Finite Group Databasebeta

← #29712078

Subgroups — #29712078

Order 15616.

1166 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455052C212yesmaximal, complemented
2#21448403C211yesmaximal, characteristic, complemented
3#21446977C212yesmaximal, complemented
4#21446866C212yesmaximal, complemented
5#21446268C212yesmaximal, complemented
6#21446188C214yesmaximal, complemented
7#21445902C211yesmaximal, characteristic, complemented
8C61×#3493C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C2×#14519807C2214yes—
11C2×#14519736C2211yescharacteristic
12#16839917—22no—
13#16839917—22no—
14#16839900—22no—
15#16839900—22no—
16C2×#14519704C2214yes—
17#16839770C2214yes—
18#16839726C2212yes—
19#16839668—22no—
20#16839667C2212yes—
21#16839667C2212yes—
22#16839623—22no—
23#16839531C2212yes—
24#16839531C2212yes—
25#16839521C2214yes—
26C61×#556—22no—
27C61×#555—22no—
28C61×#553—22no—
29C61×#552—22no—
30C61×#544C2211yescharacteristic
31C61×#538—22no—
32C61×#537C2211yescharacteristic
33C61×#529—22no—
34C61×#525C2212yes—
35C61×#520C2212yes—
36C2×#117×C61C2211yescharacteristic
37C22×Dic122—21no—
38C22×Dic122—24no—
39C22×Dic122—22no—
40C22×Dic122—22no—
41C22×Dic122—42no—
42C22×Dic122C2312yes—
43C23×Dic61—21no—
44#14519807—28no—
45#14519807—44no—
46C2×#11758202—24no—
47C2×#11758200—24no—
48C2×#11758194C2314yes—
49#14519736—42no—
50C2×C4⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×C4×Dic61C2312yes—
53#14519704—44no—
54#14519700—24no—
55#14519700—24no—
56C22⋊#92418—22no—
57C22⋊#92418—22no—
58#14519684—24no—
59#14519684—24no—
60C2×#41×C61C2312yes—
61C23×C4×C61C2311yescharacteristic
62C4⋊Q8×C61—22no—
63#126×C61—24no—
64#126×C61—24no—
65#125×C61—22no—
66#125×C61—22no—
67#124×C61—24no—
68#124×C61—22no—
69#123×C61—24no—
70#123×C61—22no—
71C22⋊Q8×C61—22no—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61D412yes—
74C22⋊Q8×C61D412yes—
75#121×C61—24no—
76#121×C61—22no—
77#121×C61—22no—
78#121×C61D412yes—
79#121×C61D412yes—
80#120×C61—22no—
81C4×Q8×C61—22no—
82C4×Q8×C61—22no—
83C4×D4×C61—22no—
84C4×D4×C61—22no—
85C4×D4×C61—24no—
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88#117×C61—24no—
89#117×C61—22no—
90#117×C61—22no—
91#117×C61—22no—
92#117×C61C2312yes—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61D412yes—
97C2×C42×C61D412yes—
98C24×C61C2×D411yescharacteristic
99#41×C61—24no—
100#41×C61—24no—
101C2×Q8×C61—22no—
102C2×Q8×C61C2×D412yes—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61C2×D412yes—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—24no—
111C22×C4×C61—22no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—24no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—42no—
122C22×Dic61—42no—
123C2×Dic122—24no—
124C2×Dic122—24no—
125C2×Dic122—24no—
126C2×Dic122—22no—
127C2×Dic122—48no—
128C2×Dic122—48no—
129C2×Dic122—42no—
130C2×Dic122—41no—
131C2×Dic122—41no—
132C2×Dic122#4114yes—
133C4⋊C4×C61—24no—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61C2×D412yes—
142C4⋊C4×C61C2×D412yes—
143C22⋊C4×C61—24no—
144C22⋊C4×C61—24no—
145C22⋊C4×C61—24no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C42×C61—24no—
152C42×C61—22no—
153C42×C61—22no—
154C42×C61—22no—
155C22⋊Dic61—42no—
156#11758202—48no—
157#11758200—48no—
158C4⋊Dic61—42no—
159C4×Dic61—24no—
160#11758194—24no—
161#11758194—24no—
162C2×#92418—24no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—22no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61#12012yes—
169C23×C61C22×D411yescharacteristic
170Q8×C61—24no—
171Q8×C61—82no—
172D4×C61—24no—
173D4×C61—42no—
174D4×C61—42no—
175D4×C61—82no—
176D4×C61—82no—
177D4×C61—82no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—24no—
181C2×C4×C61—24no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—44no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—84no—
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61C22×D412yes—
194C2×C4×C61C2×#4112yes—
195C2×C4×C61H(5,𝔽2)12yes—
196C2×Dic61—24no—
197C2×Dic61—48no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203Dic122—28no—
204Dic122—44no—
205Dic122—44no—
206Dic122—44no—
207#92418—44no—
208#23953—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—22no—
212C22×C61—22no—
213C22×C61—22no—
214C22×C61—42no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—42no—
219C22×C61C2×#12011yescharacteristic
220C22×C61#52511yescharacteristic
221C22×C61#53511yescharacteristic
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—42no—
225C4×C61—42no—
226C4×C61—42no—
227C4×C61—42no—
228C4×C61—42no—
229C4×C61#54614yes—
230Dic61—44no—
231Dic61—82no—
232#3493D6111yescharacteristic
233#3004—612no—
234#2310—611no—
235#2028—612no—
236#2007—612no—
237#1897—612no—
238#1882—614no—
239#1822—611no—
240C2×C61—21no—
241C2×C61—21no—
242C2×C61—42no—
243C2×C61—82no—
244C2×C61#240411yescharacteristic
245C2×C61#302211yescharacteristic
246C2×C61#302211yescharacteristic
247C23×Q8—611no—
248C2×#137—614no—
249#556—22no—
250#555—22no—
251#553—22no—
252#552—22no—
253#544C2×D6111yescharacteristic
254#538—22no—
255#537C2×D6111yescharacteristic
256#529—22no—
257#525C2×D6112yes—
258#520C2×D6112yes—
259C2×#117—611no—
260C2×#117C2×D6111yescharacteristic
261#451—1222no—
262#451—1222no—
263#448—1222no—
264#448—1222no—
265C2×#104—614no—
266#419—614no—
267#407—612no—
268#391—612no—
269#391—612no—
270#391—1222no—
271#381—1222no—
272#343—612no—
273#343—612no—
274#339—614no—
275C61#2395311yescharacteristic, complemented, Sylow
276C2×#41C22×D6112yes—
277C22×Q8—612no—
278C22×Q8—1224no—
279C22×Q8—1221no—
280C22×Q8—1222no—
281C22×Q8—1222no—
282C22×Q8—2442no—
283C23×C4—1221no—
284C23×C4C22×D6111yescharacteristic
285#137—1228no—
286#137—2444no—
287C2×Q16—1224no—
288C2×SD16—1224no—
289C2×M16—614no—
290C4⋊Q8—22no—
291#126—24no—
292#126—24no—
293#125—22no—
294#125—22no—
295#124—22no—
296#124—24no—
297#123—24no—
298#123—22no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8C22⋊D6112yes—
302C22⋊Q8C22⋊D6112yes—
303#121—24no—
304#121—22no—
305#121—22no—
306#121C22⋊D6112yes—
307#121C22⋊D6112yes—
308#120—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—24no—
314C4×D4—22no—
315C4×D4—22no—
316#117—24no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—2442no—
321#117C22×D6112yes—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4C22×D6111yescharacteristic
325C2×C4⋊C4C22×D6111yescharacteristic
326C2×C22⋊C4—1221no—
327C2×C22⋊C4C22⋊D6112yes—
328C2×C42—612no—
329C2×C42C22⋊D6112yes—
330#104—2444no—
331#103—1224no—
332#103—1224no—
333C22⋊C8—1222no—
334C22⋊C8—1222no—
335#95—1224no—
336#95—1224no—
337C24C2×C22⋊D6111yescharacteristic
338#41—24no—
339#41—24no—
340C2×Q8—22no—
341C2×Q8—614no—
342C2×Q8—1224no—
343C2×Q8—1222no—
344C2×Q8—1224no—
345C2×Q8—1224no—
346C2×Q8—2448no—
347C2×Q8—2448no—
348C2×Q8—2442no—
349C2×Q8—2441no—
350C2×Q8—2441no—
351C2×Q8C2×C22⋊D6112yes—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4C2×C22⋊D6112yes—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—1224no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4C2×C22⋊D6112yes—
367C22×C4C2×C22⋊D6112yes—
368C22×C4C2×C22⋊D6111yescharacteristic
369C22×C4C2×C22⋊D6112yes—
370C22×C4C2×C22⋊D6112yes—
371C22×C4C23×D6111yescharacteristic
372Q16—2448no—
373SD16—2448no—
374M16—1224no—
375M16—1224no—
376C2×C8—1224no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—2442no—
386C4⋊C4C2×C22⋊D6112yes—
387C4⋊C4C2×C22⋊D6112yes—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—22no—
391C22⋊C4—24no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—2442no—
397C42—24no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—1224no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—41no—
406C23—41no—
407C23#1451974612yes—
408C23C22×C22⋊D6111yescharacteristic
409Q8—24no—
410Q8—82no—
411Q8—1228no—
412Q8—2444no—
413Q8—2444no—
414Q8—2444no—
415D4—24no—
416D4—42no—
417D4—42no—
418D4—82no—
419D4—82no—
420D4—82no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—84no—
434C2×C4—1224no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4—2442no—
438C2×C4—2448no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4#1451974611yescharacteristic
442C2×C4#1451974611yescharacteristic
443C2×C4C22×C22⋊D6112yes—
444C2×C4#41×D6112yes—
445C2×C4#1451982312yes—
446C8—2444no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22C2×#1451974611yescharacteristic
458C22#1684026911yescharacteristic
459C22#1684032611yescharacteristic
460C4—21no—
461C4—21no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—4882no—
468C4—2444no—
469C4#1684040014yes—
470C2—21no—
471C2—21no—
472C2—42no—
473C2—82no—
474C2#2144913411yescharacteristic
475C2#2145519811yescharacteristic
476C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3493C2
socleC22×C61C2×#120
hypercenter#3493D61
hypocenterC61#23953
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3493

Complements

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

NH ≅ G/Ntype
1. #21455052C2semidirect
2. #21448403C2semidirect
3. #21446977C2semidirect
4. #21446866C2semidirect
5. #21446268C2semidirect
6. #21446188C2semidirect
7. #21445902C2semidirect
275. C61#23953semidirect

Maximal subgroups

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

subgroupindex
#214550522
#214484032
#214469772
#214468662
#214462682
#214461882
#214459022
C61×#34932
#2395361