Finite Group Databasebeta

← #29595902

Subgroups — #29595902

Order 15616.

784 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16840492C211yesmaximal, characteristic
2C2×#16839561C211yesmaximal, characteristic
3C2×#16839560C212yesmaximal
4#21446212C212yesmaximal
5C61×#1886C211yesmaximal, characteristic
6#16840492—24no—
7#16840492—22no—
8#16840492C414yescomplemented
9C2×#14519776—22no—
10C2×#14519775—21no—
11C2×#14519775C2212yes—
12C2×C4×Dic122—21no—
13C2×#14519742—22no—
14#16839770—22no—
15C2×#14519696C2212yes—
16#16839561—22no—
17#16839560—24no—
18#16839521—22no—
19C2×C4⋊Q8×C61C2211yescharacteristic
20C61×#519C2211yescharacteristic
21C61×#419—21no—
22C2×#101×C61C2211yescharacteristic
23C61×#339—21no—
24C22×Dic122—21no—
25C22×Dic122—21no—
26C22×Dic122D412yes—
27C2×#11758194—22no—
28#14519776—28no—
29#14519776—28no—
30#14519775—24no—
31#14519775—24no—
32#14519775—22no—
33#14519775—22no—
34#14519775C2×C414yes—
35#14519775D414yes—
36C4×Dic122—24no—
37C4×Dic122—24no—
38#14519742—28no—
39#14519742—28no—
40C2×C4⋊Dic61—21no—
41C2×#11758196—22no—
42C2×#11758196—22no—
43C2×#11758196—22no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61—22no—
46C2×C4×Dic61D412yes—
47#14519700—44no—
48#14519696—22no—
49#14519696—22no—
50C22×Q8×C61D411yescharacteristic
51C22×Q8×C61C2311yescharacteristic
52C2×M16×C61—21no—
53C4⋊Q8×C61—22no—
54C4⋊Q8×C61C2×C412yes—
55C4⋊Q8×C61D412yes—
56C4×Q8×C61—42no—
57#117×C61—21no—
58#117×C61—22no—
59C2×C4⋊C4×C61—22no—
60C2×C42×C61D411yescharacteristic
61#103×C61—42no—
62#101×C61—22no—
63C2×Q8×C61—21no—
64C2×Q8×C61—41no—
65C2×Q8×C61—41no—
66C2×Q8×C61—41no—
67C2×Q8×C61—41no—
68C2×Q8×C61C22⋊C412yes—
69C2×Q8×C61C22×C411yescharacteristic
70C2×Q8×C61C2×D411yescharacteristic
71C2×Q8×C61C2×D412yes—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C2×D412yes—
75C22×C4×C61C2×D411yescharacteristic
76C22×Dic61—22no—
77C22×Dic61—22no—
78C2×Dic122—28no—
79C2×Dic122—24no—
80C2×Dic122—24no—
81C2×Dic122—22no—
82C2×Dic122—42no—
83C2×Dic122—42no—
84C2×Dic122—81no—
85C2×Dic122—81no—
86C2×Dic122C22⋊C414yes—
87M16×C61—41no—
88M16×C61—41no—
89C2×C8×C61—41no—
90C4⋊C4×C61—24no—
91C4⋊C4×C61—24no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61—24no—
94C22⋊C4×C61—41no—
95C42×C61—21no—
96C42×C61—42no—
97C42×C61—41no—
98C42×C61C22⋊C412yes—
99C4⋊Dic61—24no—
100#11758196—28no—
101#11758196—24no—
102#11758196—24no—
103#11758196—24no—
104#11758196—24no—
105C4×Dic61—24no—
106C4×Dic61—28no—
107C4×Dic61—22no—
108C4×Dic61C22⋊C414yes—
109#11758194—42no—
110#11758194—42no—
111C2×#92418—42no—
112C23×C61#12011yescharacteristic
113Q8×C61—41no—
114Q8×C61—41no—
115Q8×C61—42no—
116Q8×C61—42no—
117Q8×C61—81no—
118C2×C4×C61—24no—
119C2×C4×C61—22no—
120C2×C4×C61—22no—
121C2×C4×C61—21no—
122C2×C4×C61—22no—
123C2×C4×C61—21no—
124C2×C4×C61—22no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×C4×C61C2×C22⋊C412yes—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61#12012yes—
132C2×Dic61—28no—
133C2×Dic61—24no—
134C2×Dic61—24no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137Dic122—216no—
138Dic122—44no—
139Dic122—82no—
140Dic122—84no—
141C8×C61—81no—
142#92418—82no—
143#12777—611nomaximal, Sylow
144C22×C61—22no—
145C22×C61#37911yescharacteristic
146C22×C61B(4,𝔽2)11yescharacteristic
147C22×C61#45811yescharacteristic
148C4×C61—24no—
149C4×C61—22no—
150C4×C61—21no—
151C4×C61—22no—
152C4×C61—21no—
153C4×C61—42no—
154C4×C61—82no—
155Dic61—28no—
156Dic61—44no—
157C2×Q82—611no—
158C2×#356—612no—
159C2×#356—611no—
160#1886—612no—
161#1886D6111yescharacteristic
162C2×C61—21no—
163C2×C61—21no—
164C2×C61#179011yescharacteristic
165C2×C61#220512yes—
166Q82—614no—
167Q82—1224no—
168Q82—1222no—
169C2×C4⋊Q8—612no—
170C2×C4⋊Q8—1222no—
171C2×C4⋊Q8—1221no—
172C2×C4⋊Q8C2×D6111yescharacteristic
173#519C2×D6111yescharacteristic
174C2×C4×Q8—1222no—
175C2×C4×Q8—1221no—
176#419—21no—
177#419—1222no—
178C2×#101—612no—
179C2×#101C2×D6111yescharacteristic
180#356—1224no—
181#356—1222no—
182#339—21no—
183#339—1222no—
184C61#1277711yescharacteristic, complemented, Sylow
185C22×Q8—612no—
186C22×Q8—1221no—
187C22×Q8—1221no—
188C22×Q8C22⋊D6111yescharacteristic
189C22×Q8C22×D6111yescharacteristic
190C2×M16—21no—
191C2×M16—1222no—
192C4⋊Q8—22no—
193C4⋊Q8—614no—
194C4⋊Q8—614no—
195C4⋊Q8—1228no—
196C4⋊Q8—1228no—
197C4⋊Q8—1224no—
198C4⋊Q8—1224no—
199C4⋊Q8—1222no—
200C4⋊Q8—1222no—
201C4⋊Q8C4×D6112yes—
202C4⋊Q8D24412yes—
203C4×Q8—42no—
204C4×Q8—1224no—
205C4×Q8—1228no—
206C4×Q8—1228no—
207C4×Q8—1224no—
208#117—22no—
209#117—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—1222no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—1222no—
214C2×C4⋊C4—1222no—
215C2×C42—612no—
216C2×C42—1222no—
217C2×C42—1221no—
218C2×C42C22⋊D6111yescharacteristic
219#103—42no—
220#103—2444no—
221#101—22no—
222#101—1222no—
223#101—1222no—
224C2×Q8—21no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—614no—
230C2×Q8—1224no—
231C2×Q8—1228no—
232C2×Q8—1224no—
233C2×Q8—1222no—
234C2×Q8—4881no—
235C2×Q8—4881no—
236C2×Q8—2442no—
237C2×Q8—2442no—
238C2×Q8#1175819812yes—
239C2×Q8C2×C4×D6111yescharacteristic
240C2×Q8C2×D24411yescharacteristic
241C2×Q8D4×D6112yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—1222no—
245C22×C4—1222no—
246C22×C4D4×D6112yes—
247C22×C4C2×C22⋊D6111yescharacteristic
248M16—41no—
249M16—41no—
250M16—2442no—
251M16—2442no—
252C2×C8—41no—
253C2×C8—2442no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—24no—
258C4⋊C4—1228no—
259C4⋊C4—1224no—
260C4⋊C4—1224no—
261C4⋊C4—1224no—
262C4⋊C4—1224no—
263C4⋊C4—1224no—
264C22⋊C4—41no—
265C42—21no—
266C42—41no—
267C42—42no—
268C42—614no—
269C42—1228no—
270C42—1224no—
271C42—1222no—
272C42#1175819812yes—
273C23#1451974511yescharacteristic
274Q8—41no—
275Q8—41no—
276Q8—42no—
277Q8—42no—
278Q8—81no—
279Q8—12216no—
280Q8—4882no—
281Q8—4884no—
282Q8—2444no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—1228no—
294C2×C4—1224no—
295C2×C4—1224no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4C22⋊C4×D6112yes—
299C2×C4C2×#1175819811yescharacteristic
300C2×C4#1451974511yescharacteristic
301C2×C4#1451974712yes—
302C8—81no—
303C8—4882no—
304C22—22no—
305C22#1683961711yescharacteristic
306C22#1683994811yescharacteristic
307C22#1683995111yescharacteristic
308C4—24no—
309C4—21no—
310C4—22no—
311C4—22no—
312C4—21no—
313C4—42no—
314C4—82no—
315C4—1228no—
316C4—2444no—
317C2—21no—
318C2—21no—
319C2#2144576611yescharacteristic
320C2#2144784812yes—
subgroupG/H
centerC22#16839948
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1886C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
cosocleC22×Q8×C61C23
hypercenter#1886D61
hypocenterC61#12777
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1886

Complements

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

NH ≅ G/Ntype
8. #16840492C4semidirect
184. C61#12777semidirect

Maximal subgroups

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

subgroupindex
C2×#168404922
C2×#168395612
C2×#168395602
#214462122
C61×#18862
#1277761