Finite Group Databasebeta

← #29568585

Subgroups — #29568585

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#21454168C211yesmaximal, characteristic
2#21445886C211yesmaximal, characteristic
3#21445580C211yesmaximal, characteristic
4#21443645C211yesmaximal, characteristic, complemented
5#21443644C211yesmaximal, characteristic, complemented
6#21443644C211yesmaximal, characteristic, complemented
7C61×#1272C211yesmaximal, characteristic, complemented
8C2×#14519775C411yescharacteristic
9C2×#14519757C411yescharacteristic
10C2×#14519753C411yescharacteristic
11C2×C42×D61C2211yescharacteristic
12C2×#14519689—21no—
13C2×#14519689—21no—
14C2×#14519687—21no—
15#16839668C2211yescharacteristic
16#16839659—21no—
17#16839659—21no—
18#16839631C411yescharacteristic
19#16839621—21no—
20#16839621—21no—
21#16839517C2211yescharacteristic
22#16839517C2211yescharacteristic
23#16839516—21no—
24C61×#391C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×Dic122C2×C411yescharacteristic
28C22×C4×D61C2×C411yescharacteristic
29C22×C4×D61D411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×#92418—21no—
32C22×#92418—21no—
33#14519775—42no—
34#14519757—42no—
35#14519753—42no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48C42×D61—42no—
49#14519689—24no—
50#14519689—42no—
51#14519687—42no—
52#14519684—21no—
53#14519684—21no—
54C22×Q8×C61C2×C411yescharacteristic
55C22×C8×C61—21no—
56C22×C8×C61—21no—
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C42×C61C2311yescharacteristic
59#95×C61—21no—
60#95×C61—21no—
61C23×D61C22⋊C411yescharacteristic
62C2×Q8×C61—21no—
63C2×Q8×C61—21no—
64C2×Q8×C61—41no—
65C2×Q8×C61—41no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C22×C411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C22⋊C411yescharacteristic
76C2×C4×D61—22no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—22no—
84C2×C4×D61—22no—
85C2×C4×D61—41no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×Dic122—21no—
89C2×Dic122—21no—
90C2×Dic122—41no—
91C2×Dic122—41no—
92C2×C8×C61—22no—
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C2×C8×C61—41no—
96C2×C8×C61—41no—
97C2×C8×C61—41no—
98C4⋊C4×C61—41no—
99C42×C61—41no—
100#11758198—42no—
101#11758198—42no—
102C4⋊Dic61—42no—
103C4⋊Dic61—41no—
104#11758196—42no—
105#11758196—42no—
106#11758196—42no—
107#11758196—42no—
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#10012yes—
126Q8×C61—81no—
127Q8×C61—81no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61#13111yescharacteristic
141C2×C4×C61#13111yescharacteristic
142C2×C4×C61#13711yescharacteristic
143C2×C4×C61#13711yescharacteristic
144C2×C4×C61#13711yescharacteristic
145C2×C4×C61#13711yescharacteristic
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61#9912yes—
160C2×Dic61#10412yes—
161C4×D61—42no—
162C4×D61—42no—
163C4×D61—42no—
164C4×D61—42no—
165C4×D61—42no—
166C4×D61—42no—
167Dic122—81no—
168Dic122—81no—
169C8×C61—42no—
170C8×C61—81no—
171#92418—42no—
172#92418—81no—
173#7707—611nomaximal, Sylow
174C22×C61#40811yescharacteristic
175C22×C61C2×#9911yescharacteristic
176C22×C61C2×#10011yescharacteristic
177C22×C61#41911yescharacteristic
178C22×C61#41911yescharacteristic
179C22×C61C2×#10411yescharacteristic
180C22×C61#42111yescharacteristic
181C2×D61—22no—
182C2×D61—22no—
183C2×D61—22no—
184C2×D61—41no—
185C2×D61—41no—
186C2×D61—41no—
187C2×D61—41no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—81no—
195C4×C61—81no—
196Dic61—22no—
197Dic61—22no—
198Dic61—41no—
199Dic61—81no—
200Dic61—81no—
201#2857—611no—
202#1817—611no—
203#1749—611no—
204#1272—611no—
205#1272—611no—
206#1272—611no—
207#1272D6111yescharacteristic, complemented
208C2×C61#147111yescharacteristic
209C2×C61#148511yescharacteristic
210C2×C61#149711yescharacteristic
211C2×C61#150811yescharacteristic
212C2×C61#151111yescharacteristic
213C2×C61#153011yescharacteristic
214C2×C61#155811yescharacteristic
215D61—42no—
216C2×C4⋊Q8—611no—
217C2×C22⋊Q8—611no—
218C2×C22⋊Q8—611no—
219C22×C42—611no—
220C2×C22⋊C8—1221no—
221C2×C22⋊C8—1221no—
222C2×#97—1221no—
223#391—611no—
224#391C2×D6111yescharacteristic
225#389—1221no—
226#389—1221no—
227#384—611no—
228#380—1221no—
229#380—1221no—
230#336—611no—
231#336—611no—
232#336—1221no—
233#336C2×D6111yescharacteristic
234#336C2×D6111yescharacteristic
235C61#770711yescharacteristic, complemented, Sylow
236C22×Q8—611no—
237C22×Q8C4×D6111yescharacteristic
238C23×C4—611no—
239C23×C4—611no—
240C23×C4—611no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—1221no—
244C22×C8—1221no—
245C4⋊Q8—2442no—
246C22⋊Q8—2442no—
247C22⋊Q8—2442no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4C4×D6111yescharacteristic
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C42—611no—
259C2×C42—611no—
260C2×C42—611no—
261C2×C42—2442no—
262C2×C42C22×D6111yescharacteristic
263C22⋊C8—1224no—
264C22⋊C8—2442no—
265#97—2442no—
266#95—21no—
267#95—21no—
268#95—1221no—
269#95—1221no—
270C24—611no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1221no—
276C2×Q8—1221no—
277C2×Q8—2441no—
278C2×Q8—2441no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—1222no—
285C22×C4—1222no—
286C22×C4—1222no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—2441no—
296C22×C4—2441no—
297C22×C4—2441no—
298C22×C4C2×C4×D6111yescharacteristic
299C22×C4D4×D6111yescharacteristic
300C22×C4D4×D6111yescharacteristic
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—1222no—
308C2×C8—2441no—
309C2×C8—2441no—
310C2×C8—2441no—
311C2×C8—2441no—
312C2×C8—2441no—
313C4⋊C4—41no—
314C4⋊C4—2441no—
315C4⋊C4—2442no—
316C4⋊C4—2442no—
317C4⋊C4—2442no—
318C4⋊C4—2442no—
319C4⋊C4—2442no—
320C4⋊C4—2442no—
321C22⋊C4—2442no—
322C22⋊C4—2442no—
323C42—41no—
324C42—1222no—
325C42—1222no—
326C42—2441no—
327C23—612no—
328C23—1221no—
329C23—1221no—
330C23—1221no—
331C23—1221no—
332C23—1221no—
333C23—1221no—
334C23C22⋊C4×D6111yescharacteristic
335Q8—81no—
336Q8—81no—
337Q8—4881no—
338Q8—4881no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—612no—
352C2×C4—612no—
353C2×C4—1221no—
354C2×C4—1221no—
355C2×C4—1221no—
356C2×C4—1221no—
357C2×C4—1221no—
358C2×C4—1221no—
359C2×C4—1221no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4—2442no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4#1451978311yescharacteristic
373C2×C4#1451978511yescharacteristic
374C2×C4#1451980611yescharacteristic
375C2×C4#1451980611yescharacteristic
376C2×C4#1451980911yescharacteristic
377C2×C4#1451980911yescharacteristic
378C8—42no—
379C8—81no—
380C8—4881no—
381C8—2442no—
382C22—1222no—
383C22—1222no—
384C22—1222no—
385C22—2441no—
386C22—2441no—
387C22—2441no—
388C22—2441no—
389C22#1683972711yescharacteristic
390C22#99×D6111yescharacteristic
391C22#100×D6111yescharacteristic
392C22#1683976911yescharacteristic
393C22#1683976911yescharacteristic
394C22#104×D6111yescharacteristic
395C22#1683977711yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—81no—
404C4—1222no—
405C4—1222no—
406C4—4881no—
407C4—4881no—
408C4—2441no—
409C2—2442no—
410C2#2144417711yescharacteristic
411C2#2144423811yescharacteristic
412C2#2144429011yescharacteristic
413C2#2144433711yescharacteristic
414C2#2144435011yescharacteristic
415C2#2144442811yescharacteristic
416C2#2144454311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1272C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1272D61
hypocenterC61#7707
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #21443645C2semidirect
5. #21443644C2semidirect
6. #21443644C2semidirect
7. C61×#1272C2semidirect
207. #1272D61semidirect
235. C61#7707semidirect

Maximal subgroups

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

subgroupindex
#214541682
#214458862
#214455802
#214436452
#214436442
C61×#12722
#770761