Finite Group Databasebeta

← #29834358

Subgroups — #29834358

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×#16840202C211yesmaximal, characteristic
2C22×Dic488C211yesmaximal, characteristic, complemented
3#21456314C211yesmaximal, characteristic
4#21454686C211yesmaximal, characteristic, complemented
5#21447880C214yesmaximal, complemented
6#21447868C214yesmaximal, complemented
7#21447577C211yesmaximal, characteristic, complemented
8#21447538C211yesmaximal, characteristic, complemented
9C61×#2138C211yesmaximal, characteristic
10C2×#14519807—21no—
11C22×Dic244C2211yescharacteristic
12C22×Dic244C2211yescharacteristic
13#16840333—21no—
14#16840239—21no—
15#16840202—24no—
16C2×Dic488—24no—
17C2×Dic488C2214yes—
18C2×#14519723C2212yes—
19C2×#14519721C2212yes—
20C2×#14519717C2211yescharacteristic
21#16840153—22no—
22#16840145C2212yes—
23#16840140C2212yes—
24#16839868—22no—
25#16839856C2214yes—
26#16839770—21no—
27#16839717C2211yescharacteristic
28#16839580C2214yes—
29#16839565C2214yes—
30C2×#135×C61C2211yescharacteristic
31C22×C16×C61C2211yescharacteristic
32C61×#427C2211yescharacteristic
33C61×#358C2212yes—
34C61×#357C2212yes—
35C22×Dic122—21no—
36C22×Dic122—21no—
37#14519807—44no—
38C2×Dic244—21no—
39C2×Dic244—22no—
40C2×Dic244—22no—
41C2×Dic244—22no—
42C2×Dic244—21no—
43C2×Dic244D414yes—
44C2×Dic244C2312yes—
45C2×Dic244C2312yes—
46C2×#11758202—22no—
47C2×#11758200—22no—
48C2×#11758194D411yescharacteristic
49#14519775—22no—
50#14519764—22no—
51#14519758—42no—
52#14519756—42no—
53C4×Dic122—42no—
54#14519742—24no—
55#14519736D411yescharacteristic
56#14519733—42no—
57Dic488—28no—
58#14519723—24no—
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—
77#41×C61—42no—
78#41×C61—42no—
79C2×Q8×C61—21no—
80C2×D4×C61—21no—
81C2×D4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61C2×D411yescharacteristic
84C22×Dic61—41no—
85C22×Dic61—41no—
86C2×Dic122—24no—
87C2×Dic122—22no—
88C2×Dic122—22no—
89C2×Dic122—41no—
90C2×Dic122—42no—
91C2×Dic122—42no—
92C2×Dic122—42no—
93C2×Dic122—41no—
94Q16×C61—21no—
95Q16×C61—21no—
96SD16×C61—42no—
97D8×C61—21no—
98D8×C61—21no—
99C2×C8×C61C2×D411yescharacteristic
100C2×C8×C61C2×D412yes—
101C2×C8×C61#4112yes—
102C2×C8×C61C2411yescharacteristic
103C4⋊C4×C61—22no—
104C22⋊C4×C61—41no—
105C42×C61—41no—
106C22⋊Dic61—44no—
107C22⋊Dic61D812yes—
108#11758202—44no—
109#11758200—44no—
110C4⋊Dic61D812yes—
111#11758196—44no—
112#11758196—44no—
113C4×Dic61—44no—
114C4×Dic61C2×D412yes—
115#11758194—22no—
116C2×#92418C2×D412yes—
117Dic244—24no—
118Dic244—22no—
119Dic244—22no—
120Dic244—22no—
121Dic244—42no—
122Dic244C2×D414yes—
123C16×C61—22no—
124#11758185—22no—
125C23×C61—41no—
126C23×C61C2×D811yescharacteristic
127Q8×C61—41no—
128Q8×C61—41no—
129D4×C61—41no—
130D4×C61—41no—
131D4×C61—82no—
132C2×C4×C61—42no—
133C2×C4×C61—41no—
134C2×C4×C61—82no—
135C2×C4×C61#12112yes—
136C2×C4×C61C2×D811yescharacteristic
137C2×C4×C61AGL(1,ℤ/8)12yes—
138C2×C4×C61C22×D411yescharacteristic
139C2×Dic61—42no—
140C2×Dic61—44no—
141C2×Dic61—42no—
142C2×Dic61—82no—
143C2×Dic61C2×D814yes—
144Dic122—24no—
145Dic122—42no—
146Dic122—44no—
147Dic122—42no—
148Dic122—42no—
149Dic122—82no—
150C8×C61#12112yes—
151C8×C61C22×D411yescharacteristic
152C8×C61C2×#4111yescharacteristic
153#92418—22no—
154#33588—611nomaximal, Sylow
155C22×C61—22no—
156C22×C61—82no—
157C22×C61—81no—
158C22×C61#45912yes—
159C22×C61C22×D811yescharacteristic
160C4×C61—82no—
161C4×C61—81no—
162C4×C61#45912yes—
163C4×C61C2×#12111yescharacteristic
164C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
165Dic61—24no—
166Dic61—44no—
167Dic61—82no—
168C2×#510—611no—
169C22×Q32—611no—
170#3156—611no—
171#2950—611no—
172#2211—614no—
173#2209—614no—
174#2151—611no—
175#2141—611no—
176#2138D6111yescharacteristic
177C2×C61—22no—
178C2×C61—161no—
179C2×C61C2×#45911yescharacteristic
180C2×C61#341712yes—
181C2×#137—1221no—
182C2×#135C2×D6111yescharacteristic
183C22×Q16—611no—
184C22×Q16—611no—
185#536—1221no—
186#519—1221no—
187#510—1224no—
188C2×Q32—612no—
189C2×Q32—614no—
190C2×Q32—1224no—
191C2×SD32—612no—
192C2×M32—611no—
193C22×C16C2×D6111yescharacteristic
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—614no—
204#358C2×D6112yes—
205#357C2×D6112yes—
206C61#3358811yescharacteristic, complemented, Sylow
207C2×#41—21no—
208C22×Q8—1221no—
209C22×Q8—1221no—
210#137—2444no—
211#135—22no—
212#135—22no—
213C2×Q16—614no—
214C2×Q16—612no—
215C2×Q16—612no—
216C2×Q16—1222no—
217C2×Q16—1222no—
218C2×Q16—1222no—
219C2×Q16—1221no—
220C2×Q16—1222no—
221C2×Q16—1221no—
222C2×Q16C22×D6111yescharacteristic
223C2×SD16—21no—
224C2×SD16—1222no—
225C2×D8C22×D6111yescharacteristic
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—1228no—
238Q32—1224no—
239SD32—1224no—
240M32—1222no—
241C2×C16—22no—
242C2×C16—612no—
243C2×C16C22×D6112yes—
244#107C22×D6112yes—
245#106—21no—
246#103—1224no—
247#97—612no—
248C4×C8—612no—
249#41—42no—
250#41—42no—
251C2×Q8—21no—
252C2×Q8—1222no—
253C2×Q8—1222no—
254C2×Q8—1224no—
255C2×Q8—2442no—
256C2×Q8—2442no—
257C2×Q8—2441no—
258C2×Q8—2441no—
259C2×Q8—2442no—
260C2×D4—21no—
261C2×D4—41no—
262C22×C4—41no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4D4×D6111yescharacteristic
266Q16—21no—
267Q16—21no—
268Q16—614no—
269Q16—1222no—
270Q16—1222no—
271Q16—1222no—
272Q16—1224no—
273Q16—2444no—
274Q16—2442no—
275SD16—42no—
276SD16—2444no—
277D8—21no—
278D8—21no—
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—41no—
298C23D8×D6111yescharacteristic
299Q8—41no—
300Q8—41no—
301Q8—1224no—
302Q8—4882no—
303Q8—2442no—
304Q8—2442no—
305Q8—2444no—
306Q8—2442no—
307D4—41no—
308D4—41no—
309D4—82no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—82no—
313C2×C4—614no—
314C2×C4—4882no—
315C2×C4—2442no—
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—82no—
328C22—81no—
329C22#1683995612yes—
330C22C2×D8×D6111yescharacteristic
331C4—82no—
332C4—81no—
333C4—1224no—
334C4—4882no—
335C4—2444no—
336C4#1683995612yes—
337C4C2×#1451975011yescharacteristic
338C4C2×#1451980411yescharacteristic
339C2—22no—
340C2—161no—
341C2C2×#1683995611yescharacteristic
342C2#2145896912yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
prime radicalC2×#135×C61C22
hypercenter#2138D61
hypocenterC61#33588
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
2. C22×Dic488C2semidirect
4. #21454686C2semidirect
5. #21447880C2semidirect
6. #21447868C2semidirect
7. #21447577C2semidirect
8. #21447538C2semidirect
206. C61#33588semidirect

Maximal subgroups

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

subgroupindex
C2×#168402022
C22×Dic4882
#214563142
#214546862
#214478802
#214478682
#214475772
#214475382
C61×#21382
#3358861