Finite Group Databasebeta

← #29569022

Subgroups — #29569022

Order 15616.

436 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 85 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#21453783C211yesmaximal, characteristic
2#21447144C211yesmaximal, characteristic
3#21446919C211yesmaximal, characteristic
4#21443645C211yesmaximal, characteristic
5#21443644C211yesmaximal, characteristic
6#21443641C211yesmaximal, characteristic
7C61×#2024C211yesmaximal, characteristic
8C22×#11758202—21no—
9C2×#14519710—21no—
10C2×#14519701—21no—
11#16839695—21no—
12#16839694—21no—
13#16839683—21no—
14#16839669—21no—
15#16839668C2211yescharacteristic
16#16839660C2211yescharacteristic
17#16839659—21no—
18#16839645—21no—
19#16839524—21no—
20#16839517C2211yescharacteristic
21#16839517C2211yescharacteristic
22C2×C4⋊Q8×C61C2211yescharacteristic
23C2×#103×C61—21no—
24C61×#391C2211yescharacteristic
25C61×#340—21no—
26C61×#336C2211yescharacteristic
27C22×Dic122D411yescharacteristic
28C2×#11758202—42no—
29C2×#11758202—42no—
30C2×#11758202—42no—
31C22×#92418—21no—
32C22×#92418—21no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×C4⋊Dic61D411yescharacteristic
37C2×C4⋊Dic61D411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×C4×Dic61—21no—
42#14519710—42no—
43#14519701—42no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50C22×Q8×C61—21no—
51C22×Q8×C61D411yescharacteristic
52C22×C8×C61—21no—
53C4⋊Q8×C61—42no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61D411yescharacteristic
57C2×C42×C61C2311yescharacteristic
58#103×C61—42no—
59#95×C61—21no—
60#95×C61—21no—
61C2×Q8×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×Q8×C61—41no—
69C2×Q8×C61—41no—
70C2×Q8×C61Q1612yescomplemented
71C22×C4×C61—21no—
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—
81C2×Dic122—21no—
82C2×Dic122—21no—
83C2×Dic122—41no—
84C2×Dic122—41no—
85C2×C8×C61—41no—
86C2×C8×C61—41no—
87C2×C8×C61—41no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C42×C61—41no—
94#11758202—48no—
95C4⋊Dic61—21no—
96C4⋊Dic61—41no—
97C4⋊Dic61—41no—
98C4⋊Dic61—81no—
99C4⋊Dic61—81no—
100C4⋊Dic61Q1612yescomplemented
101#11758196—41no—
102#11758196—41no—
103#11758196—81no—
104#11758196—81no—
105C4×Dic61—81no—
106C2×#92418—22no—
107C2×#92418—22no—
108C2×#92418—41no—
109C2×#92418—41no—
110C2×#92418—41no—
111C2×#92418—41no—
112C23×C61#12011yescharacteristic
113Q8×C61—42no—
114Q8×C61—42no—
115Q8×C61—42no—
116Q8×C61—81no—
117Q8×C61—81no—
118Q8×C61—81no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—22no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—81no—
133C2×C4×C61—81no—
134C2×C4×C61C2×Q1611yescharacteristic
135C2×C4×C61C2×Q1611yescharacteristic
136C2×C4×C61#13511yescharacteristic
137C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C61#13711yescharacteristic
139C2×C4×C61#13711yescharacteristic
140C2×Dic61—22no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152Dic122—42no—
153Dic122—81no—
154C8×C61—81no—
155#92418—42no—
156#92418—42no—
157#7784—611nomaximal, Sylow
158C22×C61#44911yescharacteristic
159C22×C61#45111yescharacteristic
160C22×C61#45111yescharacteristic
161C22×C61#45311yescharacteristic
162C22×C61#45511yescharacteristic
163C22×C61#45611yescharacteristic
164C22×C61B(4,𝔽2)11yescharacteristic
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—42no—
172C4×C61—82no—
173C4×C61—81no—
174Dic61—42no—
175Dic61—81no—
176Dic61—81no—
177Dic61—81no—
178#2806—611no—
179#2057—611no—
180#2024D6111yescharacteristic
181#2016—611no—
182#1272—611no—
183#1272—611no—
184#1271—611no—
185C2×C61#160011yescharacteristic
186C2×C61#160211yescharacteristic
187C2×C61#160511yescharacteristic
188C2×C61#163311yescharacteristic
189C2×C61#164411yescharacteristic
190C2×C61#164811yescharacteristic
191C2×C61#166111yescharacteristic
192C22×Q16—1221no—
193C2×C4⋊Q8C2×D6111yescharacteristic
194C2×#107—1221no—
195C2×#103—21no—
196C2×#103—1221no—
197#398—1221no—
198#398—1221no—
199#395—1221no—
200#391—611no—
201#391—1221no—
202#391C2×D6111yescharacteristic
203#389—611no—
204#389—1221no—
205#387—1221no—
206#340—21no—
207#340—1221no—
208#336—611no—
209#336—611no—
210#336C2×D6111yescharacteristic
211C61#778411yescharacteristic, complemented, Sylow
212C22×Q8—21no—
213C22×Q8—611no—
214C22×Q8C22⋊D6111yescharacteristic
215C2×Q16—2442no—
216C2×Q16—2442no—
217C2×Q16—2442no—
218C22×C8—21no—
219C22×C8—1221no—
220C22×C8—1221no—
221C4⋊Q8—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—611no—
226C2×C4⋊C4—611no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4C22⋊D6111yescharacteristic
233C2×C42—1221no—
234C2×C42C22×D6111yescharacteristic
235#107—2442no—
236#103—42no—
237#103—2442no—
238#95—21no—
239#95—21no—
240#95—1221no—
241#95—1221no—
242#95—1221no—
243#95—1221no—
244#95—1221no—
245#95—1221no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—1221no—
256C2×Q8—1221no—
257C2×Q8—2441no—
258C2×Q8—2441no—
259C2×Q8#1175820212yescomplemented
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4D4×D6111yescharacteristic
268C22×C4D4×D6111yescharacteristic
269C22×C4C2×C22⋊D6111yescharacteristic
270Q16—2448no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—1222no—
275C2×C8—1222no—
276C2×C8—2441no—
277C2×C8—2441no—
278C2×C8—2441no—
279C2×C8—2441no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—612no—
286C4⋊C4—1221no—
287C4⋊C4—4881no—
288C4⋊C4—4881no—
289C4⋊C4—4881no—
290C4⋊C4—4881no—
291C4⋊C4—2441no—
292C4⋊C4—2441no—
293C4⋊C4—2441no—
294C4⋊C4—2441no—
295C42—41no—
296C42—4881no—
297C23#1451974511yescharacteristic
298Q8—42no—
299Q8—42no—
300Q8—42no—
301Q8—81no—
302Q8—81no—
303Q8—81no—
304Q8—4881no—
305Q8—2442no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—1222no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4Q16×D6111yescharacteristic
334C2×C4C2×#1175820211yescharacteristic
335C2×C4#1451979811yescharacteristic
336C2×C4#1451980411yescharacteristic
337C2×C4#1451980711yescharacteristic
338C2×C4#1451980911yescharacteristic
339C8—81no—
340C8—2442no—
341C8—2442no—
342C22#1683990611yescharacteristic
343C22#1683991611yescharacteristic
344C22#1683991711yescharacteristic
345C22#1683992811yescharacteristic
346C22#1683993911yescharacteristic
347C22#1683994311yescharacteristic
348C22#1683994811yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—81no—
357C4—82no—
358C4—4881no—
359C4—4881no—
360C4—4881no—
361C4—2442no—
362C2#2144476411yescharacteristic
363C2#2144477511yescharacteristic
364C2#2144479111yescharacteristic
365C2#2144495811yescharacteristic
366C2#2144503011yescharacteristic
367C2#2144506011yescharacteristic
368C2#2144513511yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2024C2
socle, prime radicalC23×C61#120
hypercenter#2024D61
hypocenterC61#7784
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
70. C2×Q8×C61Q16semidirect
100. C4⋊Dic61Q16semidirect
211. C61#7784semidirect
259. C2×Q8#11758202semidirect

Maximal subgroups

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

subgroupindex
#214537832
#214471442
#214469192
#214436452
#214436442
#214436412
C61×#20242
#778461