Finite Group Databasebeta

← #29580841

Subgroups — #29580841

Order 15616.

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

normal subgroups only

This group has 534 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
2#21455949C211yesmaximal, characteristic, complemented
3#21446762C211yesmaximal, characteristic, complemented
4#21444417C211yesmaximal, characteristic
5#21444266C211yesmaximal, characteristic, complemented
6#21443675C211yesmaximal, characteristic, complemented
7C61×#2592C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840157—21no—
10#16840121—21no—
11#16840092—21no—
12#16840047—21no—
13#16839981—22no—
14#16839972—21no—
15#16839918—21no—
16#16839918—21no—
17#16839913—21no—
18#16839867—22no—
19#16839819C2211yescharacteristic
20#16839819C2211yescharacteristic
21C4×#11758194C2211yescharacteristic
22#16839522—22no—
23#16839502—22no—
24Q8⋊#92418—22no—
25C2×#124×C61C2211yescharacteristic
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#397—21no—
28C61×#393—21no—
29C61×#391—21no—
30C61×#390—21no—
31C61×#388—22no—
32C4×C4⋊C4×C61C2211yescharacteristic
33C2×#11758202—22no—
34C2×#11758202—22no—
35C2×#11758200—22no—
36C2×#11758194—21no—
37#14519777D411yescharacteristic
38#14519777D411yescharacteristic
39#14519769—21no—
40#14519756—21no—
41#14519756—21no—
42C4×Dic122—21no—
43C4×Dic122—21no—
44#14519736—21no—
45#14519708—22no—
46C4⋊#92418—21no—
47C4⋊#92418D412yes—
48#14519701—22no—
49#14519701—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×Q8×C61D411yescharacteristic
59C22×Q8×C61D411yescharacteristic
60C22×D4×C61D411yescharacteristic
61#124×C61—21no—
62#124×C61—21no—
63#124×C61—21no—
64#124×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×Q8×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C22⋊C4×C61—22no—
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—22no—
76C2×C42×C61—21no—
77C2×C42×C61D412yes—
78C2×C42×C61C2311yescharacteristic
79#95×C61—22no—
80#95×C61—22no—
81#95×C61—21no—
82C24×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—42no—
88C2×Q8×C61—42no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—42no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61—21no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C2×Dic122—21no—
100C2×Dic122—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—42no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C22⋊C4×C61—44no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—82no—
112C42×C61—42no—
113C42×C61—41no—
114C42×C61—41no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119C22⋊Dic61—41no—
120#11758202—44no—
121#11758202—82no—
122#11758200—82no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127#11758196—41no—
128#11758196—41no—
129C4×Dic61—41no—
130#11758194—22no—
131#11758194—41no—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418—41no—
135C23×C61—42no—
136C23×C61—42no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61#12011yescharacteristic
141Q8×C61—22no—
142Q8×C61—22no—
143Q8×C61—41no—
144Q8×C61—41no—
145Q8×C61—42no—
146Q8×C61—81no—
147D4×C61—42no—
148D4×C61—81no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12011yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12711yescharacteristic
170C2×Dic61—41no—
171C2×Dic61—41no—
172Dic122—42no—
173Dic122—42no—
174#92418—42no—
175#92418—42no—
176#92418—42no—
177#10010—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—82no—
181C22×C61—82no—
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—42no—
194C4×C61—82no—
195C4×C61—81no—
196C4×C61—81no—
197C4×C61#45512yes—
198Dic61—81no—
199Dic61—81no—
200#3215—611no—
201#3113—611no—
202#2592D6111yescharacteristic
203#1986—611no—
204#1527—611no—
205#1491—611no—
206#1285—611no—
207C2×C61—21no—
208C2×C61—21no—
209C2×C61—161no—
210C2×C61#200411yescharacteristic
211C2×C61#200711yescharacteristic
212C2×C61#201011yescharacteristic
213#533—611no—
214C2×#124C2×D6111yescharacteristic
215C2×C4×Q8C2×D6111yescharacteristic
216#497—1221no—
217#489—1221no—
218#483—1221no—
219#475—1221no—
220#464—1222no—
221#462—1221no—
222#451—1221no—
223#451—1221no—
224#450—1221no—
225#441—1222no—
226#432—611no—
227#432—611no—
228C4×M16—611no—
229#397—21no—
230#393—21no—
231#391—21no—
232#390—21no—
233#388—22no—
234C4×C4⋊C4C2×D6111yescharacteristic
235#339—1222no—
236#329—1222no—
237Q8⋊C8—1222no—
238C61#1001011yescharacteristic, complemented, Sylow
239C22×Q8C22⋊D6111yescharacteristic
240C22×Q8C22⋊D6111yescharacteristic
241C22×D4C22⋊D6111yescharacteristic
242C2×Q16—1222no—
243C2×Q16—1222no—
244C2×SD16—1222no—
245C2×M16—1221no—
246C4⋊Q8—611no—
247C4⋊Q8—611no—
248#125—1221no—
249#124—21no—
250#124—21no—
251#124—21no—
252#124—21no—
253C22⋊Q8—1221no—
254C22⋊Q8—1221no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—1221no—
260C4×Q8—1221no—
261#117—1221no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C22⋊D6111yescharacteristic
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C42—21no—
270C2×C42C22⋊D6112yes—
271C2×C42C22×D6111yescharacteristic
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—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—1221no—
296C2×Q8—1221no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4C2×C22⋊D6111yescharacteristic
305C22×C4C2×C22⋊D6111yescharacteristic
306C22×C4C2×C22⋊D6111yescharacteristic
307Q16—4882no—
308Q16—2444no—
309SD16—4882no—
310M16—1222no—
311M16—2441no—
312C2×C8—1222no—
313C2×C8—1222no—
314C2×C8—2441no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C4⋊C4—1221no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C22⋊C4—44no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—82no—
332C22⋊C4—2441no—
333C42—42no—
334C42—41no—
335C42—41no—
336C42—2441no—
337C42D4×D6111yescharacteristic
338C42D4×D6111yescharacteristic
339C42D4×D6111yescharacteristic
340C42#1175822411yescharacteristic
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23#1451974611yescharacteristic
347Q8—22no—
348Q8—22no—
349Q8—41no—
350Q8—41no—
351Q8—42no—
352Q8—81no—
353Q8—2442no—
354Q8—2442no—
355D4—42no—
356D4—81no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4#1451974511yescharacteristic
375C2×C4#1451974511yescharacteristic
376C2×C4#1451974811yescharacteristic
377C2×C4#1451974911yescharacteristic
378C2×C4#1451974911yescharacteristic
379C2×C4#1451977311yescharacteristic
380C8—2442no—
381C8—2442no—
382C8—2442no—
383C22—21no—
384C22—21no—
385C22—82no—
386C22—82no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22#1683967011yescharacteristic
391C22#1683994011yescharacteristic
392C22#1683994511yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—4881no—
403C4—4881no—
404C4#1683993912yes—
405C2—21no—
406C2—21no—
407C2—161no—
408C2#2144684811yescharacteristic
409C2#2144686611yescharacteristic
410C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2592C2
socleC22×C61#392
prime radicalC22×D4×C61D4
hypercenter#2592D61
hypocenterC61#10010
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2592

Complements

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

NH ≅ G/Ntype
2. #21455949C2semidirect
3. #21446762C2semidirect
5. #21444266C2semidirect
6. #21443675C2semidirect
238. C61#10010semidirect

Maximal subgroups

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

subgroupindex
#214568222
#214559492
#214467622
#214444172
#214442662
#214436752
C61×#25922
#1001061