Finite Group Databasebeta

← #29568589

Subgroups — #29568589

Order 15616.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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#21454170C211yesmaximal, characteristic
2#21445886C211yesmaximal, characteristic
3#21445580C211yesmaximal, characteristic
4#21443643C211yesmaximal, characteristic, complemented
5#21443642C211yesmaximal, characteristic, complemented
6#21443641C211yesmaximal, characteristic, complemented
7C61×#1271C211yesmaximal, characteristic, complemented
8C2×#14519776C411yescharacteristic
9C2×#14519755—21no—
10C2×C42×D61C2211yescharacteristic
11C2×#14519689—21no—
12C2×#14519689—21no—
13C2×#14519687—21no—
14#16839666—21no—
15#16839660C2211yescharacteristic
16#16839657—21no—
17#16839631C411yescharacteristic
18#16839621—21no—
19#16839619C411yescharacteristic
20#16839619C411yescharacteristic
21#16839517C2211yescharacteristic
22#16839517C2211yescharacteristic
23#16839516—21no—
24C61×#389C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×Dic122—21no—
28C22×C4×D61C2×C411yescharacteristic
29C22×C4×D61D411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×#92418—21no—
32C22×#92418—21no—
33#14519776—42no—
34#14519755—42no—
35#14519755—42no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C42×D61—42no—
48#14519689—24no—
49#14519689—42no—
50#14519687—42no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54C22×C8×C61—21no—
55C22×C8×C61—21no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C42×C61C2311yescharacteristic
60#95×C61—21no—
61C23×D61C22⋊C411yescharacteristic
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C22⋊C411yescharacteristic
72C2×C4×D61—22no—
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—22no—
80C2×C4×D61—22no—
81C2×C4×D61—41no—
82C2×C4×D61—41no—
83C2×C4×D61—41no—
84C2×Dic122—42no—
85C2×Dic122—42no—
86C2×Dic122—42no—
87C2×C8×C61—22no—
88C2×C8×C61—41no—
89C2×C8×C61—41no—
90C2×C8×C61—41no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C42×C61—41no—
99#11758198—42no—
100#11758198—42no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—41no—
104#11758196—42no—
105#11758196—42no—
106#11758196—41no—
107#11758196—41no—
108#11758196—42no—
109C4×Dic61—22no—
110C4×Dic61—22no—
111C4×Dic61—41no—
112C2×#92418—22no—
113C2×#92418—41no—
114C2×#92418—41no—
115C2×#92418—41no—
116C2×#92418—41no—
117C2×#92418—41no—
118C23×C61C2×C22⋊C411yescharacteristic
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61#10112yes—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61#13111yescharacteristic
139C2×C4×C61#13111yescharacteristic
140C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C61#13711yescharacteristic
143C2×C4×C61#13711yescharacteristic
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61#9912yes—
158C2×Dic61#10412yes—
159C4×D61—42no—
160C4×D61—42no—
161C4×D61—42no—
162C4×D61—42no—
163C4×D61—42no—
164C4×D61—42no—
165Dic122—84no—
166C8×C61—42no—
167C8×C61—81no—
168#92418—42no—
169#92418—81no—
170#7708—611nomaximal, Sylow
171C22×C61#40811yescharacteristic
172C22×C61C2×#9911yescharacteristic
173C22×C61C2×#10111yescharacteristic
174C22×C61#41811yescharacteristic
175C22×C61#41911yescharacteristic
176C22×C61C2×#10411yescharacteristic
177C22×C61#42111yescharacteristic
178C2×D61—22no—
179C2×D61—22no—
180C2×D61—22no—
181C2×D61—41no—
182C2×D61—41no—
183C2×D61—41no—
184C2×D61—41no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—81no—
192C4×C61—81no—
193Dic61—22no—
194Dic61—22no—
195Dic61—41no—
196Dic61—81no—
197Dic61—81no—
198#2857—611no—
199#1817—611no—
200#1749—611no—
201#1271—611no—
202#1271—611no—
203#1271—611no—
204#1271D6111yescharacteristic, complemented
205C2×C61#147211yescharacteristic
206C2×C61#148411yescharacteristic
207C2×C61#149711yescharacteristic
208C2×C61#150811yescharacteristic
209C2×C61#151011yescharacteristic
210C2×C61#153111yescharacteristic
211C2×C61#155611yescharacteristic
212D61—42no—
213C2×C4⋊Q8—611no—
214C2×C22⋊Q8—1221no—
215C22×C42—611no—
216C2×C22⋊C8—1221no—
217C2×C22⋊C8—1221no—
218C2×#97—1221no—
219#391—1221no—
220#389—611no—
221#389—1221no—
222#389C2×D6111yescharacteristic
223#384—611no—
224#380—611no—
225#380—611no—
226#380—1221no—
227#336—611no—
228#336—611no—
229#336—1221no—
230#336C2×D6111yescharacteristic
231#336C2×D6111yescharacteristic
232C61#770811yescharacteristic, complemented, Sylow
233C22×Q8—1221no—
234C23×C4—611no—
235C23×C4—611no—
236C23×C4—611no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—1221no—
240C22×C8—1221no—
241C4⋊Q8—2442no—
242C22⋊Q8—2442no—
243C22⋊Q8—2442no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—611no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4C4×D6111yescharacteristic
252C2×C4⋊C4C4×D6111yescharacteristic
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C42—611no—
256C2×C42—611no—
257C2×C42—611no—
258C2×C42—2442no—
259C2×C42C22×D6111yescharacteristic
260C22⋊C8—1224no—
261C22⋊C8—2442no—
262#97—2442no—
263#95—21no—
264#95—1221no—
265#95—1221no—
266#95—1221no—
267C24—611no—
268C2×Q8—2442no—
269C2×Q8—2442no—
270C2×Q8—2442no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—611no—
274C22×C4—611no—
275C22×C4—611no—
276C22×C4—1222no—
277C22×C4—1222no—
278C22×C4—1222no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—2441no—
288C22×C4—2441no—
289C22×C4—2441no—
290C22×C4C2×C4×D6111yescharacteristic
291C22×C4D4×D6111yescharacteristic
292C22×C4D4×D6111yescharacteristic
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—1222no—
300C2×C8—2441no—
301C2×C8—2441no—
302C2×C8—2441no—
303C2×C8—2441no—
304C2×C8—2441no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—1221no—
311C4⋊C4—1221no—
312C4⋊C4—2441no—
313C4⋊C4—2442no—
314C4⋊C4—2442no—
315C4⋊C4—2442no—
316C4⋊C4—2441no—
317C4⋊C4—2441no—
318C22⋊C4—2442no—
319C22⋊C4—2442no—
320C42—41no—
321C42—1222no—
322C42—1222no—
323C42—2441no—
324C23—612no—
325C23—1221no—
326C23—1221no—
327C23—1221no—
328C23—1221no—
329C23—1221no—
330C23—1221no—
331C23C22⋊C4×D6111yescharacteristic
332Q8—4884no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—612no—
346C2×C4—612no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4—2442no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4#1451978311yescharacteristic
367C2×C4#1451978511yescharacteristic
368C2×C4#1451980111yescharacteristic
369C2×C4#1451980411yescharacteristic
370C2×C4#1451980611yescharacteristic
371C2×C4#1451980911yescharacteristic
372C8—42no—
373C8—81no—
374C8—4881no—
375C8—2442no—
376C22—1222no—
377C22—1222no—
378C22—1222no—
379C22—2441no—
380C22—2441no—
381C22—2441no—
382C22—2441no—
383C22#1683972711yescharacteristic
384C22#99×D6111yescharacteristic
385C22#101×D6111yescharacteristic
386C22#1683976511yescharacteristic
387C22#1683976911yescharacteristic
388C22#104×D6111yescharacteristic
389C22#1683977711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—81no—
398C4—1222no—
399C4—1222no—
400C4—4881no—
401C4—4881no—
402C4—2441no—
403C2—2442no—
404C2#2144418111yescharacteristic
405C2#2144423411yescharacteristic
406C2#2144429011yescharacteristic
407C2#2144433711yescharacteristic
408C2#2144434611yescharacteristic
409C2#2144443211yescharacteristic
410C2#2144453411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1271C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1271D61
hypocenterC61#7708
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #21443643C2semidirect
5. #21443642C2semidirect
6. #21443641C2semidirect
7. C61×#1271C2semidirect
204. #1271D61semidirect
232. C61#7708semidirect

Maximal subgroups

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

subgroupindex
#214541702
#214458862
#214455802
#214436432
#214436422
#214436412
C61×#12712
#770861