Finite Group Databasebeta

← #29595892

Subgroups — #29595892

Order 15616.

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

normal subgroups only

This group has 740 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×#16840512C211yesmaximal, characteristic
2C2×#16839560C212yesmaximal, complemented
3C2×#16839559C211yesmaximal, characteristic
4#21446212C212yesmaximal, complemented
5C61×#1888C211yesmaximal, characteristic
6#16840512—24no—
7#16840512—22no—
8#16840512C414yescomplemented
9C2×#14519775C2212yes—
10C2×#14519770—22no—
11C2×#14519769—21no—
12C2×#14519758—21no—
13C2×#14519754—22no—
14#16839770—22no—
15C2×#14519696C2212yes—
16#16839560—24no—
17#16839559—22no—
18#16839521—22no—
19C2×#124×C61C2211yescharacteristic
20C61×#519C2211yescharacteristic
21C61×#418—21no—
22C2×#101×C61C2211yescharacteristic
23C61×#339—21no—
24C22×Dic122D412yes—
25C23×Dic61—21no—
26C2×#11758194—22no—
27#14519775—22no—
28#14519775—22no—
29#14519775C2×C414yes—
30#14519775D414yes—
31#14519770—28no—
32#14519770—28no—
33#14519769—24no—
34#14519769—24no—
35#14519758—24no—
36#14519758—24no—
37#14519754—28no—
38#14519754—28no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×#11758196—22no—
42C2×#11758196—22no—
43C2×#11758196—22no—
44C2×C22⋊Dic61—22no—
45C2×C22⋊Dic61—21no—
46C2×C4×Dic61D412yes—
47#14519700—44no—
48#14519696—22no—
49#14519696—22no—
50C22×Q8×C61C2311yescharacteristic
51C22×D4×C61D411yescharacteristic
52C2×M16×C61—21no—
53#124×C61—22no—
54#124×C61C2×C412yes—
55#124×C61D412yes—
56C4×Q8×C61—42no—
57#117×C61—21no—
58#117×C61—22no—
59C2×C22⋊C4×C61—22no—
60C2×C42×C61D411yescharacteristic
61#102×C61—42no—
62#101×C61—22no—
63C24×C61—21no—
64C2×Q8×C61—41no—
65C2×Q8×C61—41no—
66C2×Q8×C61C22×C411yescharacteristic
67C2×Q8×C61C2×D411yescharacteristic
68C2×Q8×C61C2×D412yes—
69C2×D4×C61—21no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61C22⋊C412yes—
73C22×C4×C61—21no—
74C22×C4×C61C2×D412yes—
75C22×C4×C61C2×D411yescharacteristic
76C22×Dic61—22no—
77C22×Dic61—22no—
78C22×Dic61—24no—
79C22×Dic61—81no—
80C22×Dic61—81no—
81C2×Dic122—22no—
82C2×Dic122—42no—
83C2×Dic122—42no—
84C2×Dic122C22⋊C414yes—
85M16×C61—41no—
86M16×C61—41no—
87C2×C8×C61—41no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—24no—
90C22⋊C4×C61—24no—
91C22⋊C4×C61—24no—
92C22⋊C4×C61—41no—
93C42×C61—21no—
94C42×C61—42no—
95C42×C61—41no—
96C42×C61C22⋊C412yes—
97C22⋊Dic61—28no—
98C22⋊Dic61—24no—
99C4⋊Dic61—24no—
100C4⋊Dic61—24no—
101#11758196—28no—
102#11758196—24no—
103#11758196—24no—
104#11758196—24no—
105#11758196—24no—
106C4×Dic61—22no—
107C4×Dic61C22⋊C414yes—
108#11758194—42no—
109#11758194—42no—
110C2×#92418—42no—
111C23×C61—22no—
112C23×C61—44no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61#12011yescharacteristic
116Q8×C61—41no—
117Q8×C61—41no—
118Q8×C61—42no—
119D4×C61—42no—
120D4×C61—81no—
121C2×C4×C61—22no—
122C2×C4×C61—21no—
123C2×C4×C61—22no—
124C2×C4×C61—21no—
125C2×C4×C61—22no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61C2×C22⋊C412yes—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61#12012yes—
132C2×Dic61—24no—
133C2×Dic61—24no—
134C2×Dic61—44no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—84no—
138Dic122—44no—
139Dic122—82no—
140C8×C61—81no—
141#92418—82no—
142#12775—611nomaximal, Sylow
143C22×C61—22no—
144C22×C61—44no—
145C22×C61—42no—
146C22×C61—82no—
147C22×C61—82no—
148C22×C61#37911yescharacteristic
149C22×C61B(4,𝔽2)11yescharacteristic
150C22×C61#45811yescharacteristic
151C4×C61—22no—
152C4×C61—21no—
153C4×C61—22no—
154C4×C61—21no—
155C4×C61—42no—
156C4×C61—82no—
157Dic61—44no—
158Dic61—44no—
159C2×#563—611no—
160C2×#356—612no—
161C2×#355—611no—
162#1888D6111yescharacteristic
163#1886—612no—
164C2×C61—21no—
165C2×C61—21no—
166C2×C61—82no—
167C2×C61#179011yescharacteristic
168C2×C61#220312yes—
169#563—614no—
170#563—1224no—
171#563—1222no—
172C2×C4⋊Q8—612no—
173C2×#126—1222no—
174C2×#125—1221no—
175C2×#124C2×D6111yescharacteristic
176C2×#123—1221no—
177C2×C22⋊Q8—1222no—
178#519C2×D6111yescharacteristic
179#419—1222no—
180#418—21no—
181C2×#101—612no—
182C2×#101C2×D6111yescharacteristic
183#356—1224no—
184#355—1222no—
185#339—21no—
186#339—1222no—
187C61#1277511yescharacteristic, complemented, Sylow
188C22×Q8—612no—
189C22×Q8C22×D6111yescharacteristic
190C22×D4C22⋊D6111yescharacteristic
191C23×C4—1221no—
192C2×M16—21no—
193C2×M16—1222no—
194C4⋊Q8—614no—
195C4⋊Q8—614no—
196C4⋊Q8—1222no—
197C4⋊Q8—1222no—
198#126—1228no—
199#126—1228no—
200#125—1224no—
201#125—1224no—
202#124—22no—
203#124C4×D6112yes—
204#124D24412yes—
205#123—1224no—
206#123—1224no—
207C22⋊Q8—1228no—
208C22⋊Q8—1228no—
209C4×Q8—42no—
210#117—22no—
211#117—21no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—1222no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4—1222no—
216C2×C4⋊C4—1222no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—1222no—
219C2×C22⋊C4—1221no—
220C2×C42—612no—
221C2×C42C22⋊D6111yescharacteristic
222#103—2444no—
223#102—42no—
224#101—22no—
225#101—1222no—
226#101—1222no—
227C24—21no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—614no—
231C2×Q8—1222no—
232C2×Q8—2442no—
233C2×Q8—2442no—
234C2×Q8C2×C4×D6111yescharacteristic
235C2×Q8C2×D24411yescharacteristic
236C2×Q8D4×D6112yes—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4#1175819812yes—
241C22×C4—21no—
242C22×C4—1224no—
243C22×C4—1222no—
244C22×C4—1222no—
245C22×C4—4881no—
246C22×C4—4881no—
247C22×C4D4×D6112yes—
248C22×C4C2×C22⋊D6111yescharacteristic
249M16—41no—
250M16—41no—
251M16—2442no—
252M16—2442no—
253C2×C8—41no—
254C2×C8—2442no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—1228no—
258C4⋊C4—1224no—
259C4⋊C4—1224no—
260C4⋊C4—1224no—
261C4⋊C4—1224no—
262C4⋊C4—1224no—
263C4⋊C4—1224no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—41no—
267C22⋊C4—1228no—
268C22⋊C4—1224no—
269C42—21no—
270C42—41no—
271C42—42no—
272C42—614no—
273C42—1222no—
274C42#1175819812yes—
275C23—22no—
276C23—44no—
277C23—41no—
278C23—41no—
279C23#1451974511yescharacteristic
280Q8—41no—
281Q8—41no—
282Q8—42no—
283Q8—4882no—
284Q8—2444no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—1224no—
295C2×C4—1224no—
296C2×C4—4884no—
297C2×C4—2444no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4C22⋊C4×D6112yes—
301C2×C4C2×#1175819811yescharacteristic
302C2×C4#1451974511yescharacteristic
303C2×C4#1451974712yes—
304C8—81no—
305C8—4882no—
306C22—22no—
307C22—44no—
308C22—42no—
309C22—82no—
310C22—82no—
311C22#1683961711yescharacteristic
312C22#1683994811yescharacteristic
313C22#1683995111yescharacteristic
314C4—21no—
315C4—22no—
316C4—22no—
317C4—21no—
318C4—42no—
319C4—82no—
320C4—2444no—
321C4—2444no—
322C2—21no—
323C2—21no—
324C2—82no—
325C2#2144576611yescharacteristic
326C2#2144783912yes—
subgroupG/H
centerC22#16839948
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1888C2
socleC22×C61B(4,𝔽2)
prime radicalC22×D4×C61D4
cosocleC22×Q8×C61C23
hypercenter#1888D61
hypocenterC61#12775
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1888

Complements

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

NH ≅ G/Ntype
2. C2×#16839560C2semidirect
4. #21446212C2semidirect
8. #16840512C4semidirect
187. C61#12775semidirect

Maximal subgroups

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

subgroupindex
C2×#168405122
C2×#168395602
C2×#168395592
#214462122
C61×#18882
#1277561