Finite Group Databasebeta

← #29665315

Subgroups — #29665315

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456767C211yesmaximal, characteristic
2#21455626C211yesmaximal, characteristic
3C2×#16840044C211yesmaximal, characteristic
4#21454899C211yesmaximal, characteristic
5#21445276C214yesmaximal, complemented
6#21445188C214yesmaximal, complemented
7#21444595C211yesmaximal, characteristic
8#21444474C211yesmaximal, characteristic
9C61×#1549C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11C2×C4×Dic122C2211yescharacteristic
12#16840141C2212yes—
13#16840112C2212yes—
14#16840070—22no—
15#16840058C2212yes—
16#16840048C2212yes—
17#16840044—22no—
18#16840044C2214yes—
19#16839969—22no—
20#16839918—22no—
21C8×Dic122—22no—
22#16839886C2214yes—
23#16839815C2211yescharacteristic
24#16839812—22no—
25#16839804—21no—
26C2×#14519708—21no—
27C2×C4⋊#92418C2211yescharacteristic
28#16839772—21no—
29C2×#14519700—21no—
30#16839514C2214yes—
31#16839510C2212yes—
32#16839509C2212yes—
33C2×C4⋊Q8×C61C2211yescharacteristic
34C61×#431C2211yescharacteristic
35C61×#423C2211yescharacteristic
36C61×#332C2214yes—
37C22×Dic122D411yescharacteristic
38C2×#11758202—24no—
39C22×#92418—21no—
40#14519777C2311yescharacteristic
41#14519777C2311yescharacteristic
42#14519769—21no—
43#14519756—21no—
44#14519756—21no—
45C4×Dic122—21no—
46C4×Dic122—21no—
47C4×Dic122—21no—
48C4×Dic122—21no—
49C4×Dic122D412yes—
50C4×Dic122C2312yes—
51#14519736—21no—
52C2×C4⋊Dic61D411yescharacteristic
53C2×#11758196—21no—
54C2×C4×Dic61—21no—
55#14519711—22no—
56#14519709—22no—
57#14519708—24no—
58C4⋊#92418—21no—
59C4⋊#92418D412yes—
60C4⋊#92418C2312yes—
61#14519706—24no—
62#14519702—22no—
63#14519702—22no—
64#14519701—24no—
65#14519700—24no—
66C22⋊#92418D412yes—
67C4×#92418C2312yes—
68C8×Dic61—24no—
69C22×Q8×C61—21no—
70C2×M16×C61—21no—
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61C2312yes—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61—21no—
79C4⋊C8×C61C2312yes—
80C4⋊C8×C61C2312yes—
81C22⋊C8×C61—21no—
82C4×C8×C61—21no—
83C2×Q8×C61—22no—
84C2×Q8×C61—41no—
85C2×Q8×C61—82no—
86C22×C4×C61—41no—
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—21no—
92C2×Dic122—21no—
93C2×Dic122—22no—
94C2×Dic122—21no—
95C2×Dic122—21no—
96C2×Dic122Q1614yescomplemented
97C2×Dic122C2×D412yes—
98M16×C61—42no—
99C2×C8×C61—22no—
100C2×C8×C61—22no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—41no—
105C42×C61C2×D411yescharacteristic
106C42×C61C2×D411yescharacteristic
107C42×C61#4111yescharacteristic
108C42×C61C2411yescharacteristic
109C22⋊Dic61—41no—
110#11758202—84no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61C2×D412yes—
117#11758196—22no—
118#11758196—22no—
119#11758196—41no—
120#11758196—41no—
121C4×Dic61—22no—
122C4×Dic61—22no—
123C4×Dic61—41no—
124C2×#92418—22no—
125C2×#92418—42no—
126C2×#92418C2×D414yes—
127C23×C61#12111yescharacteristic
128Q8×C61—44no—
129Q8×C61—82no—
130C2×C4×C61—22no—
131C2×C4×C61—41no—
132C2×C4×C61—42no—
133C2×C4×C61—82no—
134C2×C4×C61#12111yescharacteristic
135C2×C4×C61#12111yescharacteristic
136C2×C4×C61#12111yescharacteristic
137C2×C4×C61C2×Q1612yes—
138C2×C4×C61#13512yes—
139C2×C4×C61C22×D411yescharacteristic
140C2×C4×C61C22×D411yescharacteristic
141C2×C4×C61C2×#4111yescharacteristic
142C2×Dic61—22no—
143C2×Dic61—24no—
144C2×Dic61—22no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—81no—
148Dic122—22no—
149Dic122—24no—
150Dic122—42no—
151Dic122—81no—
152Dic122C2×Q1614yes—
153C8×C61—42no—
154C8×C61—42no—
155#92418—24no—
156#92418—42no—
157#19399—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#46712yes—
161C22×C61C2×#12111yescharacteristic
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—82no—
166C4×C61#46712yes—
167C4×C61C22×Q1611yescharacteristic
168C4×C61C2×#13511yescharacteristic
169C4×C61#57712yes—
170Dic61—24no—
171Dic61—42no—
172Dic61—81no—
173Dic61—81no—
174#3209—611no—
175#3078—611no—
176C2×#475—611no—
177#2982—611no—
178#1684—614no—
179#1670—614no—
180#1571—611no—
181#1549D6111yescharacteristic
182#1542—611no—
183C2×C61—22no—
184C2×C61#304311yescharacteristic
185C2×C61C2×#46711yescharacteristic
186C2×C61#306911yescharacteristic
187#533—611no—
188C2×C4⋊Q8C2×D6111yescharacteristic
189C2×C4×Q8—611no—
190#494—612no—
191#487—612no—
192#479—1222no—
193#477—612no—
194#475—614no—
195#475—612no—
196#475—1222no—
197#462—1222no—
198#451—1222no—
199C8×Q8—614no—
200C8×Q8—1222no—
201#431—611no—
202#431—1222no—
203#431C2×D6111yescharacteristic
204#428—1221no—
205C2×#106—1221no—
206#423C2×D6111yescharacteristic
207C2×C4⋊C8—611no—
208#419—1221no—
209C2×#103—1221no—
210#335—614no—
211#333—612no—
212#332—612no—
213#332C2×D6114yes—
214C61#1939911yescharacteristic, complemented, Sylow
215C22×Q8—21no—
216C22×Q8—611no—
217C2×Q16—1224no—
218C2×M16—21no—
219C22×C8—1221no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—611no—
224C4⋊Q8—611no—
225C4⋊Q8C22×D6112yes—
226#125—1221no—
227C22⋊Q8—1221no—
228C22⋊Q8—1221no—
229C4×Q8—612no—
230C4×Q8—612no—
231C4×Q8—1221no—
232C4×Q8—1221no—
233C4×Q8—1221no—
234C4×Q8—1221no—
235#117—1221no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—611no—
239C2×C4⋊C4—1221no—
240C2×C42—1221no—
241C2×C42C22×D6111yescharacteristic
242#107—1222no—
243#106—1224no—
244#106—1222no—
245C4⋊C8—21no—
246C4⋊C8—612no—
247C4⋊C8—612no—
248C4⋊C8—1224no—
249C4⋊C8—1221no—
250C4⋊C8C22×D6112yes—
251C4⋊C8C22×D6112yes—
252#103—1224no—
253#103—1222no—
254#103—1222no—
255#103—1224no—
256C22⋊C8—21no—
257C22⋊C8—612no—
258C4×C8—21no—
259C4×C8—612no—
260C4×C8—1224no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—82no—
264C2×Q8—612no—
265C2×Q8—614no—
266C2×Q8—1221no—
267C2×Q8—1222no—
268C2×Q8—1221no—
269C2×Q8—1221no—
270C22×C4—41no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4#1175822211yescharacteristic
274C22×C4D4×D6111yescharacteristic
275C22×C4D4×D6111yescharacteristic
276Q16—4884no—
277M16—42no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—614no—
281C2×C8—1222no—
282C2×C8—2442no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—612no—
288C4⋊C4—1222no—
289C4⋊C4—1221no—
290C4⋊C4—1222no—
291C4⋊C4—1221no—
292C4⋊C4—1221no—
293C4⋊C4—1221no—
294C4⋊C4—1221no—
295C4⋊C4—2441no—
296C4⋊C4—2441no—
297C22⋊C4—2441no—
298C42—1222no—
299C42—1222no—
300C42—2441no—
301C42#1175822211yescharacteristic
302C42D4×D6111yescharacteristic
303C42D4×D6111yescharacteristic
304C42C23×D6111yescharacteristic
305C23#1451975011yescharacteristic
306Q8—44no—
307Q8—82no—
308Q8—614no—
309Q8—1224no—
310Q8—1222no—
311Q8—4881no—
312Q8—2442no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—82no—
317C2×C4—1222no—
318C2×C4—1224no—
319C2×C4—1222no—
320C2×C4—4881no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4#1451975011yescharacteristic
324C2×C4#1451975011yescharacteristic
325C2×C4#1451975011yescharacteristic
326C2×C4Q16×D6112yes—
327C2×C4#1451979812yes—
328C2×C4C2×D4×D6111yescharacteristic
329C2×C4C2×D4×D6111yescharacteristic
330C2×C4C2×#1175822211yescharacteristic
331C8—42no—
332C8—42no—
333C8—1224no—
334C8—2442no—
335C22—21no—
336C22—21no—
337C22#1684000212yes—
338C22C2×#1451975011yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—82no—
343C4—1224no—
344C4—4881no—
345C4—4881no—
346C4—2442no—
347C4#1684000212yes—
348C4C2×Q16×D6111yescharacteristic
349C4C2×#1451979811yescharacteristic
350C4#1684057212yes—
351C2—22no—
352C2#2145533911yescharacteristic
353C2C2×#1684000211yescharacteristic
354C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C2×#121
prime radicalC23×C61#121
hypercenter#1549D61
hypocenterC61#19399
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
5. #21445276C2semidirect
6. #21445188C2semidirect
96. C2×Dic122Q16semidirect
214. C61#19399semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214556262
C2×#168400442
#214548992
#214452762
#214451882
#214445952
#214444742
C61×#15492
#1939961