Finite Group Databasebeta

← #29599160

Subgroups — #29599160

Order 15616.

540 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 77 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
1#21445777C212yesmaximal
2#21445777C211yesmaximal, characteristic
3#21443946C211yesmaximal, characteristic, complemented
4#21443946C212yesmaximal, complemented
5C61×#2809C211yesmaximal, characteristic, complemented
6C2×#14519825C2211yescharacteristic
7C2×#14519705—22no—
8C2×#14519705—21no—
9#16839761—22no—
10#16839761—21no—
11C2×#14519696C2212yes—
12C2×#14519696C2211yescharacteristic
13C61×#400—21no—
14C61×#398—22no—
15C61×#391C2211yescharacteristic
16C61×#391C2212yes—
17C61×#389—21no—
18C61×#387—22no—
19#14519825—22no—
20#14519825—21no—
21#14519825—42no—
22#14519825Q812yes—
23C2×#11758226—41no—
24C2×#11758222—22no—
25C2×#11758222—21no—
26C2×Q8×D61—41no—
27C2×#11758194—22no—
28C2×#11758194—21no—
29#14519705—42no—
30#14519705—44no—
31#14519700—42no—
32#14519700—41no—
33#14519697—42no—
34#14519697—41no—
35#14519696—22no—
36#14519696—22no—
37#14519696—22no—
38C22×Q8×C61C2311yescharacteristic
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61—21no—
42C2×C4⋊C4×C61—21no—
43C2×C4⋊C4×C61D412yes—
44C2×C4⋊C4×C61D411yescharacteristic
45C2×C42×C61D412yes—
46C2×C42×C61D411yescharacteristic
47#95×C61—21no—
48#95×C61—22no—
49#95×C61—22no—
50#95×C61—22no—
51#95×C61—22no—
52C2×Q8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61C2×Q811yescharacteristic
55C2×Q8×C61#4112yes—
56C2×Q8×C61#4111yescharacteristic
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—22no—
60C22×C4×C61—22no—
61C22×C4×C61C2×D412yes—
62C22×C4×C61C2×D411yescharacteristic
63C2×C22⋊D61—41no—
64#11758226—42no—
65#11758226—42no—
66#11758226—44no—
67Q8×D61—42no—
68Q8×D61—44no—
69Q8×D61—42no—
70#11758222—22no—
71#11758222—24no—
72#11758222—24no—
73#11758222—22no—
74#11758222—41no—
75#11758222—42no—
76#11758222—42no—
77#11758222—41no—
78C2×D244—22no—
79C2×D244—21no—
80C2×C4×D61—42no—
81C2×C4×D61—41no—
82C2×Dic122—22no—
83C2×Dic122—21no—
84C4⋊C4×C61—44no—
85C4⋊C4×C61—42no—
86C4⋊C4×C61—42no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—81no—
89C4⋊C4×C61—81no—
90C42×C61—41no—
91C42×C61—42no—
92#11758194—41no—
93#11758194—42no—
94#11758194—42no—
95#11758194—41no—
96C2×#92418—42no—
97C2×#92418—41no—
98C23×C61#12011yescharacteristic
99C22×D61—41no—
100Q8×C61—41no—
101Q8×C61—41no—
102Q8×C61—42no—
103C2×C4×C61—22no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—41no—
117C2×C4×C61—42no—
118C2×C4×C61C22⋊Q812yes—
119C2×C4×C61C22⋊Q811yescharacteristic
120C2×C4×C61#12412yes—
121C2×C4×C61#12411yescharacteristic
122C22⋊D61—42no—
123C22⋊D61—42no—
124C2×Dic61—41no—
125D244—24no—
126D244—22no—
127D244—42no—
128D244—41no—
129C4×D61—44no—
130C4×D61—42no—
131C4×D61—44no—
132C4×D61—42no—
133Dic122—24no—
134Dic122—22no—
135Dic122—42no—
136Dic122—41no—
137#92418—82no—
138#92418—81no—
139#13379—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61#39311yescharacteristic
143C22×C61B(4,𝔽2)11yescharacteristic
144C22×C61B(4,𝔽2)11yescharacteristic
145C2×D61—42no—
146C2×D61—81no—
147C2×D61—81no—
148C4×C61—22no—
149C4×C61—21no—
150C4×C61—22no—
151C4×C61—21no—
152C4×C61—42no—
153C4×C61—82no—
154C4×C61—81no—
155C4×C61—82no—
156Dic61—42no—
157#2809D6111yescharacteristic, complemented
158#1793—611no—
159#1793—612no—
160#1394—611no—
161#1394—612no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61#202911yescharacteristic
165C2×C61#220311yescharacteristic
166C2×C61#220411yescharacteristic
167D61—82no—
168C2×2−1+4—611no—
169C2×#104—1222no—
170C2×#104—1221no—
171#417—1222no—
172#417—1221no—
173C2×#101—612no—
174C2×#101—611no—
175#400—21no—
176#398—22no—
177#391C2×D6111yescharacteristic
178#391C2×D6112yes—
179#389—21no—
180#387—22no—
181C61#1337911yescharacteristic, complemented, Sylow
1822−1+4—612no—
1832−1+4—1222no—
1842−1+4—1221no—
1852−1+4—2442no—
186C2×#41—1222no—
187C2×#41—1221no—
188C2×#41—2441no—
189C22×Q8—2441no—
190C22×Q8C22×D6111yescharacteristic
191C2×M16—1222no—
192C2×M16—1221no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22⋊D6112yes—
198C2×C4⋊C4C22⋊D6111yescharacteristic
199C2×C42C22⋊D6112yes—
200C2×C42C22⋊D6111yescharacteristic
201#104—2442no—
202#104—2444no—
203#103—2442no—
204#103—2441no—
205#102—2442no—
206#102—2441no—
207#101—1222no—
208#101—1222no—
209#101—1222no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#41—1222no—
216#41—1224no—
217#41—1224no—
218#41—1222no—
219#41—2441no—
220#41—2442no—
221#41—2442no—
222#41—2444no—
223#41—2442no—
224#41—2442no—
225#41—2441no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—1222no—
229C2×Q8—1221no—
230C2×Q8—2442no—
231C2×Q8—2444no—
232C2×Q8—2442no—
233C2×Q8Q8×D6111yescharacteristic
234C2×Q8#1175822612yes—
235C2×Q8#1175822611yescharacteristic
236C2×D4—1222no—
237C2×D4—1221no—
238C2×D4—2441no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—2442no—
244C22×C4—2441no—
245C22×C4C2×C22⋊D6112yes—
246C22×C4C2×C22⋊D6111yescharacteristic
247M16—2441no—
248M16—2442no—
249M16—2442no—
250M16—2441no—
251C2×C8—2442no—
252C2×C8—2441no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—81no—
258C4⋊C4—81no—
259C42—41no—
260C42—42no—
261C23—2441no—
262C23#1451974611yescharacteristic
263Q8—41no—
264Q8—42no—
265Q8—41no—
266Q8—1222no—
267Q8—1224no—
268Q8—2441no—
269Q8—2442no—
270D4—1222no—
271D4—1224no—
272D4—2441no—
273D4—2442no—
274D4—2442no—
275D4—2442no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—2442no—
292C2×C4—2444no—
293C2×C4—2442no—
294C2×C4—2444no—
295C2×C4—2441no—
296C2×C4#1451975312yes—
297C2×C4#1451975311yescharacteristic
298C2×C4#1451976612yes—
299C2×C4#1451976611yescharacteristic
300C8—4881no—
301C8—4882no—
302C22—21no—
303C22—21no—
304C22—4881no—
305C22—4881no—
306C22—2442no—
307C22#1683967311yescharacteristic
308C22#1683994911yescharacteristic
309C22#1683994911yescharacteristic
310C4—21no—
311C4—22no—
312C4—21no—
313C4—22no—
314C4—42no—
315C4—82no—
316C4—81no—
317C4—82no—
318C4—2442no—
319C2—21no—
320C2—21no—
321C2—4882no—
322C2#2144698211yescharacteristic
323C2#2144784011yescharacteristic
324C2#2144784511yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC22×Q8×C61C23
FrattiniC22×Q8C22×D61
FittingC61×#2809C2
socleC22×C61B(4,𝔽2)
prime radicalC2×#14519825C22
hypercenter#2809D61
hypocenterC61#13379
LCSG ▷ C22×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2809

Complements

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

NH ≅ G/Ntype
3. #21443946C2semidirect
4. #21443946C2semidirect
5. C61×#2809C2semidirect
157. #2809D61semidirect
181. C61#13379semidirect

Maximal subgroups

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

subgroupindex
#214457772
#214439462
C61×#28092
#1337961