Finite Group Databasebeta

← #29657939

Subgroups — #29657939

Order 15616.

946 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 179 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#21457197C212yesmaximal, complemented
2#21456144C211yesmaximal, characteristic, complemented
3#21444970C214yesmaximal, complemented
4#21444908C214yesmaximal, complemented
5#21444557C211yesmaximal, characteristic, complemented
6#21444270C212yesmaximal, complemented
7C61×#3548C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840119—22no—
10#16840114C2212yes—
11#16840110C2212yes—
12#16840052C2214yes—
13#16840049—22no—
14#16840013—24no—
15#16839957C2214yes—
16#16839954—22no—
17#16839786C2212yes—
18#16839756—22no—
19C2×C4×#92418C2211yescharacteristic
20#16839508C2214yes—
21D4⋊#92418C2218yes—
22C61×#562—22no—
23C61×#554—22no—
24C61×#550—22no—
25C61×#547C2214yes—
26C61×#540—22no—
27C61×#537—24no—
28C61×#533C2211yescharacteristic
29C61×#532C2212yes—
30C2×#11758222—21no—
31C2×#11758199—24no—
32C2×#11758194—22no—
33C22×#92418—21no—
34#14519777C2311yescharacteristic
35#14519774C2311yescharacteristic
36#14519765—21no—
37#14519751—22no—
38C4×D244—22no—
39#14519708—24no—
40C4⋊#92418—22no—
41C4⋊#92418C2314yes—
42#14519700—24no—
43#14519698—24no—
44#14519697—24no—
45#14519697—24no—
46C4×#92418—21no—
47C4×#92418C2312yes—
48C2×#41×C61—24no—
49C2×#41×C61—22no—
50C4⋊Q8×C61D412yes—
51C4⋊Q8×C61C2312yes—
52#127×C61—22no—
53#127×C61D412yes—
54#125×C61—22no—
55#125×C61D412yes—
56#124×C61—22no—
57#123×C61—28no—
58C22⋊Q8×C61—28no—
59C22⋊Q8×C61—22no—
60C22⋊Q8×C61—24no—
61#121×C61—28no—
62#121×C61—28no—
63#121×C61—24no—
64#121×C61—22no—
65#121×C61D414yes—
66C4×Q8×C61—22no—
67C4×D4×C61—24no—
68C4×D4×C61—22no—
69C4×D4×C61C2314yes—
70#117×C61D412yes—
71C2×C4⋊C4×C61—24no—
72C2×C42×C61C2311yescharacteristic
73#41×C61—82no—
74#41×C61—82no—
75#41×C61—84no—
76#41×C61—84no—
77C2×Q8×C61—22no—
78C2×Q8×C61—22no—
79C2×D4×C61—28no—
80C2×D4×C61—24no—
81C2×D4×C61—22no—
82C2×D4×C61—22no—
83C2×D4×C61C2×D414yes—
84C22×C4×C61—28no—
85C22×C4×C61—24no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D412yes—
88C2×C22⋊D61—41no—
89#11758222—44no—
90C2×D244—21no—
91C2×D244—22no—
92C2×C4×D61—41no—
93C2×Dic122—21no—
94C4⋊C4×C61—28no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61C2×D414yes—
99C4⋊C4×C61C2×D414yes—
100C22⋊C4×C61—28no—
101C22⋊C4×C61—22no—
102C22⋊C4×C61—24no—
103C22⋊C4×C61—24no—
104C42×C61—22no—
105C42×C61C2×D412yes—
106C42×C61C2×D411yescharacteristic
107C42×C61C2411yescharacteristic
108#11758199—84no—
109#11758198—42no—
110C4⋊Dic61—22no—
111#11758194—82no—
112C2×#92418—22no—
113C2×#92418—24no—
114C2×#92418—42no—
115C23×C61—24no—
116C23×C61#12011yescharacteristic
117C22×D61—41no—
118Q8×C61—42no—
119Q8×C61—42no—
120Q8×C61—82no—
121D4×C61—24no—
122D4×C61—24no—
123D4×C61—44no—
124D4×C61—82no—
125D4×C61—84no—
126D4×C61—88no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—24no—
131C2×C4×C61—24no—
132C2×C4×C61—44no—
133C2×C4×C61—44no—
134C2×C4×C61—88no—
135C2×C4×C61#12012yes—
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61C22×D411yescharacteristic
138C2×C4×C61C22×D412yes—
139C22⋊D61—82no—
140C2×Dic61—41no—
141D244—42no—
142D244—42no—
143D244—42no—
144C4×D61—82no—
145Dic122—42no—
146#92418—42no—
147#92418—44no—
148#18696—611nomaximal, Sylow
149C22×C61—22no—
150C22×C61—21no—
151C22×C61—44no—
152C22×C61—44no—
153C22×C61—44no—
154C22×C61C2×#12011yescharacteristic
155C2×D61—82no—
156C2×D61—81no—
157C4×C61—22no—
158C4×C61—21no—
159C4×C61—44no—
160C4×C61—44no—
161C4×C61#45412yes—
162C4×C61#57612yes—
163C4×C61#57712yes—
164Dic61—81no—
165#3548D6111yescharacteristic, complemented
166#3253—612no—
167#3131—611no—
168#1635—614no—
169#1625—614no—
170#1561—611no—
171#1492—612no—
172C2×C61—41no—
173C2×C61—84no—
174C2×C61#301212yes—
175C2×C61C2×#45411yescharacteristic
176D61—161no—
177#562—22no—
178#554—22no—
179#550—22no—
180#547C2×D6114yes—
181#540—22no—
182#537—24no—
183#533C2×D6111yescharacteristic
184#532—611no—
185#532C2×D6112yes—
186#488—1222no—
187#487—612no—
188#486—612no—
189#476—614no—
190#476—1222no—
191#469—1224no—
192#459—614no—
193#459—1222no—
194#423—612no—
195#416—1222no—
196C2×C4×C8—611no—
197#332—614no—
198D4⋊C8—618no—
199C61#1869611yescharacteristic, complemented, Sylow
200C2×#41—24no—
201C2×#41—22no—
202C2×#41—1221no—
203C2×D8—1224no—
204C2×M16—1222no—
205C22×C8—1221no—
206C4⋊Q8—611no—
207C4⋊Q8C22⋊D6112yes—
208C4⋊Q8C22×D6112yes—
209#127—22no—
210#127—611no—
211#127C22⋊D6112yes—
212#125—22no—
213#125C22⋊D6112yes—
214#124—22no—
215#124—1221no—
216#123—28no—
217C22⋊Q8—28no—
218C22⋊Q8—22no—
219C22⋊Q8—24no—
220#121—28no—
221#121—28no—
222#121—24no—
223#121—22no—
224#121—1222no—
225#121C22⋊D6114yes—
226C4×Q8—22no—
227C4×D4—24no—
228C4×D4—22no—
229C4×D4—1222no—
230C4×D4C22×D6114yes—
231#117C22⋊D6112yes—
232C2×C4⋊C4—24no—
233C2×C42C22×D6111yescharacteristic
234#106—1224no—
235C4⋊C8—614no—
236C4⋊C8—1222no—
237#103—1224no—
238#102—1224no—
239#102—1224no—
240#102—1224no—
241C4×C8—612no—
242C4×C8—1221no—
243#41—82no—
244#41—82no—
245#41—84no—
246#41—84no—
247#41—2444no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—1221no—
251C2×D4—28no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—1221no—
256C2×D4—1222no—
257C2×D4—2441no—
258C2×D4C2×C22⋊D6114yes—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—2441no—
262C22×C4C2×C22⋊D6111yescharacteristic
263C22×C4C2×C22⋊D6112yes—
264D8—4884no—
265M16—4882no—
266C2×C8—1222no—
267C2×C8—1224no—
268C2×C8—2442no—
269C4⋊C4—28no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—1222no—
274C4⋊C4C2×C22⋊D6114yes—
275C4⋊C4C2×C22⋊D6114yes—
276C22⋊C4—28no—
277C22⋊C4—22no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—2442no—
281C42—22no—
282C42C2×C22⋊D6111yescharacteristic
283C42C2×C22⋊D6112yes—
284C42C23×D6111yescharacteristic
285C23—24no—
286C23—2441no—
287C23#1451974611yescharacteristic
288Q8—42no—
289Q8—42no—
290Q8—82no—
291Q8—2442no—
292D4—24no—
293D4—24no—
294D4—44no—
295D4—82no—
296D4—84no—
297D4—88no—
298D4—4882no—
299D4—2442no—
300D4—2442no—
301D4—2442no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—88no—
310C2×C4—4882no—
311C2×C4—2441no—
312C2×C4#1451974612yes—
313C2×C4#1451974611yescharacteristic
314C2×C4C22×C22⋊D6111yescharacteristic
315C2×C4C22×C22⋊D6112yes—
316C8—2442no—
317C8—2444no—
318C22—22no—
319C22—21no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—4882no—
324C22—4881no—
325C22C2×#1451974611yescharacteristic
326C4—22no—
327C4—21no—
328C4—44no—
329C4—44no—
330C4—4881no—
331C4#1683993512yes—
332C4#1684056512yes—
333C4#1684057312yes—
334C2—41no—
335C2—84no—
336C2—9761no—
337C2#2145512312yes—
338C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3548C2
socleC22×C61C2×#120
hypercenter#3548D61
hypocenterC61#18696
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #21457197C2semidirect
2. #21456144C2semidirect
3. #21444970C2semidirect
4. #21444908C2semidirect
5. #21444557C2semidirect
6. #21444270C2semidirect
7. C61×#3548C2semidirect
165. #3548D61semidirect
199. C61#18696semidirect

Maximal subgroups

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

subgroupindex
#214571972
#214561442
#214449702
#214449082
#214445572
#214442702
C61×#35482
#1869661