Finite Group Databasebeta

← #29568923

Subgroups — #29568923

Order 15616.

460 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 83 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#21452749C211yesmaximal, characteristic
2#21447149C211yesmaximal, characteristic, complemented
3#21447062C211yesmaximal, characteristic, complemented
4#21443643C211yesmaximal, characteristic, complemented
5#21443643C211yesmaximal, characteristic, complemented
6#21443640C211yesmaximal, characteristic
7C61×#2023C211yesmaximal, characteristic
8C2×#14519758—21no—
9C2×#14519708—21no—
10C2×#14519701—21no—
11C2×#14519701—21no—
12#16839692—21no—
13#16839687—21no—
14#16839687—21no—
15#16839660C2211yescharacteristic
16#16839660C2211yescharacteristic
17#16839637—21no—
18#16839637—21no—
19#16839524—21no—
20#16839517C2211yescharacteristic
21#16839517C2211yescharacteristic
22C2×C4⋊Q8×C61C2211yescharacteristic
23C2×#102×C61—21no—
24C61×#390C2211yescharacteristic
25C61×#340—21no—
26C61×#336C2211yescharacteristic
27C23×Dic61—21no—
28C22×#92418—21no—
29C22×#92418—21no—
30#14519758—42no—
31#14519758—42no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61D411yescharacteristic
34C2×C4⋊Dic61D411yescharacteristic
35C2×C4⋊Dic61D411yescharacteristic
36C2×C4⋊Dic61D411yescharacteristic
37C2×#11758196—21no—
38C2×#11758196—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42#14519708—42no—
43#14519701—42no—
44#14519701—42no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50C22×Q8×C61—21no—
51C22×D4×C61D411yescharacteristic
52C22×C8×C61—21no—
53C4⋊Q8×C61—42no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61D411yescharacteristic
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61—21no—
59C2×C42×C61C2311yescharacteristic
60#102×C61—42no—
61C24×C61—21no—
62C2×Q8×C61—22no—
63C2×Q8×C61—41no—
64C2×Q8×C61—41no—
65C2×Q8×C61—41no—
66C2×Q8×C61—41no—
67C2×Q8×C61—41no—
68C2×D4×C61—21no—
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C2×D4×C61SD1612yes—
72C22×C4×C61—21no—
73C22×C4×C61—41no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—42no—
82C22×Dic61—81no—
83C22×Dic61—81no—
84C2×C8×C61—41no—
85C2×C8×C61—41no—
86C2×C8×C61—41no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—41no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C42×C61—41no—
97C22⋊Dic61—42no—
98C22⋊Dic61—42no—
99C22⋊Dic61—42no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—41no—
105C4⋊Dic61—41no—
106C4⋊Dic61—81no—
107#11758196—41no—
108#11758196—41no—
109#11758196—41no—
110#11758196—41no—
111C2×#92418—22no—
112C2×#92418—41no—
113C2×#92418—41no—
114C2×#92418—41no—
115C2×#92418—41no—
116C2×#92418—41no—
117C23×C61—22no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61#12011yescharacteristic
125Q8×C61—42no—
126Q8×C61—42no—
127Q8×C61—81no—
128Q8×C61—81no—
129D4×C61—42no—
130D4×C61—81no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—81no—
142C2×C4×C61—81no—
143C2×C4×C61C2×SD1611yescharacteristic
144C2×C4×C61#13511yescharacteristic
145C2×C4×C61#13511yescharacteristic
146C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C61#13711yescharacteristic
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—84no—
162C8×C61—81no—
163#92418—42no—
164#92418—81no—
165#7770—611nomaximal, Sylow
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61#44911yescharacteristic
174C22×C61#45011yescharacteristic
175C22×C61#45211yescharacteristic
176C22×C61#45411yescharacteristic
177C22×C61#45511yescharacteristic
178C22×C61#45611yescharacteristic
179C22×C61B(4,𝔽2)11yescharacteristic
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—82no—
187C4×C61—81no—
188Dic61—81no—
189Dic61—81no—
190Dic61—81no—
191Dic61—81no—
192#2724—611no—
193#2058—611no—
194#2043—611no—
195#2023D6111yescharacteristic
196#1271—611no—
197#1271—611no—
198#1270—611no—
199C2×C61—82no—
200C2×C61#159711yescharacteristic
201C2×C61#160111yescharacteristic
202C2×C61#160311yescharacteristic
203C2×C61#163511yescharacteristic
204C2×C61#164011yescharacteristic
205C2×C61#165411yescharacteristic
206C2×C61#166411yescharacteristic
207C2×C4⋊Q8C2×D6111yescharacteristic
208C2×#123—1221no—
209C2×#106—1221no—
210C2×#103—1221no—
211C2×#103—1221no—
212C2×#102—21no—
213#398—1221no—
214#397—1221no—
215#397—1221no—
216#390C2×D6111yescharacteristic
217#389—611no—
218#389—611no—
219#385—1221no—
220#385—1221no—
221#340—21no—
222#340—1221no—
223#336—611no—
224#336—611no—
225#336C2×D6111yescharacteristic
226C61#777011yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×D4C22⋊D6111yescharacteristic
229C23×C4—1221no—
230C22×C8—21no—
231C22×C8—1221no—
232C22×C8—1221no—
233C4⋊Q8—42no—
234#123—2442no—
235#123—2442no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—611no—
239C2×C4⋊C4—611no—
240C2×C4⋊C4—611no—
241C2×C4⋊C4—611no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4C22⋊D6111yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1221no—
249C2×C22⋊C4—1221no—
250C2×C22⋊C4—1221no—
251C2×C42C22×D6111yescharacteristic
252#106—2442no—
253#103—2442no—
254#103—2442no—
255#102—42no—
256#95—1221no—
257#95—1221no—
258#95—1221no—
259#95—1221no—
260#95—1221no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4#1175820112yes—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—4881no—
279C22×C4—4881no—
280C22×C4—2442no—
281C22×C4D4×D6111yescharacteristic
282C22×C4D4×D6111yescharacteristic
283C22×C4C2×C22⋊D6111yescharacteristic
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—1222no—
288C2×C8—2441no—
289C2×C8—2441no—
290C2×C8—2441no—
291C2×C8—2441no—
292C2×C8—2441no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—1221no—
299C4⋊C4—1221no—
300C4⋊C4—1221no—
301C4⋊C4—1221no—
302C4⋊C4—4881no—
303C4⋊C4—2441no—
304C4⋊C4—2441no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C4⋊C4—2441no—
308C4⋊C4—2441no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—2442no—
314C22⋊C4—2442no—
315C22⋊C4—2442no—
316C42—41no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#1451974511yescharacteristic
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329D4—42no—
330D4—81no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—4884no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4C2×#1175820111yescharacteristic
357C2×C4#1451979611yescharacteristic
358C2×C4#1451979811yescharacteristic
359C2×C4#1451980111yescharacteristic
360C2×C4#1451980211yescharacteristic
361C2×C4#1451980611yescharacteristic
362C8—81no—
363C8—4881no—
364C8—2442no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#1683990411yescharacteristic
373C22#1683991211yescharacteristic
374C22#1683992111yescharacteristic
375C22#1683993311yescharacteristic
376C22#1683993911yescharacteristic
377C22#1683994311yescharacteristic
378C22#1683994811yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—82no—
386C4—81no—
387C4—4881no—
388C4—4881no—
389C4—4881no—
390C4—4881no—
391C2—82no—
392C2#2144474511yescharacteristic
393C2#2144477011yescharacteristic
394C2#2144477911yescharacteristic
395C2#2144496811yescharacteristic
396C2#2144500211yescharacteristic
397C2#2144510211yescharacteristic
398C2#2144515211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2023C2
socleC23×C61#120
prime radicalC22×D4×C61D4
hypercenter#2023D61
hypocenterC61#7770
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
2. #21447149C2semidirect
3. #21447062C2semidirect
4. #21443643C2semidirect
5. #21443643C2semidirect
226. C61#7770semidirect

Maximal subgroups

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

subgroupindex
#214527492
#214471492
#214470622
#214436432
#214436402
C61×#20232
#777061