—
13#16839679—24no—
14#16839671—24no—
15#16839664—22no—
16#16839663—22no—
17#16839625C2212yes—
18#16839617C2214yes—
19#16839617C2212yes—
20#16839617C2212yes—
21#16839529C2212yes—
22#16839529C2212yes—
23#16839529C2214yes—
24C61×#552—24no—
25C61×#551—24no—
26C61×#543C2211yescharacteristic
27C61×#539—22no—
28C61×#534C2211yescharacteristic
29C61×#529—22no—
30C2×#123×C61C2212yes—
31C61×#518C2212yes—
32C2×#117×C61C2211yescharacteristic
33C24×D61C2312yes—
34C22×D244—24no—
35C22×D244—21no—
36C22×D244—21no—
37C22×D244—24no—
38C22×D244—42no—
39C22×C22⋊D61—24no—
40#14519745—28no—
41#14519745—44no—
42#14519736—42no—
43C2×C4⋊Dic61—21no—
44C2×C22⋊Dic61C2312yes—
45C2×C22⋊Dic61C2314yes—
46C2×#11758198—24no—
47C2×#11758198—24no—
48C2×#11758198—24no—
49C2×#11758198—28no—
50C2×C4×Dic61—21no—
51#14519692—44no—
52#14519684—24no—
53C22×D4×C61C2312yes—
54C23×C4×C61C2311yescharacteristic
55#126×C61—24no—
56#126×C61—24no—
57#125×C61—24no—
58#125×C61—22no—
59#124×C61—22no—
60#123×C61—24no—
61#123×C61—22no—
62#123×C61—22no—
63#123×C61—24no—
64#123×C61—22no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—24no—
68#121×C61—22no—
69#120×C61—22no—
70C4×Q8×C61—22no—
71C4×D4×C61—22no—
72C4×D4×C61—24no—
73C4×D4×C61—24no—
74#117×C61—24no—
75#117×C61—24no—
76#117×C61—22no—
77C2×C4⋊C4×C61D412yescomplemented
78C2×C4⋊C4×C61D414yescomplemented
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61D414yescomplemented
81C2×C22⋊C4×C61C2311yescharacteristic
82C2×C22⋊C4×C61C2312yes—
83C2×C22⋊C4×C61C2311yescharacteristic
84C2×C42×C61D412yescomplemented
85C24×C61C2×D412yes—
86C24×C61C2411yescharacteristic
87C23×D61—24no—
88C23×D61—22no—
89C23×D61—22no—
90C23×D61—24no—
91C23×D61—84no—
92C23×D61#4114yes—
93C2×Q8×C61—22no—
94C2×D4×C61—22no—
95C2×D4×C61—22no—
96C2×D4×C61—24no—
97C2×D4×C61—24no—
98C2×D4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—24no—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D414yes—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D414yes—
109C2×C22⋊D61—48no—
110C2×C22⋊D61—88no—
111C22×Dic61—24no—
112C22×Dic61—22no—
113C2×D244—24no—
114C2×D244—24no—
115C2×D244—416no—
116C2×D244—42no—
117C2×D244—42no—
118C2×D244—42no—
119C2×D244—42no—
120C4⋊C4×C61—24no—
121C4⋊C4×C61—24no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—24no—
127C22⋊C4×C61—24no—
128C22⋊C4×C61—24no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—22no—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C22⋊C4×C61—24no—
134C22⋊C4×C61—24no—
135C42×C61—24no—
136C42×C61—22no—
137C42×C61—22no—
138C22⋊Dic61—24no—
139C22⋊Dic61—24no—
140C22⋊Dic61—24no—
141#11758198—48no—
142#11758198—84no—
143#11758198—84no—
144#11758198—88no—
145C4⋊Dic61—42no—
146C4×Dic61—42no—
147C23×C61—22no—
148C23×C61—24no—
149C23×C61—42no—
150C23×C61—44no—
151C23×C61C22×D411yescharacteristic
152C23×C61C22×D412yes—
153C23×C61C2×#4112yes—
154C23×C61H(5,𝔽2)12yes—
155C22×D61—24no—
156C22×D61—28no—
157C22×D61—44no—
158C22×D61—44no—
159C22×D61—48no—
160C22×D61—48no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—44no—
164C22×D61—84no—
165C22×D61—88no—
166C22×D61—84no—
167Q8×C61—82no—
168D4×C61—42no—
169D4×C61—42no—
170D4×C61—44no—
171D4×C61—82no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—28no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—24no—
181C2×C4×C61—24no—
182C2×C4×C61—42no—
183C2×C4×C61—44no—
184C2×C4×C61—42no—
185C2×C4×C61—84no—
186C2×C4×C61#12012yes—
187C2×C4×C61#12012yes—
188C22⋊D61—816no—
189C2×Dic61—44no—
190C2×Dic61—48no—
191C2×Dic61—42no—
192C2×Dic61—44no—
193D244—48no—
194D244—48no—
195#23988—611nomaximal, Sylow
196C22×C61—22no—
197C22×C61—24no—
198C22×C61—42no—
199C22×C61—42no—
200C22×C61—42no—
201C22×C61—44no—
202C22×C61—84no—
203C22×C61C2×#12011yescharacteristic
204C22×C61#52511yescharacteristic
205C22×C61#53511yescharacteristic
206C22×C61#54614yes—
207C2×D61—48no—
208C2×D61—48no—
209C2×D61—44no—
210C2×D61—44no—
211C2×D61—44no—
212C2×D61—84no—
213C2×D61—88no—
214C2×D61—84no—
215C4×C61—22no—
216C4×C61—22no—
217C4×C61—42no—
218C4×C61—42no—
219C4×C61—44no—
220C4×C61—42no—
221Dic61—82no—
222Dic61—84no—
223#3491D6111yescharacteristic, complemented
224#2613—612no—
225#2309—611no—
226#2029—612no—
227#2021—612no—
228#1899—612no—
229#1889—614no—
230#1820—611no—
231C2×C61—21no—
232C2×C61—24no—
233C2×C61—21no—
234C2×C61—82no—
235C2×C61#240411yescharacteristic
236C2×C61#302712yes—
237D61—84no—
238C23×D4—611no—
239#552—24no—
240#551—24no—
241#543C2×D6111yescharacteristic
242#539—22no—
243#534C2×D6111yescharacteristic
244#529—22no—
245C2×#123C2×D6112yes—
246C2×#120—614no—
247#518C2×D6112yes—
248C2×#117—611no—
249C2×#117C2×D6111yescharacteristic
250C2×#99—614no—
251#394—1224no—
252#392—1224no—
253#390—1222no—
254#390—1222no—
255#381—612no—
256#379—612no—
257#379—612no—
258#379—614no—
259#342—612no—
260#342—612no—
261#342—614no—
262C61#2398811yescharacteristic, complemented, Sylow
263C25—612no—
264C22×D4—1224no—
265C22×D4—1224no—
266C22×D4—1221no—
267C22×D4—1221no—
268C22×D4—1224no—
269C22×D4—2442no—
270C22×D4C22×D6112yes—
271C23×C4C22×D6111yescharacteristic
272#126—24no—
273#126—24no—
274#125—22no—
275#125—24no—
276#124—22no—
277#123—24no—
278#123—22no—
279#123—22no—
280#123—24no—
281#123—22no—
282C22⋊Q8—24no—
283C22⋊Q8—22no—
284C22⋊Q8—24no—
285#121—22no—
286#120—22no—
287#120—1228no—
288#120—2444no—
289C4×Q8—22no—
290C4×D4—22no—
291C4×D4—24no—
292C4×D4—24no—
293#117—24no—
294#117—24no—
295#117—22no—
296#117—2442no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4C22⋊D6112yescomplemented
299C2×C4⋊C4C22⋊D6114yescomplemented
300C2×C22⋊C4—22no—
301C2×C22⋊C4—612no—
302C2×C22⋊C4—614no—
303C2×C22⋊C4—1224no—
304C2×C22⋊C4—1224no—
305C2×C22⋊C4—1228no—
306C2×C22⋊C4—1224no—
307C2×C22⋊C4C22⋊D6114yescomplemented
308C2×C22⋊C4C22×D6111yescharacteristic
309C2×C22⋊C4C22×D6111yescharacteristic
310C2×C22⋊C4C22×D6112yes—
311C2×C42—1221no—
312C2×C42C22⋊D6112yescomplemented
313#99—2444no—
314#95—1224no—
315C24—614no—
316C24—1222no—
317C24—1224no—
318C24—1222no—
319C24—1224no—
320C24—4884no—
321C24C2×C22⋊D6112yes—
322C24C23×D6111yescharacteristic
323C2×Q8—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—1224no—
330C2×D4—1224no—
331C2×D4—4888no—
332C2×D4—24416no—
333C2×D4—2442no—
334C2×D4—2442no—
335C2×D4—2448no—
336C2×D4—2442no—
337C2×D4—2442no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—1224no—
342C22×C4—1222no—
343C22×C4C2×C22⋊D6112yes—
344C22×C4C2×C22⋊D6114yes—
345C22×C4C2×C22⋊D6111yescharacteristic
346C22×C4C2×C22⋊D6111yescharacteristic
347C22×C4C2×C22⋊D6112yes—
348C22×C4C2×C22⋊D6112yes—
349C22×C4C2×C22⋊D6114yes—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—2442no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—1224no—
367C22⋊C4—1224no—
368C22⋊C4—1224no—
369C22⋊C4—4884no—
370C22⋊C4—4884no—
371C22⋊C4—4888no—
372C22⋊C4—2448no—
373C42—24no—
374C42—22no—
375C42—22no—
376C42—2442no—
377C23—22no—
378C23—24no—
379C23—42no—
380C23—44no—
381C23—1224no—
382C23—1228no—
383C23—4884no—
384C23—4888no—
385C23—4884no—
386C23—2444no—
387C23—2444no—
388C23—2448no—
389C23—2448no—
390C23—2442no—
391C23—2442no—
392C23—2444no—
393C23C22×C22⋊D6111yescharacteristic
394C23C22×C22⋊D6112yes—
395C23#41×D6112yes—
396C23#1451982312yes—
397Q8—82no—
398D4—42no—
399D4—42no—
400D4—44no—
401D4—82no—
402D4—48816no—
403D4—2448no—
404D4—2448no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—28no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4—2444no—
420C2×C4—2448no—
421C2×C4—2442no—
422C2×C4—2444no—
423C2×C4#1451974612yes—
424C2×C4#1451974612yes—
425C22—22no—
426C22—24no—
427C22—42no—
428C22—42no—
429C22—44no—
430C22—42no—
431C22—84no—
432C22—4884no—
433C22—4888no—
434C22—4884no—
435C22—2448no—
436C22—2448no—
437C22—2444no—
438C22—2444no—
439C22—2444no—
440C22C2×#1451974611yescharacteristic
441C22#1684026911yescharacteristic
442C22#1684032611yescharacteristic
443C22#1684040014yes—
444C4—22no—
445C4—22no—
446C4—42no—
447C4—44no—
448C4—42no—
449C4—42no—
450C4—4882no—
451C4—4884no—
452C2—21no—
453C2—24no—
454C2—21no—
455C2—82no—
456C2—4884no—
457C2#2144913411yescharacteristic
458C2#2145522112yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3491C2
socleC22×C61C2×#120
hypercenter#3491D61
hypocenterC61#23988
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3491

Complements

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

NH ≅ G/Ntype
1. #21451535C2semidirect
2. #21448399C2semidirect
3. #21446982C2semidirect
4. #21446943C2semidirect
5. #21446278C2semidirect
6. #21446227C2semidirect
7. #21445896C2semidirect
8. C61×#3491C2semidirect
77. C2×C4⋊C4×C61D4semidirect
78. C2×C4⋊C4×C61D4semidirect
80. C2×C22⋊C4×C61D4semidirect
84. C2×C42×C61D4semidirect
223. #3491D61semidirect
262. C61#23988semidirect
298. C2×C4⋊C4C22⋊D61semidirect
299. C2×C4⋊C4C22⋊D61semidirect
307. C2×C22⋊C4C22⋊D61semidirect
312. C2×C42C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214515352
#214483992
#214469822
#214469432
#214462782
#214462272
#214458962
C61×#34912
#2398861