Finite Group Databasebeta

← #29580834

Subgroups — #29580834

Order 15616.

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

normal subgroups only

This group has 528 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#21457025C211yesmaximal, characteristic, complemented
2#21455918C211yesmaximal, characteristic
3#21446762C211yesmaximal, characteristic, complemented
4#21444421C211yesmaximal, characteristic, complemented
5#21444262C211yesmaximal, characteristic
6#21443675C211yesmaximal, characteristic, complemented
7C61×#2562C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840172—21no—
10#16840132—21no—
11#16840097—21no—
12#16840097—21no—
13#16840092—21no—
14#16840066—22no—
15#16840042—21no—
16#16839967—21no—
17#16839913—21no—
18#16839862—22no—
19#16839819C2211yescharacteristic
20#16839819C2211yescharacteristic
21C4×#11758194C2211yescharacteristic
22#16839522—22no—
23#16839504—22no—
24D4⋊#92418—22no—
25C2×#125×C61C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#400—21no—
28C61×#394—21no—
29C61×#388—22no—
30C61×#380—21no—
31C61×#380—21no—
32C4×C4⋊C4×C61C2211yescharacteristic
33C2×#11758200—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⋊#92418D412yes—
48#14519700—22no—
49#14519700—22no—
50#14519700—22no—
51#14519698—22no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54#14519688—21no—
55#14519688—21no—
56#14519688D412yes—
57C4×#92418—21no—
58C22×D4×C61D411yescharacteristic
59C23×C4×C61—21no—
60#125×C61—21no—
61#125×C61—21no—
62#125×C61—21no—
63#125×C61—21no—
64C4×D4×C61—21no—
65C4×D4×C61—21no—
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61D411yescharacteristic
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×C61D412yes—
78C2×C42×C61C2311yescharacteristic
79#95×C61—22no—
80#95×C61—22no—
81#95×C61—21no—
82C24×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61—42no—
86C22×C4×C61—21no—
87C22×C4×C61—22no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—42no—
91C22×C4×C61—42no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C2×Dic122—21no—
98C2×Dic122—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—42no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C22⋊C4×C61—44no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C22⋊C4×C61—82no—
114C42×C61—42no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119C22⋊Dic61—41no—
120#11758200—44no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125#11758196—41no—
126#11758196—41no—
127C4×Dic61—41no—
128#11758194—22no—
129#11758194—41no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—41no—
133C23×C61—22no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61#12011yescharacteristic
141D4×C61—22no—
142D4×C61—22no—
143D4×C61—41no—
144D4×C61—41no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—82no—
161C2×C4×C61—82no—
162C2×C4×C61#12011yescharacteristic
163C2×C4×C61#12011yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12711yescharacteristic
168C2×Dic61—41no—
169C2×Dic61—41no—
170Dic122—42no—
171Dic122—42no—
172#92418—42no—
173#92418—42no—
174#92418—42no—
175#10009—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61#39211yescharacteristic
186C22×C61#45511yescharacteristic
187C22×C61#45611yescharacteristic
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—82no—
194C4×C61—81no—
195C4×C61—81no—
196C4×C61—81no—
197C4×C61#45412yes—
198Dic61—81no—
199Dic61—81no—
200#3236—611no—
201#3110—611no—
202#2562D6111yescharacteristic
203#1986—611no—
204#1528—611no—
205#1490—611no—
206#1285—611no—
207C2×C61—21no—
208C2×C61—21no—
209C2×C61—82no—
210C2×C61#200311yescharacteristic
211C2×C61#200711yescharacteristic
212C2×C61#201011yescharacteristic
213#533—611no—
214C2×#125C2×D6111yescharacteristic
215C2×C4×D4C2×D6111yescharacteristic
216#501—1221no—
217#491—1221no—
218#484—1221no—
219#484—1221no—
220#483—1221no—
221#479—1222no—
222#474—1221no—
223#461—1221no—
224#450—1221no—
225#440—1222no—
226#432—611no—
227#432—611no—
228C4×M16—611no—
229#400—21no—
230#394—21no—
231#388—22no—
232#380—21no—
233#380—21no—
234C4×C4⋊C4C2×D6111yescharacteristic
235#339—1222no—
236#330—1222no—
237D4⋊C8—1222no—
238C61#1000911yescharacteristic, complemented, Sylow
239C22×D4C22⋊D6111yescharacteristic
240C23×C4—21no—
241C2×SD16—1222no—
242C2×M16—1221no—
243C4⋊Q8—611no—
244C4⋊Q8—611no—
245#125—21no—
246#125—21no—
247#125—21no—
248#125—21no—
249#125—1221no—
250C22⋊Q8—1221no—
251C22⋊Q8—1221no—
252C4×Q8—1221no—
253C4×Q8—1221no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258#117—1221no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C22⋊D6111yescharacteristic
263C2×C4⋊C4C22⋊D6111yescharacteristic
264C2×C4⋊C4C22⋊D6111yescharacteristic
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C42C22⋊D6112yes—
269C2×C42C22×D6111yescharacteristic
270#107—1222no—
271#107—1222no—
272#106—1222no—
273C4⋊C8—612no—
274C4⋊C8—1221no—
275#103—1222no—
276#103—1222no—
277#103—1222no—
278#102—1222no—
279C22⋊C8—1221no—
280C22⋊C8—1221no—
281#97—612no—
282#97—1221no—
283#97—1221no—
284C4×C8—1221no—
285#95—22no—
286#95—22no—
287#95—21no—
288C24—21no—
289C2×Q8—1221no—
290C2×Q8—1221no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—42no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C2×C22⋊D6111yescharacteristic
303C22×C4C2×C22⋊D6111yescharacteristic
304C22×C4C2×C22⋊D6111yescharacteristic
305SD16—2444no—
306M16—1222no—
307M16—2441no—
308C2×C8—1222no—
309C2×C8—1222no—
310C2×C8—2441no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—1221no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C22⋊C4—44no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—82no—
332C22⋊C4—2441no—
333C42—42no—
334C42—2441no—
335C42D4×D6111yescharacteristic
336C42D4×D6111yescharacteristic
337C42D4×D6111yescharacteristic
338C42#1175822411yescharacteristic
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23#1451974611yescharacteristic
347Q8—2442no—
348Q8—2442no—
349D4—22no—
350D4—22no—
351D4—41no—
352D4—41no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4#1451974511yescharacteristic
373C2×C4#1451974511yescharacteristic
374C2×C4#1451974811yescharacteristic
375C2×C4#1451974911yescharacteristic
376C2×C4#1451974911yescharacteristic
377C2×C4#1451977311yescharacteristic
378C8—2442no—
379C8—2442no—
380C8—2442no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22#1683967011yescharacteristic
391C22#1683994011yescharacteristic
392C22#1683994511yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—4881no—
403C4—4881no—
404C4#1683993412yes—
405C2—21no—
406C2—21no—
407C2—82no—
408C2#2144684111yescharacteristic
409C2#2144686611yescharacteristic
410C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2562C2
socleC22×C61#392
prime radicalC22×D4×C61D4
hypercenter#2562D61
hypocenterC61#10009
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2562

Complements

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

NH ≅ G/Ntype
1. #21457025C2semidirect
3. #21446762C2semidirect
4. #21444421C2semidirect
6. #21443675C2semidirect
238. C61#10009semidirect

Maximal subgroups

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

subgroupindex
#214570252
#214559182
#214467622
#214444212
#214442622
#214436752
C61×#25622
#1000961