Finite Group Databasebeta

← #29580827

Subgroups — #29580827

Order 15616.

654 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 85 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#21456822C211yesmaximal, characteristic, complemented
2#21455918C211yesmaximal, characteristic, complemented
3#21446762C211yesmaximal, characteristic, complemented
4#21444417C211yesmaximal, characteristic, complemented
5#21444262C211yesmaximal, characteristic, complemented
6#21443675C211yesmaximal, characteristic, complemented
7C61×#2540C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840157—21no—
10#16840121—21no—
11#16840097—21no—
12#16840042—21no—
13#16839981—22no—
14#16839967—21no—
15#16839918—21no—
16#16839913—21no—
17#16839913—21no—
18#16839862—22no—
19#16839819C2211yescharacteristic
20#16839819C2211yescharacteristic
21C4×#11758194C2211yescharacteristic
22#16839522—22no—
23#16839502—22no—
24D4⋊#92418—22no—
25C2×#124×C61C2211yescharacteristic
26C2×C22⋊Q8×C61—21no—
27C2×#121×C61—21no—
28C2×C4×D4×C61C2211yescharacteristic
29C61×#399—21no—
30C61×#392—21no—
31C61×#390—22no—
32C4×C4⋊C4×C61C2211yescharacteristic
33C2×#11758202—22no—
34C2×#11758200—22no—
35C2×#11758200—22no—
36C2×#11758194—21no—
37#14519777D411yescharacteristic, complemented
38#14519777D411yescharacteristic, complemented
39#14519769—21no—
40#14519756—21no—
41#14519756—21no—
42C4×Dic122—21no—
43C4×Dic122—21no—
44#14519736—21no—
45#14519710—22no—
46C4⋊#92418—21no—
47C4⋊#92418D412yescomplemented
48#14519701—22no—
49#14519700—22no—
50#14519698—22no—
51#14519698—22no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54#14519688—21no—
55#14519688—21no—
56#14519688D412yescomplemented
57C4×#92418—21no—
58C22×Q8×C61D411yescharacteristic
59C22×D4×C61—22no—
60C22×D4×C61D411yescharacteristic
61C22×D4×C61D411yescharacteristic
62C23×C4×C61—21no—
63#124×C61—21no—
64#124×C61—21no—
65#124×C61—21no—
66#124×C61—21no—
67C22⋊Q8×C61—44no—
68#121×C61—44no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—21no—
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—21no—
81C2×C42×C61D412yes—
82C2×C42×C61C2311yescharacteristic
83#95×C61—21no—
84C24×C61—21no—
85C24×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×Q8×C61—42no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—42no—
94C2×D4×C61—44no—
95C2×D4×C61—42no—
96C2×D4×C61—44no—
97C2×D4×C61—82no—
98C22×C4×C61—22no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—42no—
102C22×C4×C61—42no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C2×Dic122—21no—
109C2×Dic122—21no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—44no—
114C22⋊C4×C61—44no—
115C22⋊C4×C61—44no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C42×C61—42no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127C22⋊Dic61—41no—
128#11758202—82no—
129#11758200—44no—
130#11758200—82no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135#11758196—41no—
136#11758196—41no—
137C4×Dic61—41no—
138#11758194—22no—
139#11758194—41no—
140C2×#92418—22no—
141C2×#92418—22no—
142C2×#92418—41no—
143C23×C61—22no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—42no—
148C23×C61—42no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—41no—
155C23×C61#12011yescharacteristic
156Q8×C61—42no—
157Q8×C61—81no—
158D4×C61—22no—
159D4×C61—22no—
160D4×C61—41no—
161D4×C61—41no—
162D4×C61—42no—
163D4×C61—81no—
164D4×C61—88no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—22no—
168C2×C4×C61—24no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—82no—
178C2×C4×C61—82no—
179C2×C4×C61#12011yescharacteristic
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12711yescharacteristic
185C2×Dic61—41no—
186C2×Dic61—41no—
187Dic122—42no—
188Dic122—42no—
189#92418—42no—
190#92418—42no—
191#92418—42no—
192#10008—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61—42no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61—82no—
201C22×C61—82no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61#39211yescharacteristic
208C22×C61#45511yescharacteristic
209C22×C61#45611yescharacteristic
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—44no—
216C4×C61—81no—
217C4×C61—81no—
218C4×C61#45312yes—
219Dic61—81no—
220Dic61—81no—
221#3215—611no—
222#3110—611no—
223#2540D6111yescharacteristic
224#1986—611no—
225#1527—611no—
226#1490—611no—
227#1285—611no—
228C2×C61—21no—
229C2×C61—21no—
230C2×C61—82no—
231C2×C61—161no—
232C2×C61#200211yescharacteristic
233C2×C61#200711yescharacteristic
234C2×C61#201011yescharacteristic
235#533—611no—
236C2×#124C2×D6111yescharacteristic
237C2×C22⋊Q8—21no—
238C2×#121—21no—
239C2×C4×D4C2×D6111yescharacteristic
240#497—1221no—
241#489—1221no—
242#484—1221no—
243#474—1221no—
244#464—1222no—
245#461—1221no—
246#451—1221no—
247#450—1221no—
248#450—1221no—
249#440—1222no—
250#432—611no—
251#432—611no—
252C4×M16—611no—
253#399—21no—
254#392—21no—
255#390—22no—
256C4×C4⋊C4C2×D6111yescharacteristic
257#339—1222no—
258#329—1222no—
259D4⋊C8—1222no—
260C61#1000811yescharacteristic, complemented, Sylow
261C22×Q8C22⋊D6111yescharacteristic
262C22×D4—22no—
263C22×D4C22⋊D6111yescharacteristic
264C22×D4C22⋊D6111yescharacteristic
265C23×C4—21no—
266C2×Q16—1222no—
267C2×SD16—1222no—
268C2×SD16—1222no—
269C2×M16—1221no—
270C4⋊Q8—611no—
271C4⋊Q8—611no—
272#125—1221no—
273#124—21no—
274#124—21no—
275#124—21no—
276#124—21no—
277C22⋊Q8—44no—
278C22⋊Q8—1221no—
279C22⋊Q8—1221no—
280#121—44no—
281C4×Q8—1221no—
282C4×Q8—1221no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287#117—1221no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4C22⋊D6111yescharacteristic
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C42C22⋊D6112yes—
297C2×C42C22×D6111yescharacteristic
298#107—1222no—
299C4⋊C8—612no—
300C4⋊C8—1221no—
301#103—1222no—
302#103—1222no—
303#102—1222no—
304#102—1222no—
305C22⋊C8—1221no—
306C22⋊C8—1221no—
307#97—612no—
308#97—1221no—
309#97—1221no—
310C4×C8—1221no—
311#95—21no—
312C24—21no—
313C24—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—42no—
317C2×Q8—1221no—
318C2×Q8—1221no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—82no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4C2×C22⋊D6111yescharacteristic
336C22×C4C2×C22⋊D6111yescharacteristic
337C22×C4C2×C22⋊D6111yescharacteristic
338Q16—4882no—
339SD16—4882no—
340SD16—2444no—
341M16—1222no—
342M16—2441no—
343C2×C8—1222no—
344C2×C8—1222no—
345C2×C8—2441no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—44no—
350C4⋊C4—1221no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4—2441no—
355C4⋊C4—2441no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—2441no—
365C42—42no—
366C42—2441no—
367C42D4×D6111yescharacteristic
368C42D4×D6111yescharacteristic
369C42D4×D6111yescharacteristic
370C42#1175822411yescharacteristic
371C23—22no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23#1451974611yescharacteristic
384Q8—42no—
385Q8—81no—
386Q8—2442no—
387Q8—2442no—
388D4—22no—
389D4—22no—
390D4—41no—
391D4—41no—
392D4—42no—
393D4—81no—
394D4—88no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4#1451974511yescharacteristic
412C2×C4#1451974511yescharacteristic
413C2×C4#1451974811yescharacteristic
414C2×C4#1451974911yescharacteristic
415C2×C4#1451974911yescharacteristic
416C2×C4#1451977311yescharacteristic
417C8—2442no—
418C8—2442no—
419C8—2442no—
420C22—21no—
421C22—21no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—82no—
428C22—82no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22#1683967011yescharacteristic
435C22#1683994011yescharacteristic
436C22#1683994511yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—44no—
443C4—81no—
444C4—81no—
445C4—4881no—
446C4—4881no—
447C4#1683992912yes—
448C2—21no—
449C2—21no—
450C2—82no—
451C2—161no—
452C2#2144683411yescharacteristic
453C2#2144686611yescharacteristic
454C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#2540C2
socleC22×C61#392
hypercenter#2540D61
hypocenterC61#10008
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2540

Complements

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

NH ≅ G/Ntype
1. #21456822C2semidirect
2. #21455918C2semidirect
3. #21446762C2semidirect
4. #21444417C2semidirect
5. #21444262C2semidirect
6. #21443675C2semidirect
37. #14519777D4semidirect
38. #14519777D4semidirect
47. C4⋊#92418D4semidirect
56. #14519688D4semidirect
260. C61#10008semidirect

Maximal subgroups

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

subgroupindex
#214568222
#214559182
#214467622
#214444172
#214442622
#214436752
C61×#25402
#1000861