Finite Group Databasebeta

← #29652853

Subgroups — #29652853

Order 15616.

1198 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 183 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#21455906C212yesmaximal, complemented
2#21449102C211yesmaximal, characteristic, complemented
3#21444856C214yesmaximal, complemented
4#21444754C214yesmaximal, complemented
5#21444327C212yesmaximal, complemented
6#21444163C211yesmaximal, characteristic, complemented
7C61×#3497C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16840078C2214yes—
10#16840049—22no—
11#16839954—22no—
12#16839895C2214yes—
13#16839872—24no—
14#16839819C2212yes—
15#16839701C2212yes—
16#16839688—21no—
17#16839679C2212yes—
18#16839673—21no—
19C4×#11758198—22no—
20#16839608C2211yescharacteristic
21#16839496C2218yes—
22#16839488C2214yes—
23C2×2−1+4×C61—21no—
24C61×#550—22no—
25D4×Q8×C61—22no—
26C61×#548—24no—
27C61×#537—21no—
28C61×#537C2214yes—
29C61×#532—22no—
30C61×#532C2212yes—
31C2×C4×Q8×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic
33C22×C4×D61—21no—
34C2×#11758199—24no—
35C4×D244—22no—
36C4×D244—22no—
37C2×C4⋊Dic61C2311yescharacteristic
38C2×#11758198—21no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×C4×Dic61—21no—
42#14519708—24no—
43C4⋊#92418—22no—
44#14519698—24no—
45#14519697—24no—
46C22⋊#92418C2314yes—
47#14519688—22no—
48#14519684—22no—
49#14519684—21no—
50#14519684—22no—
51#14519684C2312yes—
522−1+4×C61—48no—
53C2×#41×C61—24no—
54C2×#41×C61—24no—
55C2×#41×C61—22no—
56C22×Q8×C61—24no—
57C22×Q8×C61D411yescharacteristic
58C4⋊Q8×C61—22no—
59C4⋊Q8×C61D412yes—
60#127×C61—22no—
61#127×C61D412yes—
62#124×C61—24no—
63#124×C61—22no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61—28no—
66#121×C61—24no—
67#121×C61—28no—
68#121×C61—22no—
69#121×C61—24no—
70#121×C61C2314yes—
71C4×Q8×C61—22no—
72C4×Q8×C61—22no—
73C4×D4×C61—24no—
74C4×D4×C61—24no—
75C4×D4×C61—22no—
76C4×D4×C61D414yes—
77C2×C4⋊C4×C61D411yescharacteristic
78C2×C4⋊C4×C61C2312yes—
79C2×C42×C61D412yes—
80C2×C42×C61C2311yescharacteristic
81C23×D61—21no—
82#41×C61—48no—
83#41×C61—48no—
84#41×C61—416no—
85#41×C61—48no—
86C2×Q8×C61—24no—
87C2×Q8×C61—22no—
88C2×Q8×C61—22no—
89C2×Q8×C61—24no—
90C2×Q8×C61—416no—
91C2×D4×C61—24no—
92C2×D4×C61—24no—
93C2×D4×C61—24no—
94C2×D4×C61—22no—
95C2×D4×C61—22no—
96C2×D4×C61C2×D414yes—
97C22×C4×C61—28no—
98C22×C4×C61—24no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2411yescharacteristic
104C22×Dic61—21no—
105C22×Dic61—22no—
106C2×D244—22no—
107C2×D244—42no—
108C2×C4×D61—44no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—24no—
115C4⋊C4×C61C2×D414yes—
116C22⋊C4×C61—28no—
117C22⋊C4×C61—24no—
118C22⋊C4×C61—24no—
119C42×C61—22no—
120C42×C61—22no—
121C42×C61C2×D412yes—
122C42×C61C2×D412yes—
123#11758199—48no—
124#11758198—44no—
125#11758198—42no—
126#11758198—44no—
127C4⋊Dic61—22no—
128C4×Dic61—42no—
129C2×#92418—24no—
130C23×C61—24no—
131C23×C61C22×D411yescharacteristic
132C22×D61—22no—
133C22×D61—42no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—42no—
137Q8×C61—28no—
138Q8×C61—44no—
139Q8×C61—48no—
140D4×C61—44no—
141D4×C61—44no—
142D4×C61—48no—
143D4×C61—48no—
144D4×C61—44no—
145D4×C61—84no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—24no—
152C2×C4×C61—41no—
153C2×C4×C61—48no—
154C2×C4×C61—416no—
155C2×C4×C61—42no—
156C2×C4×C61#12012yes—
157C2×C4×C61#12011yescharacteristic
158C2×C4×C61#12011yescharacteristic
159C2×C4×C61AGL(1,ℤ/8)14yes—
160C2×C4×C61C22×D412yes—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—41no—
165C2×Dic61—42no—
166D244—24no—
167D244—42no—
168C4×D61—82no—
169C4×D61—82no—
170#92418—44no—
171#18186—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—44no—
175C22×C61—44no—
176C22×C61—44no—
177C22×C61C2×#12011yescharacteristic
178C22×C61C2×AGL(1,ℤ/8)12yes—
179C2×D61—44no—
180C2×D61—42no—
181C2×D61—82no—
182C2×D61—81no—
183C2×D61—81no—
184C4×C61—22no—
185C4×C61—24no—
186C4×C61—21no—
187C4×C61—22no—
188C4×C61—44no—
189C4×C61—42no—
190C4×C61#45412yes—
191Dic61—82no—
192Dic61—81no—
193#3497D6111yescharacteristic, complemented
194#3109—612no—
195#2398—611no—
196#1616—614no—
197#1598—614no—
198#1505—612no—
199#1467—611no—
200C2×C61—22no—
201C2×C61—84no—
202C2×C61#300311yescharacteristic
203C2×C61C2×#45411yescharacteristic
204C2×C61#302511yescharacteristic
205D61—82no—
206C2×2−1+4—21no—
207#550—22no—
208D4×Q8—22no—
209#548—24no—
210#537—21no—
211#537C2×D6114yes—
212#532—22no—
213#532C2×D6112yes—
214C2×C4×Q8C2×D6111yescharacteristic
215C2×C4×D4—611no—
216#481—614no—
217#476—1222no—
218#459—1222no—
219#447—614no—
220#442—1224no—
221#432—612no—
222#400—612no—
223#397—1221no—
224#394—612no—
225#393—1221no—
226C4×C22⋊C4—1222no—
227#376—611no—
228#327—618no—
229#324—614no—
230C61#1818611yescharacteristic, complemented, Sylow
2312−1+4—48no—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—22no—
235C22×Q8—24no—
236C22×Q8C22⋊D6111yescharacteristic
237C22×D4—611no—
238C23×C4—1221no—
239C2×D8—1224no—
240C4⋊Q8—22no—
241C4⋊Q8C22⋊D6112yes—
242#127—22no—
243#127C22⋊D6112yes—
244#124—24no—
245#124—22no—
246C22⋊Q8—24no—
247C22⋊Q8—28no—
248#121—24no—
249#121—28no—
250#121—22no—
251#121—24no—
252#121C22×D6114yes—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—22no—
258C4×D4—1222no—
259C4×D4—1222no—
260C4×D4C22⋊D6114yes—
261C2×C4⋊C4—611no—
262C2×C4⋊C4C22⋊D6111yescharacteristic
263C2×C4⋊C4C22×D6112yes—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1222no—
266C2×C22⋊C4—1222no—
267C2×C42—1221no—
268C2×C42C22⋊D6112yes—
269C2×C42C22×D6111yescharacteristic
270#106—1224no—
271C4⋊C8—1222no—
272#102—1224no—
273#102—1224no—
274C22⋊C8—614no—
275#97—1222no—
276#95—612no—
277#95—1221no—
278#95—1222no—
279#95—1222no—
280C24—1221no—
281#41—48no—
282#41—48no—
283#41—416no—
284#41—48no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—416no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—1222no—
296C2×D4—2442no—
297C2×D4C2×C22⋊D6114yes—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—1222no—
301C22×C4—1221no—
302C22×C4—2444no—
303C22×C4—2441no—
304C22×C4—2441no—
305C22×C4C2×C22⋊D6111yescharacteristic
306C22×C4C2×C22⋊D6111yescharacteristic
307C22×C4C2×C22⋊D6112yes—
308C22×C4C2×C22⋊D6112yes—
309C22×C4C23×D6111yescharacteristic
310D8—2448no—
311C2×C8—1224no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—1222no—
317C4⋊C4C2×C22⋊D6114yes—
318C22⋊C4—28no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—2444no—
322C22⋊C4—2442no—
323C22⋊C4—2444no—
324C42—22no—
325C42—22no—
326C42—2442no—
327C42C2×C22⋊D6112yes—
328C42C2×C22⋊D6112yes—
329C23—24no—
330C23—1222no—
331C23—2442no—
332C23—2441no—
333C23—2441no—
334C23—2442no—
335C23C22×C22⋊D6111yescharacteristic
336Q8—28no—
337Q8—44no—
338Q8—48no—
339D4—44no—
340D4—44no—
341D4—48no—
342D4—48no—
343D4—44no—
344D4—84no—
345D4—1224no—
346D4—2442no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—41no—
354C2×C4—48no—
355C2×C4—416no—
356C2×C4—42no—
357C2×C4—4882no—
358C2×C4—4882no—
359C2×C4—2442no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4—2441no—
363C2×C4—2442no—
364C2×C4#1451974611yescharacteristic
365C2×C4#1451974612yes—
366C2×C4#1451974611yescharacteristic
367C2×C4#1451980314yes—
368C2×C4C22×C22⋊D6112yes—
369C8—2444no—
370C22—21no—
371C22—21no—
372C22—44no—
373C22—44no—
374C22—44no—
375C22—4882no—
376C22—4881no—
377C22—4881no—
378C22—2444no—
379C22—2442no—
380C22C2×#1451974611yescharacteristic
381C22C2×#1451980312yes—
382C4—21no—
383C4—22no—
384C4—22no—
385C4—24no—
386C4—42no—
387C4—44no—
388C4—4882no—
389C4—4881no—
390C4#1683993512yes—
391C2—22no—
392C2—84no—
393C2—4882no—
394C2#2145504611yescharacteristic
395C2C2×#1683993511yescharacteristic
396C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3497C2
socleC22×C61C2×#120
hypercenter#3497D61
hypocenterC61#18186
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21455906C2semidirect
2. #21449102C2semidirect
3. #21444856C2semidirect
4. #21444754C2semidirect
5. #21444327C2semidirect
6. #21444163C2semidirect
7. C61×#3497C2semidirect
193. #3497D61semidirect
230. C61#18186semidirect

Maximal subgroups

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

subgroupindex
#214559062
#214491022
#214448562
#214447542
#214443272
#214441632
C61×#34972
#1818661