Finite Group Databasebeta

← #29595560

Subgroups — #29595560

Order 15616.

1156 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 135 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#21447511C211yesmaximal, characteristic, complemented
2#21447491C211yesmaximal, characteristic, complemented
3#21447460C212yesmaximal, complemented
4#21445789C212yesmaximal, complemented
5C61×#3484C211yesmaximal, characteristic
6#16839779—22no—
7#16839767—22no—
8#16839746C2212yescomplemented
9#16839736C2212yescomplemented
10#16839559—22no—
11#16839555—22no—
12#16839552—22no—
13#16839549—22no—
14C2×H(5,𝔽2)×C61C2211yescharacteristic
15C23×D4×C61C411yescharacteristic, complemented
16C61×#546—22no—
17C61×#546—22no—
18D42×C61—22no—
19D42×C61—22no—
20C61×#535C411yescharacteristic, complemented
21C61×#535C2211yescharacteristic
22C61×#534—22no—
23C61×#525C411yescharacteristic, complemented
24C61×#525C2211yescharacteristic
25C2×#120×C61—22no—
26C61×#518C411yescharacteristic, complemented
27C2×#11758194—22no—
28#14519736—22no—
29#14519704—44no—
30#14519698—44no—
31#14519696—22no—
32#14519694—22no—
33#14519692—22no—
34#14519692—22no—
35C25×C61—21no—
36H(5,𝔽2)×C61—44no—
37C2×#41×C61—21no—
38C2×#41×C61C2×C411yescharacteristic
39C2×#41×C61D412yes—
40C2×#41×C61C2311yescharacteristic
41C22×D4×C61—24no—
42C22×D4×C61—22no—
43C22×D4×C61—22no—
44C22×D4×C61—22no—
45C22×D4×C61—21no—
46C22×D4×C61—21no—
47C22×D4×C61—44no—
48C22×D4×C61C2×C411yescharacteristic
49C22×D4×C61C2×C411yescharacteristic
50C22×D4×C61D412yes—
51C23×C4×C61C2×C411yescharacteristic
52#127×C61—21no—
53#127×C61D412yes—
54#124×C61—21no—
55#124×C61D412yes—
56#123×C61—24no—
57#123×C61—21no—
58#123×C61D412yes—
59C22⋊Q8×C61—21no—
60C22⋊Q8×C61—21no—
61#121×C61—24no—
62#121×C61—24no—
63#121×C61—22no—
64#121×C61—21no—
65#121×C61—21no—
66#121×C61—22no—
67#120×C61—24no—
68#120×C61—24no—
69#120×C61—24no—
70#120×C61—21no—
71#120×C61—22no—
72#120×C61—22no—
73#120×C61D412yes—
74C4×D4×C61—22no—
75C4×D4×C61—22no—
76C4×D4×C61—22no—
77C4×D4×C61—22no—
78#117×C61C2×C411yescharacteristic
79#117×C61C2×C411yescharacteristic
80C2×C22⋊C4×C61—24no—
81C2×C22⋊C4×C61—22no—
82C24×C61—21no—
83C24×C61—24no—
84C24×C61—21no—
85C24×C61—22no—
86C24×C61—44no—
87C24×C61—84no—
88C24×C61C22⋊C411yescharacteristic
89#41×C61—24no—
90#41×C61—42no—
91#41×C61—42no—
92#41×C61—42no—
93#41×C61—44no—
94C2×Q8×C61C22⋊C411yescharacteristic
95C2×Q8×C61C2×D411yescharacteristic
96C2×D4×C61—24no—
97C2×D4×C61—24no—
98C2×D4×C61—22no—
99C2×D4×C61—24no—
100C2×D4×C61—21no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61—24no—
104C2×D4×C61—24no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—22no—
108C2×D4×C61—42no—
109C2×D4×C61—42no—
110C2×D4×C61—48no—
111C2×D4×C61—44no—
112C2×D4×C61—44no—
113C2×D4×C61—44no—
114C2×D4×C61—48no—
115C2×D4×C61—44no—
116C2×D4×C61—88no—
117C2×D4×C61C22⋊C411yescharacteristic
118C2×D4×C61C22⋊C412yes—
119C2×D4×C61C22⋊C412yes—
120C2×D4×C61C2×D411yescharacteristic
121C2×D4×C61C2×D412yes—
122C22×C4×C61—24no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—42no—
128C22×C4×C61C22⋊C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C2×D412yes—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C22⋊C4×C61—24no—
137C22⋊C4×C61—24no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—21no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—24no—
143C22⋊C4×C61C22⋊C412yes—
144C42×C61—21no—
145C42×C61C22⋊C412yes—
146C22⋊Dic61—42no—
147C4⋊Dic61—42no—
148C4×Dic61—42no—
149#11758194—42no—
150#11758194—42no—
151C2×#92418—42no—
152C23×C61—24no—
153C23×C61—21no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61—21no—
157C23×C61—44no—
158C23×C61—41no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—48no—
162C23×C61—48no—
163C23×C61—82no—
164C23×C61—82no—
165C23×C61—84no—
166C23×C61—84no—
167C23×C61—88no—
168C23×C61—88no—
169C23×C61C2×C22⋊C411yescharacteristic
170C23×C61#12012yes—
171Q8×C61—22no—
172Q8×C61—41no—
173D4×C61—24no—
174D4×C61—22no—
175D4×C61—22no—
176D4×C61—42no—
177D4×C61—41no—
178D4×C61—44no—
179D4×C61—42no—
180D4×C61—44no—
181D4×C61—48no—
182D4×C61—42no—
183D4×C61—44no—
184D4×C61—44no—
185D4×C61—42no—
186D4×C61—84no—
187D4×C61—84no—
188D4×C61#9914yes—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—44no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—84no—
202C2×C4×C61C2×C22⋊C411yescharacteristic
203C2×C4×C61C2×C22⋊C411yescharacteristic
204C2×C4×C61#12012yes—
205C2×Dic61—82no—
206#92418—82no—
207#12711—611nomaximal, Sylow
208C22×C61—22no—
209C22×C61—22no—
210C22×C61—21no—
211C22×C61—24no—
212C22×C61—42no—
213C22×C61—41no—
214C22×C61—44no—
215C22×C61—41no—
216C22×C61—42no—
217C22×C61—44no—
218C22×C61—42no—
219C22×C61—42no—
220C22×C61—41no—
221C22×C61—41no—
222C22×C61—42no—
223C22×C61—84no—
224C22×C61—816no—
225C22×C61—84no—
226C22×C61—164no—
227C22×C61#37911yescharacteristic
228C4×C61—22no—
229C4×C61—21no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—41no—
233C4×C61—41no—
234C4×C61C2×#9912yes—
235Dic61—162no—
236#3484D6111yescharacteristic
237#2134—611no—
238#2128—611no—
239#2120—612no—
240#1795—612no—
241C2×C61—21no—
242C2×C61—42no—
243C2×C61—42no—
244C2×C61—41no—
245C2×C61—44no—
246C2×C61—82no—
247C2×C61—81no—
248C2×C61#178111yescharacteristic
249C2×H(5,𝔽2)C2×D6111yescharacteristic
250C23×D4Dic6111yescharacteristic, complemented
251#546—22no—
252#546—22no—
253D42—22no—
254D42—22no—
255#535Dic6111yescharacteristic, complemented
256#535C2×D6111yescharacteristic
257#534—22no—
258#525Dic6111yescharacteristic, complemented
259#525C2×D6111yescharacteristic
260C2×#120—22no—
261#518Dic6111yescharacteristic, complemented
262#421—1222no—
263#418—1222no—
264#413—612no—
265#410—612no—
266#355—1222no—
267#353—1222no—
268#352—1222no—
269AΣL(1,𝔽16)—1222no—
270C61#1271111yescharacteristic, complemented, Sylow
271C25—21no—
272H(5,𝔽2)—44no—
273C2×#41—21no—
274C2×#41C2×Dic6111yescharacteristic
275C2×#41C22⋊D6112yes—
276C2×#41C22×D6111yescharacteristic
277C22×D4—24no—
278C22×D4—22no—
279C22×D4—22no—
280C22×D4—22no—
281C22×D4—21no—
282C22×D4—21no—
283C22×D4—44no—
284C22×D4C2×Dic6111yescharacteristic
285C22×D4C2×Dic6111yescharacteristic
286C22×D4C22⋊D6112yes—
287C23×C4C2×Dic6111yescharacteristic
288C2×M16—1222no—
289#127—21no—
290#127C22⋊D6112yes—
291#124—21no—
292#124C22⋊D6112yes—
293#123—24no—
294#123—21no—
295#123C22⋊D6112yes—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298#121—24no—
299#121—24no—
300#121—21no—
301#121—22no—
302#121—21no—
303#121—22no—
304#120—24no—
305#120—24no—
306#120—24no—
307#120—21no—
308#120—22no—
309#120—22no—
310#120C22⋊D6112yes—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315#117—1222no—
316#117C2×Dic6111yescharacteristic
317#117C2×Dic6111yescharacteristic
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320#104—2444no—
321#102—2444no—
322#101—1222no—
323#100—1222no—
324#99—1222no—
325#99—1222no—
326C24—21no—
327C24—24no—
328C24—21no—
329C24—22no—
330C24—44no—
331C24—84no—
332C24C22⋊Dic6111yescharacteristic
333#41—24no—
334#41—42no—
335#41—42no—
336#41—42no—
337#41—44no—
338C2×Q8C22⋊Dic6111yescharacteristic
339C2×Q8C2×C22⋊D6111yescharacteristic
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—48no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—88no—
361C2×D4C22⋊Dic6112yes—
362C2×D4C22⋊Dic6111yescharacteristic
363C2×D4C22⋊Dic6112yes—
364C2×D4C2×C22⋊D6111yescharacteristic
365C2×D4C2×C22⋊D6112yes—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4C22⋊Dic6111yescharacteristic
373C22×C4C22×Dic6111yescharacteristic
374C22×C4C2×C22⋊D6112yes—
375M16—2442no—
376M16—2442no—
377C2×C8—2442no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—2442no—
383C22⋊C4—22no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—22no—
387C22⋊C4—21no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—24no—
391C22⋊C4—2442no—
392C22⋊C4C22⋊Dic6112yes—
393C42—21no—
394C42—2442no—
395C42C22⋊Dic6112yes—
396C23—24no—
397C23—21no—
398C23—22no—
399C23—22no—
400C23—21no—
401C23—44no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—48no—
406C23—48no—
407C23—82no—
408C23—82no—
409C23—84no—
410C23—84no—
411C23—88no—
412C23—88no—
413C23C2×C22⋊Dic6111yescharacteristic
414C23#1451974612yes—
415Q8—22no—
416Q8—41no—
417D4—24no—
418D4—22no—
419D4—22no—
420D4—42no—
421D4—41no—
422D4—44no—
423D4—42no—
424D4—44no—
425D4—48no—
426D4—42no—
427D4—44no—
428D4—44no—
429D4—42no—
430D4—84no—
431D4—84no—
432D4#1451969214yes—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—84no—
446C2×C4—4882no—
447C2×C4C2×C22⋊Dic6111yescharacteristic
448C2×C4C2×C22⋊Dic6111yescharacteristic
449C2×C4#1451974612yes—
450C8—4882no—
451C22—22no—
452C22—21no—
453C22—22no—
454C22—24no—
455C22—42no—
456C22—41no—
457C22—44no—
458C22—41no—
459C22—42no—
460C22—44no—
461C22—42no—
462C22—42no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—84no—
467C22—816no—
468C22—84no—
469C22—164no—
470C22#1683961811yescharacteristic
471C4—22no—
472C4—21no—
473C4—42no—
474C4—42no—
475C4—41no—
476C4—41no—
477C4—9762no—
478C4C2×#1451969212yes—
479C2—21no—
480C2—42no—
481C2—42no—
482C2—41no—
483C2—44no—
484C2—82no—
485C2—81no—
486C2#2144573411yescharacteristic
subgroupG/H
centerC2#21445734
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
Fitting, prime radicalC61×#3484C2
socleC2×C61#1781
cosocleC2×#41×C61C23
hypercenter#3484D61
hypocenterC61#12711
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ #3484

Complements

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

NH ≅ G/Ntype
1. #21447511C2semidirect
2. #21447491C2semidirect
3. #21447460C2semidirect
4. #21445789C2semidirect
8. #16839746C22semidirect
9. #16839736C22semidirect
15. C23×D4×C61C4semidirect
20. C61×#535C4semidirect
23. C61×#525C4semidirect
26. C61×#518C4semidirect
250. C23×D4Dic61semidirect
255. #535Dic61semidirect
258. #525Dic61semidirect
261. #518Dic61semidirect
270. C61#12711semidirect

Maximal subgroups

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

subgroupindex
#214475112
#214474912
#214474602
#214457892
C61×#34842
#1271161