Finite Group Databasebeta

← #29580931

Subgroups — #29580931

Order 15616.

512 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 81 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#21456205C211yesmaximal, characteristic
2#21455949C211yesmaximal, characteristic, complemented
3#21444443C211yesmaximal, characteristic
4#21444369C211yesmaximal, characteristic, complemented
5#21444266C211yesmaximal, characteristic, complemented
6#21443683C211yesmaximal, characteristic, complemented
7C61×#2632C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840121—21no—
10#16840121—21no—
11#16840116—22no—
12#16840092—21no—
13#16840047—21no—
14#16839972—21no—
15#16839918—21no—
16#16839867—22no—
17#16839819C2211yescharacteristic
18#16839819C2211yescharacteristic
19#16839771—21no—
20#16839767—21no—
21C4×#11758194C2211yescharacteristic
22#16839502—22no—
23#16839498—22no—
24Q8⋊#92418—22no—
25C2×#124×C61C2211yescharacteristic
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#399—21no—
28C61×#395—21no—
29C61×#394—21no—
30C61×#393—21no—
31C61×#388—22no—
32C61×#383C2211yescharacteristic
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#14519708—22no—
44C4⋊#92418—21no—
45C4⋊#92418—21no—
46#14519701—22no—
47#14519701—22no—
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#14519688D412yes—
57C4×#92418D412yes—
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—22no—
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×C61C2311yescharacteristic
78#95×C61—22no—
79#95×C61—21no—
80#95×C61—21no—
81#95×C61D412yes—
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—22no—
94C22×C4×C61—21no—
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—41no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—82no—
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—41no—
113C42×C61—41no—
114C42×C61C2×D411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C22⋊Dic61—41no—
119#11758202—44no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124#11758196—41no—
125#11758196—41no—
126C4×Dic61—41no—
127#11758194—42no—
128C2×#92418—22no—
129C2×#92418—22no—
130C2×#92418—41no—
131C23×C61—42no—
132C23×C61—42no—
133C23×C61—41no—
134C23×C61—42no—
135C23×C61#12011yescharacteristic
136Q8×C61—22no—
137Q8×C61—22no—
138Q8×C61—41no—
139Q8×C61—41no—
140Q8×C61—42no—
141Q8×C61—81no—
142D4×C61—42no—
143D4×C61—81no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
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×Dic61—41no—
164C2×Dic61—41no—
165Dic122—42no—
166Dic122—81no—
167#92418—42no—
168#92418—42no—
169#92418—81no—
170#10024—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—84no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61#39211yescharacteristic
178C22×C61#45512yes—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—42no—
186C4×C61—82no—
187C4×C61—81no—
188C4×C61—81no—
189Dic61—81no—
190Dic61—81no—
191#3140—611no—
192#3113—611no—
193#2632D6111yescharacteristic
194#1533—611no—
195#1515—611no—
196#1491—611no—
197#1288—611no—
198C2×C61—22no—
199C2×C61—161no—
200C2×C61#200411yescharacteristic
201C2×C61#200911yescharacteristic
202C2×C61#201011yescharacteristic
203#533—611no—
204C2×#124C2×D6111yescharacteristic
205C2×C4×Q8C2×D6111yescharacteristic
206#489—1221no—
207#489—1221no—
208#488—1222no—
209#483—1221no—
210#475—1221no—
211#462—1221no—
212#451—1221no—
213#441—1222no—
214#432—611no—
215#432—611no—
216#419—1221no—
217#418—1221no—
218C4×M16—611no—
219#399—21no—
220#395—21no—
221#394—21no—
222#393—21no—
223#388—22no—
224#383C2×D6111yescharacteristic
225#329—1222no—
226#328—1222no—
227Q8⋊C8—1222no—
228C61#1002411yescharacteristic, complemented, Sylow
229C22×Q8C22⋊D6111yescharacteristic
230C22×Q8C22⋊D6111yescharacteristic
231C22×D4C22⋊D6111yescharacteristic
232C2×Q16—1222no—
233C2×M16—1221no—
234C4⋊Q8—611no—
235C4⋊Q8—611no—
236#125—1221no—
237#124—21no—
238#124—21no—
239#124—21no—
240#124—21no—
241C22⋊Q8—1221no—
242C22⋊Q8—1221no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—1221no—
248C4×Q8—1221no—
249#117—1221no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4C22⋊D6111yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C42—21no—
258C2×C42C22×D6111yescharacteristic
259#106—1222no—
260C4⋊C8—1221no—
261C4⋊C8—1221no—
262#103—1222no—
263#103—1222no—
264#103—1222no—
265#103—1222no—
266#102—1222no—
267#102—1222no—
268C22⋊C8—1221no—
269C22⋊C8—1221no—
270#97—612no—
271#97—1221no—
272#97—1221no—
273C4×C8—612no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95C22⋊D6112yes—
278C24—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—1221no—
286C2×Q8—1221no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—42no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4C2×C22⋊D6111yescharacteristic
295C22×C4C2×C22⋊D6111yescharacteristic
296C22×C4C2×C22⋊D6111yescharacteristic
297Q16—2444no—
298M16—2442no—
299C2×C8—1222no—
300C2×C8—1222no—
301C2×C8—2441no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—82no—
307C4⋊C4—1221no—
308C4⋊C4—1221no—
309C4⋊C4—1221no—
310C4⋊C4—1221no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C22⋊C4—44no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—82no—
319C22⋊C4—2441no—
320C42—41no—
321C42—41no—
322C42—2441no—
323C42D4×D6111yescharacteristic
324C42D4×D6111yescharacteristic
325C42D4×D6111yescharacteristic
326C42#1175822411yescharacteristic
327C23—42no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23#1451974611yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—42no—
337Q8—81no—
338Q8—4881no—
339Q8—2442no—
340D4—42no—
341D4—81no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4#1451974511yescharacteristic
358C2×C4#1451974511yescharacteristic
359C2×C4#1451974811yescharacteristic
360C2×C4#1451974911yescharacteristic
361C2×C4#1451974911yescharacteristic
362C2×C4#1451977311yescharacteristic
363C8—4881no—
364C8—2442no—
365C8—2442no—
366C22—21no—
367C22—21no—
368C22—84no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#1683967011yescharacteristic
373C22#1683994012yes—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—4881no—
385C4—4881no—
386C2—22no—
387C2—161no—
388C2#2144684611yescharacteristic
389C2#2144687911yescharacteristic
390C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2632C2
socleC22×C61#392
prime radicalC22×D4×C61D4
hypercenter#2632D61
hypocenterC61#10024
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2632

Complements

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

NH ≅ G/Ntype
2. #21455949C2semidirect
4. #21444369C2semidirect
5. #21444266C2semidirect
6. #21443683C2semidirect
228. C61#10024semidirect

Maximal subgroups

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

subgroupindex
#214562052
#214559492
#214444432
#214443692
#214442662
#214436832
C61×#26322
#1002461