Finite Group Databasebeta

← #29654187

Subgroups — #29654187

Order 15616.

1256 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1256 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#21456616C212yesmaximal, complemented
2#21451299C211yesmaximal, characteristic, complemented
3#21444856C214yesmaximal, complemented
4#21444754C214yesmaximal, complemented
5#21444335C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C2×C61×#537C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840080—22no—
10#16840078C2214yescomplemented
11#16840013—24no—
12#16839896—22no—
13#16839895C2214yescomplemented
14#16839726C2212yes—
15#16839701C2212yescomplemented
16#16839679C2212yescomplemented
17#16839652—22no—
18#16839641—21no—
19#16839641—21no—
20C4×C22⋊Dic61C2211yescharacteristic
21#16839496C2218yescomplemented
22#16839488C2214yescomplemented
23C22×#41×C61—21no—
24C61×#537—24no—
25C61×#537—22no—
26C61×#537C2214yes—
27C2×C22⋊Q8×C61—22no—
28C2×#121×C61—22no—
29C2×#121×C61—21no—
30C2×#121×C61C2212yes—
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic
33C22×C22⋊D61—21no—
34C2×#11758199—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#14519708—24no—
45#14519698—24no—
46#14519697—24no—
47C22⋊#92418—22no—
48C22⋊#92418C2314yes—
49#14519684—22no—
50#14519684—22no—
51#14519684C2312yes—
52C2×#41×C61—24no—
53C2×#41×C61—24no—
54C2×#41×C61—48no—
55C22×Q8×C61—21no—
56C22×D4×C61—21no—
57C22×D4×C61—22no—
58C22×D4×C61—21no—
59C22×D4×C61D412yes—
60C23×C4×C61—22no—
61C23×C4×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64C22⋊Q8×C61—28no—
65C22⋊Q8×C61—28no—
66#121×C61—24no—
67#121×C61—28no—
68#121×C61—28no—
69#121×C61—22no—
70#121×C61—22no—
71#121×C61—24no—
72#121×C61D414yes—
73#121×C61C2314yes—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61D412yes—
78C2×C4⋊C4×C61D412yes—
79C2×C4⋊C4×C61C2312yes—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61—22no—
82C24×C61—22no—
83C24×C61C2×D411yescharacteristic
84C23×D61—21no—
85#41×C61—416no—
86#41×C61—48no—
87#41×C61—48no—
88C2×Q8×C61—24no—
89C2×Q8×C61—44no—
90C2×D4×C61—24no—
91C2×D4×C61—24no—
92C2×D4×C61—24no—
93C2×D4×C61—24no—
94C2×D4×C61—22no—
95C2×D4×C61—44no—
96C2×D4×C61—48no—
97C2×D4×C61—82no—
98C2×D4×C61—82no—
99C2×D4×C61C2×D414yes—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—24no—
103C22×C4×C61—28no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—48no—
108C22×C4×C61—81no—
109C22×C4×C61—81no—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—44no—
117C2×C22⋊D61—41no—
118C2×C22⋊D61—41no—
119C22×Dic61—21no—
120C22×Dic61—22no—
121C2×D244—22no—
122C2×D244—81no—
123C4⋊C4×C61—24no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61C2×D414yes—
128C22⋊C4×C61—28no—
129C22⋊C4×C61—24no—
130C22⋊C4×C61—24no—
131C22⋊Dic61—42no—
132#11758199—84no—
133#11758198—42no—
134#11758198—44no—
135#11758198—82no—
136C4⋊Dic61—22no—
137C4×Dic61—81no—
138#11758194—82no—
139C2×#92418—24no—
140C23×C61—22no—
141C23×C61—22no—
142C23×C61—24no—
143C23×C61—41no—
144C23×C61—44no—
145C23×C61—44no—
146C23×C61—44no—
147C23×C61#12011yescharacteristic
148C23×C61#12011yescharacteristic
149C23×C61C22×D411yescharacteristic
150C22×D61—42no—
151C22×D61—42no—
152C22×D61—41no—
153C22×D61—42no—
154Q8×C61—48no—
155D4×C61—44no—
156D4×C61—48no—
157D4×C61—48no—
158D4×C61—48no—
159D4×C61—82no—
160D4×C61—84no—
161C2×C4×C61—24no—
162C2×C4×C61—24no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—24no—
167C2×C4×C61—42no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—416no—
172C2×C4×C61—44no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—81no—
176C2×C4×C61#12012yes—
177C2×C4×C61AGL(1,ℤ/8)14yes—
178C2×C4×C61C22×D412yes—
179C22⋊D61—82no—
180C22⋊D61—82no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—41no—
185C2×Dic61—42no—
186D244—44no—
187#92418—44no—
188#18324—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—44no—
197C22×C61—44no—
198C22×C61—44no—
199C22×C61—44no—
200C22×C61—88no—
201C22×C61#45412yes—
202C22×C61C2×#12011yescharacteristic
203C22×C61C2×AGL(1,ℤ/8)12yes—
204C2×D61—84no—
205C2×D61—81no—
206C2×D61—81no—
207C2×D61—81no—
208C4×C61—24no—
209C4×C61—22no—
210C4×C61—22no—
211C4×C61—44no—
212C4×C61—42no—
213Dic61—82no—
214Dic61—81no—
215C2×#537D6111yescharacteristic, complemented
216#3192—612no—
217#2591—611no—
218#1616—614no—
219#1598—614no—
220#1507—612no—
221#1463—611no—
222C2×C61—22no—
223C2×C61—22no—
224C2×C61—41no—
225C2×C61—84no—
226C2×C61#300311yescharacteristic
227C2×C61C2×#45411yescharacteristic
228C2×C61#302611yescharacteristic
229D61—161no—
230C22×#41—21no—
231#537—24no—
232#537—22no—
233#537C2×D6114yes—
234C2×C22⋊Q8—22no—
235C2×#121—22no—
236C2×#121—21no—
237C2×#121—611no—
238C2×#121C2×D6112yes—
239C22×C4⋊C4C2×D6111yescharacteristic
240#481—614no—
241#481—1222no—
242#469—1224no—
243#447—614no—
244#447—1222no—
245#407—612no—
246#400—612no—
247#394—612no—
248#388—1222no—
249#386—1221no—
250#386—1221no—
251C4×C22⋊C4—611no—
252#327—618no—
253#324—614no—
254C61#1832411yescharacteristic, complemented, Sylow
255C2×#41—24no—
256C2×#41—24no—
257C2×#41—48no—
258C22×Q8—21no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4—611no—
263C22×D4—1221no—
264C22×D4C22⋊D6112yes—
265C23×C4—22no—
266C23×C4C22⋊D6111yescharacteristic
267C23×C4C22⋊D6111yescharacteristic
268C23×C4C22×D6111yescharacteristic
269C2×D8—1224no—
270C2×M16—1222no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273#121—24no—
274#121—28no—
275#121—28no—
276#121—24no—
277#121—22no—
278#121—22no—
279#121—1222no—
280#121—1222no—
281#121C22⋊D6114yes—
282#121C22×D6114yes—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—611no—
287C2×C4⋊C4C22⋊D6112yes—
288C2×C4⋊C4C22⋊D6112yes—
289C2×C4⋊C4C22×D6112yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—1222no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4—1222no—
296C2×C42—1221no—
297#106—1224no—
298#102—1224no—
299#102—1224no—
300C22⋊C8—614no—
301C22⋊C8—1222no—
302#95—612no—
303#95—1222no—
304#95—1222no—
305C24—22no—
306C24—1221no—
307C24C2×C22⋊D6111yescharacteristic
308#41—416no—
309#41—48no—
310#41—48no—
311C2×Q8—24no—
312C2×Q8—44no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—1222no—
323C2×D4—4881no—
324C2×D4—2444no—
325C2×D4—2441no—
326C2×D4—2441no—
327C2×D4C2×C22⋊D6114yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—48no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—1222no—
339C22×C4—1221no—
340C22×C4C2×C22⋊D6112yes—
341C22×C4C2×C22⋊D6112yes—
342C22×C4C2×C22⋊D6112yes—
343C22×C4C2×C22⋊D6112yes—
344C22×C4C2×C22⋊D6111yescharacteristic
345C22×C4C23×D6111yescharacteristic
346D8—4884no—
347M16—4882no—
348C2×C8—1224no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—1222no—
354C4⋊C4C2×C22⋊D6114yes—
355C22⋊C4—28no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—4882no—
359C22⋊C4—2442no—
360C22⋊C4—2442no—
361C22⋊C4—2444no—
362C42—4881no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—41no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—2442no—
371C23—2442no—
372C23—2441no—
373C23—2442no—
374C23#1451974611yescharacteristic
375C23#1451974611yescharacteristic
376C23C22×C22⋊D6111yescharacteristic
377Q8—48no—
378D4—44no—
379D4—48no—
380D4—48no—
381D4—48no—
382D4—82no—
383D4—84no—
384D4—4882no—
385D4—4882no—
386D4—2444no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—416no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—81no—
402C2×C4—2442no—
403C2×C4—2442no—
404C2×C4—2442no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4#1451974612yes—
408C2×C4#1451980314yes—
409C2×C4C22×C22⋊D6112yes—
410C8—2444no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—88no—
423C22—4884no—
424C22—4881no—
425C22—4881no—
426C22—4881no—
427C22#1683993512yes—
428C22C2×#1451974611yescharacteristic
429C22C2×#1451980312yes—
430C4—22no—
431C4—22no—
432C4—24no—
433C4—42no—
434C4—44no—
435C4—4882no—
436C4—4881no—
437C2—22no—
438C2—22no—
439C2—41no—
440C2—84no—
441C2—9761no—
442C2#2145504611yescharacteristic
443C2C2×#1683993511yescharacteristic
444C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#537C2
socleC22×C61C2×#120
hypercenterC2×#537D61
hypocenterC61#18324
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #21456616C2semidirect
2. #21451299C2semidirect
3. #21444856C2semidirect
4. #21444754C2semidirect
5. #21444335C2semidirect
6. #21444149C2semidirect
7. C2×C61×#537C2semidirect
10. #16840078C22semidirect
13. #16839895C22semidirect
15. #16839701C22semidirect
16. #16839679C22semidirect
21. #16839496C22semidirect
22. #16839488C22semidirect
215. C2×#537D61semidirect
254. C61#18324semidirect

Maximal subgroups

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

subgroupindex
#214566162
#214512992
#214448562
#214447542
#214443352
#214441492
C2×C61×#5372
#1832461