Finite Group Databasebeta

← #29834349

Subgroups — #29834349

Order 15616.

712 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C2×#16840202C211yesmaximal, characteristic, complemented
2C22×#14519722C211yesmaximal, characteristic, complemented
3#21456306C211yesmaximal, characteristic
4#21454686C211yesmaximal, characteristic, complemented
5#21447884C214yesmaximal, complemented
6#21447858C214yesmaximal, complemented
7#21447577C211yesmaximal, characteristic, complemented
8#21447534C211yesmaximal, characteristic, complemented
9C61×#2138C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C22×Dic244C2211yescharacteristic
12C22×D488C2211yescharacteristic
13#16840326—21no—
14#16840239—21no—
15#16840202—24no—
16C2×#14519723C2212yescomplemented
17C2×#14519722—24no—
18C2×#14519722C2214yes—
19C2×#14519721C2212yescomplemented
20C2×#14519717C2211yescharacteristic
21#16840148—22no—
22#16840147C2212yes—
23#16840137C2212yes—
24#16839868—22no—
25#16839856C2214yescomplemented
26#16839770—21no—
27#16839717C2211yescharacteristic
28#16839580C2214yescomplemented
29#16839562C2214yes—
30C2×#135×C61C2211yescharacteristic
31C22×C16×C61C2211yescharacteristic
32C61×#427C2211yescharacteristic
33C61×#358C2212yescomplemented
34C61×#357C2212yescomplemented
35C22×Dic122—21no—
36C22×D244—21no—
37#14519802—44no—
38C2×Dic244—21no—
39C2×Dic244—22no—
40C2×Dic244D414yescomplemented
41C2×Dic244C2312yes—
42C2×#11758201—22no—
43C2×D488—22no—
44C2×D488—22no—
45C2×D488—21no—
46C2×D488C2312yes—
47C2×#11758199—22no—
48C2×#11758194D411yescharacteristic
49#14519773—22no—
50#14519766—22no—
51#14519751—42no—
52#14519745—42no—
53C4×Dic122—42no—
54#14519742—24no—
55#14519736D411yescharacteristic
56#14519733—42no—
57#14519723—24no—
58#14519722—28no—
59#14519721—24no—
60#14519717—22no—
61C2×#11758185C2312yes—
62#14519700—24no—
63#14519687D412yes—
64C8×Dic61C2312yes—
65C2×#41×C61—21no—
66#135×C61—22no—
67#135×C61—22no—
68C2×Q16×C61C2311yescharacteristic
69C2×SD16×C61—21no—
70C2×D8×C61C2311yescharacteristic
71C22×C8×C61C2311yescharacteristic
72#117×C61—21no—
73C2×C16×C61—22no—
74C2×C16×C61C2312yes—
75#107×C61C2312yes—
76#106×C61—21no—
77C23×D61—41no—
78#41×C61—42no—
79#41×C61—42no—
80C2×Q8×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61C2×D411yescharacteristic
85C2×C22⋊D61—44no—
86C22×Dic61—41no—
87C2×D244—22no—
88C2×D244—42no—
89C2×D244—42no—
90C2×D244—41no—
91C2×Dic122—24no—
92C2×Dic122—22no—
93C2×Dic122—41no—
94C2×Dic122—42no—
95Q16×C61—21no—
96Q16×C61—21no—
97SD16×C61—42no—
98D8×C61—21no—
99D8×C61—21no—
100C2×C8×C61C2×D411yescharacteristic
101C2×C8×C61C2×D412yes—
102C2×C8×C61#4112yes—
103C2×C8×C61C2411yescharacteristic
104C4⋊C4×C61—22no—
105C22⋊C4×C61—41no—
106C42×C61—41no—
107C22⋊Dic61D812yes—
108#11758201—44no—
109#11758199—44no—
110#11758198—44no—
111C4⋊Dic61D812yes—
112#11758196—44no—
113C4×Dic61—44no—
114C4×Dic61C2×D412yes—
115#11758194—22no—
116C2×#92418C2×D412yes—
117Dic244—24no—
118Dic244—22no—
119Dic244C2×D414yes—
120D488—22no—
121D488—22no—
122D488—42no—
123C16×C61—22no—
124#11758185—22no—
125C23×C61—41no—
126C23×C61C2×D811yescharacteristic
127C22×D61—42no—
128C22×D61—82no—
129C22×D61—162no—
130Q8×C61—41no—
131Q8×C61—41no—
132D4×C61—41no—
133D4×C61—41no—
134D4×C61—82no—
135C2×C4×C61—42no—
136C2×C4×C61—41no—
137C2×C4×C61—82no—
138C2×C4×C61#12112yes—
139C2×C4×C61C2×D811yescharacteristic
140C2×C4×C61AGL(1,ℤ/8)12yes—
141C2×C4×C61C22×D411yescharacteristic
142C22⋊D61—164no—
143C2×Dic61—42no—
144C2×Dic61—44no—
145C2×Dic61C2×D814yes—
146D244—42no—
147D244—42no—
148D244—82no—
149Dic122—24no—
150Dic122—42no—
151Dic122—44no—
152C8×C61#12112yes—
153C8×C61C22×D411yescharacteristic
154C8×C61C2×#4111yescharacteristic
155#92418—22no—
156#33587—611nomaximal, Sylow
157C22×C61—22no—
158C22×C61—82no—
159C22×C61—81no—
160C22×C61#45912yes—
161C22×C61C22×D811yescharacteristic
162C2×D61—84no—
163C2×D61—82no—
164C2×D61—164no—
165C4×C61—82no—
166C4×C61—81no—
167C4×C61#45912yes—
168C4×C61C2×#12111yescharacteristic
169C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
170Dic61—24no—
171Dic61—44no—
172C2×#510—611no—
173C22×SD32—611no—
174#3155—611no—
175#2950—611no—
176#2211—614no—
177#2207—614no—
178#2151—611no—
179#2140—611no—
180#2138D6111yescharacteristic, complemented
181C2×C61—22no—
182C2×C61—161no—
183C2×C61C2×#45911yescharacteristic
184C2×C61#341612yes—
185D61—162no—
186C2×AGL(1,ℤ/8)—1221no—
187C2×#135C2×D6111yescharacteristic
188C22×Q16—611no—
189C22×D8—611no—
190#535—1221no—
191#519—1221no—
192#510—1224no—
193C2×Q32—612no—
194C2×SD32—612no—
195C2×SD32—614no—
196C2×SD32—1224no—
197C2×M32—611no—
198C22×C16C2×D6111yescharacteristic
199#496—1222no—
200#495—612no—
201#493—612no—
202#441—1222no—
203C4×Q16—614no—
204#427C2×D6111yescharacteristic
205#419—1221no—
206#405—611no—
207C4⋊C16—614no—
208#358C2×D6112yescomplemented
209#357—614no—
210#357C2×D6112yescomplemented
211C61#3358711yescharacteristic, complemented, Sylow
212C2×#41—21no—
213C22×Q8—1221no—
214C22×D4—1221no—
215AGL(1,ℤ/8)—2444no—
216#135—22no—
217#135—22no—
218C2×Q16—614no—
219C2×Q16—612no—
220C2×Q16—1222no—
221C2×Q16—1221no—
222C2×Q16C22×D6111yescharacteristic
223C2×SD16—21no—
224C2×SD16—1222no—
225C2×D8—612no—
226C2×D8—1222no—
227C2×D8—1222no—
228C2×D8—1222no—
229C2×D8—1221no—
230C2×D8C22×D6111yescharacteristic
231C2×M16—611no—
232C22×C8C22×D6111yescharacteristic
233#127—1222no—
234#124—1222no—
235#121—2442no—
236#120—2442no—
237C4×Q8—1224no—
238C4×Q8—2442no—
239#117—21no—
240#117—611no—
241#117—2442no—
242Q32—1224no—
243SD32—1228no—
244SD32—1224no—
245M32—1222no—
246C2×C16—22no—
247C2×C16—612no—
248C2×C16C22×D6112yes—
249#107C22×D6112yes—
250#106—21no—
251#103—1224no—
252#97—612no—
253C4×C8—612no—
254C24—2441no—
255#41—42no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—1222no—
259C2×Q8—1224no—
260C2×Q8—2441no—
261C2×Q8—2442no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—1222no—
265C2×D4—2444no—
266C2×D4—2442no—
267C2×D4—2441no—
268C2×D4—2442no—
269C22×C4—41no—
270C22×C4—2441no—
271C22×C4D4×D6111yescharacteristic
272Q16—21no—
273Q16—21no—
274Q16—614no—
275Q16—1222no—
276Q16—1224no—
277SD16—42no—
278SD16—2444no—
279D8—21no—
280D8—21no—
281D8—1222no—
282D8—1222no—
283D8—2444no—
284D8—2442no—
285M16—1222no—
286C2×C8—612no—
287C2×C8#1175822212yes—
288C2×C8D4×D6112yes—
289C2×C8D4×D6111yescharacteristic
290C2×C8C23×D6111yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—612no—
293C4⋊C4—2444no—
294C22⋊C4—41no—
295C22⋊C4—612no—
296C22⋊C4—2444no—
297C42—41no—
298C42—612no—
299C42—2444no—
300C16—22no—
301C16—1222no—
302C23—41no—
303C23—9762no—
304C23—4882no—
305C23—2442no—
306C23D8×D6111yescharacteristic
307Q8—41no—
308Q8—41no—
309Q8—1224no—
310Q8—2444no—
311Q8—2442no—
312D4—41no—
313D4—41no—
314D4—82no—
315D4—9764no—
316D4—4882no—
317D4—2442no—
318D4—2442no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—82no—
322C2×C4—614no—
323C2×C4—2444no—
324C2×C4—2442no—
325C2×C4#1451975012yes—
326C2×C4D8×D6111yescharacteristic
327C2×C4#1451980412yes—
328C2×C4C2×D4×D6111yescharacteristic
329C8—1222no—
330C8#1451975012yes—
331C8C2×D4×D6111yescharacteristic
332C8C2×#1175822211yescharacteristic
333C22—22no—
334C22—82no—
335C22—81no—
336C22—9764no—
337C22—4884no—
338C22—4882no—
339C22#1683995612yes—
340C22C2×D8×D6111yescharacteristic
341C4—82no—
342C4—81no—
343C4—1224no—
344C4—2444no—
345C4#1683995612yes—
346C4C2×#1451975011yescharacteristic
347C4C2×#1451980411yescharacteristic
348C2—22no—
349C2—161no—
350C2—9762no—
351C2C2×#1683995611yescharacteristic
352C2#2145896212yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
prime radical#21456306C2
hypercenter#2138D61
hypocenterC61#33587
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. C2×#16840202C2semidirect
2. C22×#14519722C2semidirect
4. #21454686C2semidirect
5. #21447884C2semidirect
6. #21447858C2semidirect
7. #21447577C2semidirect
8. #21447534C2semidirect
9. C61×#2138C2semidirect
16. C2×#14519723C22semidirect
19. C2×#14519721C22semidirect
25. #16839856C22semidirect
28. #16839580C22semidirect
33. C61×#358C22semidirect
34. C61×#357C22semidirect
40. C2×Dic244D4semidirect
180. #2138D61semidirect
208. #358C2×D61semidirect
210. #357C2×D61semidirect
211. C61#33587semidirect

Maximal subgroups

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

subgroupindex
C2×#168402022
C22×#145197222
#214563062
#214546862
#214478842
#214478582
#214475772
#214475342
C61×#21382
#3358761