Finite Group Databasebeta

← #29580785

Subgroups — #29580785

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#21455858C211yesmaximal, characteristic, complemented
2#21455616C211yesmaximal, characteristic
3#21444382C211yesmaximal, characteristic, complemented
4#21444248C211yesmaximal, characteristic, complemented
5#21444210C211yesmaximal, characteristic
6#21443677C211yesmaximal, characteristic, complemented
7C61×#2580C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840097—21no—
10#16840062—22no—
11#16840057—21no—
12#16840052—21no—
13#16840042—21no—
14#16839972—21no—
15#16839918—21no—
16C4×#11758202—22no—
17#16839815C2211yescharacteristic
18#16839815C2211yescharacteristic
19#16839802—21no—
20#16839786C2211yescharacteristic
21#16839767—21no—
22#16839498—22no—
23Q8⋊#92418—22no—
24D4⋊#92418—22no—
25C2×C4×Q8×C61C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#399—21no—
28C61×#393—21no—
29C61×#390—21no—
30C61×#389—21no—
31C61×#385—22no—
32C61×#382C2211yescharacteristic
33C2×#11758202—22no—
34C2×#11758194—21no—
35#14519777D411yescharacteristic
36#14519777D411yescharacteristic
37#14519769—21no—
38#14519756—21no—
39#14519756—21no—
40C4×Dic122—21no—
41C4×Dic122—21no—
42#14519736—21no—
43#14519710—22no—
44#14519710—22no—
45#14519708—22no—
46C4⋊#92418—21no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418D412yes—
50#14519701—22no—
51#14519700—22no—
52#14519698—22no—
53#14519698—22no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C4×#92418—21no—
57C4×#92418D412yes—
58C22×Q8×C61D411yescharacteristic
59C22×D4×C61D411yescharacteristic
60C23×C4×C61—21no—
61C4×Q8×C61—21no—
62C4×Q8×C61—21no—
63C4×Q8×C61—21no—
64C4×Q8×C61—21no—
65C4×D4×C61—21no—
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—22no—
77C2×C42×C61—21no—
78C2×C42×C61C2311yescharacteristic
79#95×C61—22no—
80#95×C61—21no—
81#95×C61D412yes—
82C24×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—42no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61Q1612yescomplemented
92C22×C4×C61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—82no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C2×Dic122—21no—
103C2×Dic122—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—82no—
113C22⋊C4×C61—44no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—82no—
117C42×C61—41no—
118C42×C61—41no—
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C22⋊Dic61—41no—
124#11758202—44no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129#11758196—41no—
130#11758196—41no—
131C4×Dic61—41no—
132#11758194—41no—
133#11758194—41no—
134C2×#92418—22no—
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—22no—
144Q8×C61—22no—
145Q8×C61—41no—
146Q8×C61—41no—
147D4×C61—22no—
148D4×C61—22no—
149D4×C61—41no—
150D4×C61—41no—
151C2×C4×C61—22no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—82no—
165C2×C4×C61—82no—
166C2×C4×C61#12011yescharacteristic
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12711yescharacteristic
172C2×C4×C61C2×Q1611yescharacteristic
173C2×C4×C61#13511yescharacteristic
174C2×C4×C61#13711yescharacteristic
175C2×C4×C61#13711yescharacteristic
176C2×Dic61—41no—
177C2×Dic61—41no—
178Dic122—42no—
179Dic122—81no—
180#92418—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—42no—
199C4×C61—82no—
200C4×C61—81no—
201C4×C61—81no—
202C4×C61#45111yescharacteristic
203C4×C61#45211yescharacteristic
204C4×C61#46711yescharacteristic
205C4×C61#47011yescharacteristic
206Dic61—81no—
207Dic61—81no—
208#3104—611no—
209#3077—611no—
210#2580D6111yescharacteristic
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#533—611no—
222C2×C4×Q8C2×D6111yescharacteristic
223C2×C4×D4C2×D6111yescharacteristic
224#484—1221no—
225#478—1222no—
226#477—1221no—
227#476—1221no—
228#474—1221no—
229#462—1221no—
230#451—1221no—
231C4×Q16—1222no—
232#431—611no—
233#431—611no—
234#428—1221no—
235#423—611no—
236#418—1221no—
237#399—21no—
238#393—21no—
239#390—21no—
240#389—21no—
241#385—22no—
242#382C2×D6111yescharacteristic
243#328—1222no—
244Q8⋊C8—1222no—
245D4⋊C8—1222no—
246C61#1000211yescharacteristic, complemented, Sylow
247C22×Q8C22⋊D6111yescharacteristic
248C22×D4C22⋊D6111yescharacteristic
249C23×C4—21no—
250C2×Q16—1222no—
251C2×M16—1221no—
252C4⋊Q8—611no—
253C4⋊Q8—611no—
254#125—1221no—
255C22⋊Q8—1221no—
256C22⋊Q8—1221no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—1221no—
262C4×Q8—1221no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267#117—1221no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C22⋊D6111yescharacteristic
272C2×C4⋊C4C22⋊D6111yescharacteristic
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C42—21no—
277C2×C42C22×D6111yescharacteristic
278#107—1222no—
279#107—1222no—
280#106—1222no—
281C4⋊C8—612no—
282C4⋊C8—1221no—
283C4⋊C8—1221no—
284C4⋊C8—1221no—
285#103—1222no—
286#103—1222no—
287#102—1222no—
288#102—1222no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291C4×C8—612no—
292C4×C8—1221no—
293#95—22no—
294#95—21no—
295#95C22⋊D6112yes—
296C24—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×Q8—1221no—
301C2×Q8—1221no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4#1175820212yescomplemented
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4C2×C22⋊D6111yescharacteristic
316C22×C4C2×C22⋊D6111yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318Q16—2444no—
319M16—2441no—
320M16—2441no—
321C2×C8—1222no—
322C2×C8—1222no—
323C2×C8—2441no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—82no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—2441no—
338C4⋊C4—2441no—
339C22⋊C4—44no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—82no—
343C22⋊C4—2441no—
344C42—41no—
345C42—41no—
346C42—2441no—
347C42D4×D6111yescharacteristic
348C42D4×D6111yescharacteristic
349C42D4×D6111yescharacteristic
350C42#1175822411yescharacteristic
351C23—22no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23#1451974611yescharacteristic
357Q8—22no—
358Q8—22no—
359Q8—41no—
360Q8—41no—
361Q8—4881no—
362Q8—2442no—
363D4—22no—
364D4—22no—
365D4—41no—
366D4—41no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4#1451974511yescharacteristic
385C2×C4#1451974511yescharacteristic
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—4881no—
395C8—2442no—
396C8—2442no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—42no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22#1683967011yescharacteristic
405C22#1683991811yescharacteristic
406C22#1683992411yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—82no—
413C4—81no—
414C4—81no—
415C4—4881no—
416C4—4881no—
417C4#1683991711yescharacteristic
418C4#1683992211yescharacteristic
419C4#1684000111yescharacteristic
420C4#1684001811yescharacteristic
421C2—21no—
422C2—21no—
423C2—82no—
424C2#2144682711yescharacteristic
425C2#2144687211yescharacteristic
426C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2580C2
socleC22×C61#392
prime radicalC22×D4×C61D4
hypercenter#2580D61
hypocenterC61#10002
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. #21455858C2semidirect
3. #21444382C2semidirect
4. #21444248C2semidirect
6. #21443677C2semidirect
91. 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
#214558582
#214556162
#214443822
#214442482
#214442102
#214436772
C61×#25802
#1000261