Finite Group Databasebeta

← #29834354

Subgroups — #29834354

Order 15616.

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

normal subgroups only

This group has 680 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×#16840205C211yesmaximal, characteristic
2C22×#14519723C211yesmaximal, characteristic, complemented
3#21456308C211yesmaximal, characteristic
4#21454686C211yesmaximal, characteristic, complemented
5#21447884C214yesmaximal, complemented
6#21447868C214yesmaximal, complemented
7#21447579C211yesmaximal, characteristic, complemented
8#21447524C211yesmaximal, characteristic
9C61×#2141C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11C2×#14519799C2211yescharacteristic
12C22×Dic244C2211yescharacteristic
13#16840336—21no—
14#16840239—21no—
15#16840205—24no—
16C2×Dic488C2212yes—
17C2×#14519723—24no—
18C2×#14519723C2214yes—
19C2×#14519722C2212yes—
20C22×#11758185C2211yescharacteristic
21#16840155—22no—
22#16840147C2212yes—
23#16840140C2212yes—
24#16839868—22no—
25#16839856C2214yes—
26#16839770—21no—
27#16839717C2211yescharacteristic
28#16839580C2214yes—
29#16839565C2212yes—
30#16839562C2212yes—
31C22×Q16×C61C2211yescharacteristic
32C2×M32×C61C2211yescharacteristic
33C61×#427C2211yescharacteristic
34C61×#358C2214yes—
35C2×#11758222—21no—
36C22×Dic122—21no—
37#14519808—44no—
38#14519799—22no—
39#14519799—22no—
40C2×Dic244—21no—
41C2×Dic244—22no—
42C2×Dic244D414yes—
43C2×Dic244C2311yescharacteristic
44C2×Dic244C2312yes—
45C2×#11758202—22no—
46C2×#11758190—21no—
47C2×#11758201—22no—
48C2×D488C2311yescharacteristic
49C2×#11758194D411yescharacteristic
50#14519775—22no—
51#14519766—22no—
52#14519760—42no—
53#14519753—42no—
54C4×Dic122—42no—
55#14519742—24no—
56#14519736D411yescharacteristic
57#14519733—42no—
58Dic488—24no—
59#14519723—28no—
60#14519722—24no—
61C2×#11758185—22no—
62C2×#11758185C2312yes—
63#14519700—24no—
64#14519687D412yes—
65C8×Dic61C2312yes—
66C22×Q8×C61—21no—
67C2×Q16×C61—22no—
68C2×Q16×C61—21no—
69C2×Q16×C61—22no—
70C2×Q16×C61C2312yes—
71C22×C8×C61C2311yescharacteristic
72#117×C61—21no—
73M32×C61—22no—
74C2×C16×C61C2312yes—
75#107×C61C2312yes—
76#106×C61—21no—
77C2×Q8×C61—22no—
78C2×Q8×C61—41no—
79C2×Q8×C61—42no—
80C2×Q8×C61—42no—
81C22×C4×C61—41no—
82C22×C4×C61C2×D411yescharacteristic
83C2×C22⋊D61—41no—
84C22×Dic61—41no—
85#11758222—42no—
86#11758222—42no—
87C2×D244—21no—
88C2×C4×D61—41no—
89C2×Dic122—24no—
90C2×Dic122—22no—
91C2×Dic122—21no—
92C2×Dic122—41no—
93C2×Dic122—42no—
94Q16×C61—22no—
95Q16×C61—22no—
96Q16×C61—42no—
97C2×C8×C61C2×D411yescharacteristic
98C2×C8×C61C2×D412yes—
99C2×C8×C61#4112yes—
100C2×C8×C61C2411yescharacteristic
101C4⋊C4×C61—22no—
102C22⋊C4×C61—41no—
103C42×C61—41no—
104C22⋊Dic61D812yes—
105#11758202—44no—
106#11758201—44no—
107#11758198—44no—
108C4⋊Dic61D812yes—
109#11758196—44no—
110#11758196—44no—
111C4×Dic61—44no—
112C4×Dic61C2×D412yes—
113#11758194—22no—
114C2×#92418C2×D412yes—
115Dic244—24no—
116Dic244—21no—
117Dic244—22no—
118Dic244—21no—
119Dic244C2×D414yes—
120D488—21no—
121D488—21no—
122#11758190—42no—
123C16×C61—22no—
124#11758185—22no—
125C23×C61C2×D811yescharacteristic
126C22×D61—41no—
127Q8×C61—42no—
128Q8×C61—42no—
129Q8×C61—82no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—82no—
133C2×C4×C61#12112yes—
134C2×C4×C61C2×D811yescharacteristic
135C2×C4×C61AGL(1,ℤ/8)12yes—
136C2×C4×C61C22×D411yescharacteristic
137C22⋊D61—82no—
138C2×Dic61—42no—
139C2×Dic61—44no—
140C2×Dic61—41no—
141C2×Dic61C2×D814yes—
142D244—41no—
143D244—41no—
144C4×D61—82no—
145Dic122—24no—
146Dic122—42no—
147Dic122—44no—
148Dic122—41no—
149Dic122—41no—
150C8×C61#12112yes—
151C8×C61C22×D411yescharacteristic
152C8×C61C2×#4111yescharacteristic
153#92418—22no—
154#33588—611nomaximal, Sylow
155C22×C61—22no—
156C22×C61#45912yes—
157C22×C61C22×D811yescharacteristic
158C2×D61—82no—
159C2×D61—81no—
160C4×C61—82no—
161C4×C61—82no—
162C4×C61#45912yes—
163C4×C61C2×#12111yescharacteristic
164C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
165Dic61—24no—
166Dic61—44no—
167Dic61—81no—
168C2×#510—611no—
169C22×Q32—611no—
170#3156—611no—
171#2950—611no—
172#2211—614no—
173#2209—614no—
174#2151—611no—
175#2141D6111yescharacteristic, complemented
176#2138—611no—
177C2×C61—22no—
178C2×C61C2×#45911yescharacteristic
179C2×C61#341712yes—
180D61—161no—
181C2×#137—1221no—
182C2×#135—611no—
183C22×Q16—611no—
184C22×Q16C2×D6111yescharacteristic
185#536—1221no—
186#519—1221no—
187#510—1224no—
188C2×Q32—614no—
189C2×Q32—612no—
190C2×Q32—1224no—
191C2×SD32—612no—
192C2×M32C2×D6111yescharacteristic
193C22×C16—611no—
194#497—1222no—
195#495—612no—
196#494—612no—
197#441—1222no—
198C4×Q16—614no—
199#427C2×D6111yescharacteristic
200#419—1221no—
201#405—611no—
202C4⋊C16—614no—
203#358—612no—
204#358C2×D6114yes—
205#357—612no—
206C61#3358811yescharacteristic, complemented, Sylow
207C2×#41—1221no—
208C22×Q8—21no—
209C22×Q8—1221no—
210#137—2444no—
211#135—1222no—
212#135—1222no—
213C2×Q16—22no—
214C2×Q16—22no—
215C2×Q16—21no—
216C2×Q16—614no—
217C2×Q16—612no—
218C2×Q16—611no—
219C2×Q16—1222no—
220C2×Q16—1222no—
221C2×Q16—1221no—
222C2×Q16C22×D6112yes—
223C2×SD16—1222no—
224C2×SD16—1221no—
225C2×D8—611no—
226C2×M16—611no—
227C22×C8C22×D6111yescharacteristic
228C4⋊Q8—1222no—
229#124—1222no—
230#123—2442no—
231C22⋊Q8—2442no—
232C4×Q8—1224no—
233C4×Q8—2442no—
234#117—21no—
235#117—611no—
236#117—2442no—
237Q32—1224no—
238Q32—1228no—
239SD32—1224no—
240M32—22no—
241C2×C16—612no—
242C2×C16—1222no—
243C2×C16C22×D6112yes—
244#107C22×D6112yes—
245#106—21no—
246#103—1224no—
247#97—612no—
248C4×C8—612no—
249#41—2442no—
250#41—2442no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—1222no—
256C2×Q8—1221no—
257C2×Q8—1224no—
258C2×Q8—2441no—
259C2×Q8—2442no—
260C2×D4—1221no—
261C2×D4—2441no—
262C22×C4—41no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4D4×D6111yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—42no—
269Q16—614no—
270Q16—1221no—
271Q16—1222no—
272Q16—1221no—
273Q16—1224no—
274Q16—2444no—
275SD16—2444no—
276SD16—2442no—
277D8—1221no—
278D8—1221no—
279M16—1222no—
280C2×C8—612no—
281C2×C8#1175822212yes—
282C2×C8D4×D6112yes—
283C2×C8D4×D6111yescharacteristic
284C2×C8C23×D6111yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—612no—
287C4⋊C4—2444no—
288C4⋊C4—2444no—
289C22⋊C4—41no—
290C22⋊C4—612no—
291C22⋊C4—2444no—
292C42—41no—
293C42—612no—
294C42—2444no—
295C16—22no—
296C16—1222no—
297C23—2441no—
298C23D8×D6111yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—82no—
302Q8—1224no—
303Q8—2441no—
304Q8—2441no—
305Q8—2444no—
306Q8—2442no—
307D4—4882no—
308D4—2441no—
309D4—2441no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—614no—
314C2×C4—4882no—
315C2×C4—2441no—
316C2×C4—2444no—
317C2×C4—2442no—
318C2×C4#1451975012yes—
319C2×C4D8×D6111yescharacteristic
320C2×C4#1451980412yes—
321C2×C4C2×D4×D6111yescharacteristic
322C8—1222no—
323C8#1451975012yes—
324C8C2×D4×D6111yescharacteristic
325C8C2×#1175822211yescharacteristic
326C22—22no—
327C22—4882no—
328C22—4881no—
329C22#1683995612yes—
330C22C2×D8×D6111yescharacteristic
331C4—82no—
332C4—82no—
333C4—1224no—
334C4—4881no—
335C4—2444no—
336C4#1683995612yes—
337C4C2×#1451975011yescharacteristic
338C4C2×#1451980411yescharacteristic
339C2—22no—
340C2—9761no—
341C2C2×#1683995611yescharacteristic
342C2#2145897112yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2141C2
socleC22×C61C22×D8
prime radicalC2×#14519799C22
hypercenter#2141D61
hypocenterC61#33588
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
2. C22×#14519723C2semidirect
4. #21454686C2semidirect
5. #21447884C2semidirect
6. #21447868C2semidirect
7. #21447579C2semidirect
9. C61×#2141C2semidirect
175. #2141D61semidirect
206. C61#33588semidirect

Maximal subgroups

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

subgroupindex
C2×#168402052
C22×#145197232
#214563082
#214546862
#214478842
#214478682
#214475792
#214475242
C61×#21412
#3358861