Finite Group Databasebeta

← #29580783

Subgroups — #29580783

Order 15616.

548 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 101 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#21455854C211yesmaximal, characteristic, complemented
2#21451173C211yesmaximal, characteristic
3#21444383C211yesmaximal, characteristic, complemented
4#21444247C211yesmaximal, characteristic, complemented
5#21444210C211yesmaximal, characteristic
6#21443679C211yesmaximal, characteristic, complemented
7C61×#3077C211yesmaximal, characteristic
8C2×C4×Dic122C2211yescharacteristic
9#16840095—21no—
10#16840054—21no—
11#16839969—21no—
12#16839917—21no—
13C4×#11758202—22no—
14#16839815C2211yescharacteristic
15#16839815C2211yescharacteristic
16#16839698—21no—
17#16839674—21no—
18#16839665—21no—
19#16839658—21no—
20#16839635—22no—
21#16839628C2211yescharacteristic
22#16839499—22no—
23#16839492—22no—
24D4⋊#92418—22no—
25C61×#533C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#478—22no—
28C61×#476—21no—
29C61×#474—21no—
30C61×#428—21no—
31C61×#423C2211yescharacteristic
32C61×#418—21no—
33C22×Dic122D411yescharacteristic
34C2×#11758202—22no—
35C4×Dic122—21no—
36C4×Dic122—21no—
37C4×Dic122—21no—
38C4×Dic122—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×#11758196—21no—
42C2×#11758196—22no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—22no—
45C2×C4×Dic61—21no—
46#14519710—22no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49#14519701—22no—
50#14519700—22no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53C4×#92418—21no—
54C4×#92418D412yes—
55#14519684—22no—
56#14519684—21no—
57#14519684D412yes—
58C22×D4×C61D411yescharacteristic
59C23×C4×C61—21no—
60C2×M16×C61—21no—
61C4⋊Q8×C61D411yescharacteristic
62C4⋊Q8×C61D411yescharacteristic
63#125×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×D4×C61—21no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72#117×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C2×C42×C61C2311yescharacteristic
76#107×C61—22no—
77#106×C61—22no—
78C4⋊C8×C61—21no—
79C4⋊C8×C61D412yes—
80#102×C61—22no—
81#102×C61—22no—
82C24×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61Q1612yescomplemented
91C22×C4×C61—21no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—82no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×Dic61—21no—
99C22×Dic61—22no—
100C22×Dic61—21no—
101C2×Dic122—21no—
102C2×Dic122—21no—
103C2×Dic122—42no—
104M16×C61—41no—
105M16×C61—41no—
106C2×C8×C61—22no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C42×C61—41no—
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C22⋊Dic61—44no—
124C22⋊Dic61—82no—
125#11758202—44no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130#11758196—41no—
131#11758196—41no—
132#11758196—82no—
133C4×Dic61—41no—
134C4×Dic61—41no—
135C2×#92418—22no—
136C2×#92418—41no—
137C23×C61—22no—
138C23×C61—42no—
139C23×C61—42no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61#12011yescharacteristic
143Q8×C61—42no—
144Q8×C61—81no—
145D4×C61—22no—
146D4×C61—22no—
147D4×C61—41no—
148D4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—82no—
156C2×C4×C61—82no—
157C2×C4×C61#12011yescharacteristic
158C2×C4×C61#12011yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12711yescharacteristic
163C2×C4×C61C2×Q1611yescharacteristic
164C2×C4×C61#13511yescharacteristic
165C2×C4×C61#13711yescharacteristic
166C2×C4×C61#13711yescharacteristic
167C2×Dic61—22no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176Dic122—22no—
177Dic122—22no—
178Dic122—41no—
179Dic122—41no—
180C8×C61—42no—
181#92418—42no—
182#92418—81no—
183#10002—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—44no—
187C22×C61—42no—
188C22×C61—82no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61#39211yescharacteristic
192C22×C61#45111yescharacteristic
193C22×C61#45211yescharacteristic
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—81no—
199C4×C61—81no—
200C4×C61—81no—
201C4×C61#45111yescharacteristic
202C4×C61#45211yescharacteristic
203C4×C61#46711yescharacteristic
204C4×C61#47011yescharacteristic
205Dic61—42no—
206Dic61—82no—
207Dic61—81no—
208#3104—611no—
209#3077D6111yescharacteristic
210#2580—611no—
211#1518—611no—
212#1487—611no—
213#1478—611no—
214#1286—611no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—82no—
218C2×C61#200111yescharacteristic
219C2×C61#200811yescharacteristic
220C2×C61#201011yescharacteristic
221#533C2×D6111yescharacteristic
222C2×C4×Q8—611no—
223C2×C4×D4C2×D6111yescharacteristic
224#484—1221no—
225#478—22no—
226#477—1221no—
227#476—21no—
228#474—21no—
229#462—1221no—
230#451—1221no—
231C4×Q16—1222no—
232#431—611no—
233#431—611no—
234#428—21no—
235#423C2×D6111yescharacteristic
236#418—21no—
237#399—1221no—
238#393—1221no—
239#390—1221no—
240#389—1221no—
241#385—1222no—
242#382—611no—
243#328—1222no—
244Q8⋊C8—1222no—
245D4⋊C8—1222no—
246C61#1000211yescharacteristic, complemented, Sylow
247C22×Q8—611no—
248C22×D4C22⋊D6111yescharacteristic
249C23×C4—21no—
250C2×Q16—1222no—
251C2×M16—21no—
252C4⋊Q8C22⋊D6111yescharacteristic
253C4⋊Q8C22⋊D6111yescharacteristic
254#125—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—1221no—
260C4×Q8—1221no—
261C4×Q8—1221no—
262C4×Q8—1221no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267#117—21no—
268C2×C4⋊C4—611no—
269C2×C4⋊C4—1222no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4C22⋊D6111yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—1222no—
275C2×C22⋊C4—1222no—
276C2×C42—1221no—
277C2×C42C22×D6111yescharacteristic
278#107—22no—
279#107—1222no—
280#106—22no—
281C4⋊C8—21no—
282C4⋊C8—1221no—
283C4⋊C8—1221no—
284C4⋊C8C22⋊D6112yes—
285#103—1222no—
286#103—1222no—
287#102—22no—
288#102—22no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291C4×C8—612no—
292C4×C8—1221no—
293#95—612no—
294#95—1222no—
295#95—1221no—
296C24—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—1221no—
300C2×Q8—1221no—
301C2×Q8—2442no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4#1175820212yescomplemented
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—82no—
312C22×C4—1221no—
313C22×C4—1222no—
314C22×C4—1221no—
315C22×C4D4×D6111yescharacteristic
316C22×C4D4×D6111yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318Q16—2444no—
319M16—41no—
320M16—41no—
321C2×C8—22no—
322C2×C8—1222no—
323C2×C8—2441no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—4882no—
335C4⋊C4—2441no—
336C4⋊C4—2441no—
337C4⋊C4—2441no—
338C4⋊C4—2441no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—4882no—
343C22⋊C4—2444no—
344C42—41no—
345C42—2441no—
346C42—2441no—
347C42D4×D6111yescharacteristic
348C42#1175822411yescharacteristic
349C42C2×C22⋊D6111yescharacteristic
350C42C2×C22⋊D6111yescharacteristic
351C23—22no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23#1451974511yescharacteristic
357Q8—42no—
358Q8—81no—
359Q8—1222no—
360Q8—1222no—
361Q8—2441no—
362Q8—2441no—
363D4—22no—
364D4—22no—
365D4—41no—
366D4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—1222no—
376C2×C4—2441no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2441no—
380C2×C4—2442no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4#1451974511yescharacteristic
385C2×C4#1451974611yescharacteristic
386C2×C4#1451974811yescharacteristic
387C2×C4#1451974911yescharacteristic
388C2×C4#1451974911yescharacteristic
389C2×C4#1451977311yescharacteristic
390C2×C4C2×#1175820211yescharacteristic
391C2×C4#1451979711yescharacteristic
392C2×C4#1451980711yescharacteristic
393C2×C4#1451980811yescharacteristic
394C8—42no—
395C8—4881no—
396C8—2442no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—42no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22#1683967011yescharacteristic
405C22#1683991711yescharacteristic
406C22#1683992211yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—4882no—
415C4—4881no—
416C4—2442no—
417C4#1683991811yescharacteristic
418C4#1683992411yescharacteristic
419C4#1684000111yescharacteristic
420C4#1684001811yescharacteristic
421C2—21no—
422C2—21no—
423C2—82no—
424C2#2144682711yescharacteristic
425C2#2144687011yescharacteristic
426C2#2144688311yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3077C2
socleC22×C61#392
prime radicalC22×D4×C61D4
hypercenter#3077D61
hypocenterC61#10002
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #21455854C2semidirect
3. #21444383C2semidirect
4. #21444247C2semidirect
6. #21443679C2semidirect
90. C2×D4×C61Q16semidirect
246. C61#10002semidirect
307. C2×D4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214558542
#214511732
#214443832
#214442472
#214442102
#214436792
C61×#30772
#1000261