Finite Group Databasebeta

← #29663113

Subgroups — #29663113

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456949C211yesmaximal, characteristic
2#21456744C211yesmaximal, characteristic
3C2×#16840158C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic
5#21445317C214yesmaximal, complemented
6#21445193C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444501C211yesmaximal, characteristic, complemented
9C61×#1549C211yesmaximal, characteristic, complemented
10C2×#14519785—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13#16840160—21no—
14#16840158—21no—
15#16840158—21no—
16#16840158C2212yes—
17#16840158C2212yes—
18#16840136—21no—
19#16840112—21no—
20#16840109—21no—
21#16840081—22no—
22#16840053C2214yes—
23#16839964C2214yes—
24#16839901—22no—
25#16839839—21no—
26#16839839C2212yes—
27#16839827—21no—
28#16839827C2212yes—
29#16839815C2211yescharacteristic
30C2×#14519708—21no—
31C2×#14519708—21no—
32#16839730—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yes—
35#16839512C2212yes—
36#16839510C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yes—
41C2×#11758222—21no—
42C22×Dic122—21no—
43C2×#11758201—24no—
44#14519785—44no—
45C2×#11758189—22no—
46C22×#92418D411yescharacteristic
47C22×#92418D411yescharacteristic
48C2×C8×D61—22no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777D411yescharacteristic
53#14519777C2312yes—
54#14519774D411yescharacteristic
55#14519765—21no—
56#14519751—21no—
57#14519751D412yes—
58C4×D244—21no—
59C4×D244C2312yes—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62#14519709—24no—
63#14519708—22no—
64#14519708—22no—
65#14519708—22no—
66#14519708—22no—
67C4⋊#92418C2312yes—
68#14519701—24no—
69#14519699—24no—
70C22⋊#92418—21no—
71#14519689—22no—
72#14519689—22no—
73C4×#92418—21no—
74C4×#92418D411yescharacteristic
75C4×#92418D411yescharacteristic
76C4×#92418C2311yescharacteristic
77C4×#92418C2311yescharacteristic
78C22×Q8×C61—21no—
79C2×M16×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C42×C61C2311yescharacteristic
87C4⋊C8×C61—21no—
88C4⋊C8×C61C2312yes—
89C4⋊C8×C61C2312yes—
90C22⋊C8×C61—21no—
91C4×C8×C61—21no—
92C2×Q8×C61—22no—
93C2×Q8×C61—41no—
94C2×Q8×C61—82no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C2×C22⋊D61—22no—
100C22×Dic61—41no—
101#11758222—42no—
102#11758222—42no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244C2×D412yes—
106C2×C4×D61—22no—
107C2×Dic122—21no—
108C2×Dic122—22no—
109C2×Dic122—41no—
110C2×Dic122—82no—
111M16×C61—42no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C42×C61C2411yescharacteristic
122#11758201—84no—
123#11758198—22no—
124#11758198—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130C4⋊Dic61C2×D412yes—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418SD1614yescomplemented
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137C2×#92418C2×D411yescharacteristic
138#11758189—44no—
139C8×D61—44no—
140C23×C61#12111yescharacteristic
141C22×D61—22no—
142Q8×C61—44no—
143Q8×C61—82no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—42no—
147C2×C4×C61—82no—
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61C2×SD1612yes—
152C2×C4×C61#13712yes—
153C2×C4×C61C22×D411yescharacteristic
154C2×C4×C61C22×D411yescharacteristic
155C2×C4×C61C2×#4111yescharacteristic
156C22⋊D61—44no—
157C2×Dic61—22no—
158C2×Dic61—42no—
159C2×Dic61—82no—
160D244—24no—
161D244—42no—
162D244—42no—
163C4×D61—42no—
164C4×D61—42no—
165Dic122—42no—
166Dic122—82no—
167Dic122—82no—
168C8×C61—42no—
169C8×C61—42no—
170#92418—21no—
171#92418—21no—
172#92418—42no—
173#92418C2×SD1612yes—
174#92418C2×SD1612yes—
175#19191—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61#46112yes—
179C22×C61C2×#12111yescharacteristic
180C2×D61—42no—
181C2×D61—42no—
182C2×D61—42no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—82no—
186C4×C61#46112yes—
187C4×C61#47112yes—
188C4×C61C22×SD1611yescharacteristic
189C4×C61C2×#13711yescharacteristic
190C4×C61#57712yes—
191Dic61—42no—
192Dic61—82no—
193#3227—611no—
194#3206—611no—
195C2×#498—611no—
196#2931—611no—
197#1691—614no—
198#1671—614no—
199#1564—611no—
200#1549D6111yescharacteristic, complemented
201#1548—611no—
202C2×C61—22no—
203C2×C61C2×#46111yescharacteristic
204C2×C61#305711yescharacteristic
205C2×C61C2×#47111yescharacteristic
206D61—82no—
207C2×#131—1221no—
208#532—611no—
209C2×C4⋊Q8—611no—
210C2×C4⋊Q8C2×D6111yescharacteristic
211#498—612no—
212#498—612no—
213#498—1221no—
214#498—1221no—
215#498—1221no—
216#492—1221no—
217#487—1221no—
218#486—1221no—
219#481—1222no—
220#476—614no—
221#461—614no—
222#448—1222no—
223#436—612no—
224#436—1221no—
225C8×D4—612no—
226C8×D4—1221no—
227#431—611no—
228#431C2×D6111yescharacteristic
229C2×#106—1221no—
230C2×#106—1221no—
231#423C2×D6111yescharacteristic
232#408—1221no—
233C2×C4×C8—611no—
234#335—612no—
235#334—612no—
236#333—614no—
237#332C2×D6114yes—
238C61#1919111yescharacteristic, complemented, Sylow
239C2×#41—1221no—
240C22×Q8—21no—
241C22×Q8—1221no—
242C2×SD16—1224no—
243#131—2444no—
244C2×M16—21no—
245C2×M16—1222no—
246C22×C8—611no—
247C22×C8—611no—
248C22×C8—1222no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—611no—
253C4⋊Q8—612no—
254C4⋊Q8—1221no—
255C4⋊Q8—1221no—
256C4⋊Q8—1221no—
257C4⋊Q8C22×D6112yes—
258#127—611no—
259#124—1221no—
260#121—612no—
261#121—1221no—
262C4×D4—612no—
263C4×D4—1221no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C42C22×D6111yescharacteristic
269#106—1222no—
270#106—1222no—
271#106—1222no—
272#106—1222no—
273#106—1224no—
274C4⋊C8—21no—
275C4⋊C8—612no—
276C4⋊C8C22×D6112yes—
277C4⋊C8C22×D6112yes—
278#103—1224no—
279#102—1224no—
280C22⋊C8—21no—
281C22⋊C8—1222no—
282C22⋊C8—1222no—
283C22⋊C8—1221no—
284C4×C8—21no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289C4×C8—1221no—
290#41—2442no—
291#41—2442no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8—1221no—
296C2×Q8—1222no—
297C2×Q8—4882no—
298C2×Q8—2441no—
299C2×D4—612no—
300C2×D4—1222no—
301C2×D4—1221no—
302C2×D4—1221no—
303C22×C4—41no—
304C22×C4—1222no—
305C22×C4—2441no—
306C22×C4#1175822211yescharacteristic
307C22×C4D4×D6111yescharacteristic
308C22×C4D4×D6111yescharacteristic
309SD16—4884no—
310M16—42no—
311M16—2444no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—611no—
315C2×C8—611no—
316C2×C8—614no—
317C2×C8—611no—
318C2×C8—611no—
319C2×C8—1222no—
320C2×C8—1222no—
321C2×C8—2444no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—612no—
327C4⋊C4—1221no—
328C4⋊C4—1222no—
329C4⋊C4—1222no—
330C4⋊C4—2441no—
331C4⋊C4—2441no—
332C22⋊C4—1222no—
333C22⋊C4—1222no—
334C42#1175822211yescharacteristic
335C42D4×D6111yescharacteristic
336C42D4×D6111yescharacteristic
337C42C23×D6111yescharacteristic
338C23—1222no—
339C23#1451975011yescharacteristic
340Q8—44no—
341Q8—82no—
342Q8—4882no—
343Q8—4882no—
344Q8—2442no—
345D4—1224no—
346D4—2444no—
347D4—2442no—
348D4—2442no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—1222no—
354C2×C4—4882no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4#1451975011yescharacteristic
359C2×C4#1451975011yescharacteristic
360C2×C4#1451975011yescharacteristic
361C2×C4SD16×D6112yes—
362C2×C4#1451980912yes—
363C2×C4C2×D4×D6111yescharacteristic
364C2×C4C2×D4×D6111yescharacteristic
365C2×C4C2×#1175822211yescharacteristic
366C8—42no—
367C8—42no—
368C8—612no—
369C8—612no—
370C8—1221no—
371C8—1221no—
372C8—2442no—
373C22—21no—
374C22—21no—
375C22—2442no—
376C22—2442no—
377C22—2442no—
378C22#1683996612yes—
379C22C2×#1451975011yescharacteristic
380C4—21no—
381C4—21no—
382C4—82no—
383C4—4882no—
384C4—2442no—
385C4#1683996612yes—
386C4#1684002812yes—
387C4C2×SD16×D6111yescharacteristic
388C4C2×#1451980911yescharacteristic
389C4#1684057212yes—
390C2—22no—
391C2—4882no—
392C2C2×#1683996611yescharacteristic
393C2#2145545711yescharacteristic
394C2C2×#1684002811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1549D61
hypocenterC61#19191
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
3. C2×#16840158C2semidirect
5. #21445317C2semidirect
6. #21445193C2semidirect
7. #21444565C2semidirect
8. #21444501C2semidirect
9. C61×#1549C2semidirect
133. C2×#92418SD16semidirect
200. #1549D61semidirect
238. C61#19191semidirect

Maximal subgroups

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

subgroupindex
#214569492
#214567442
C2×#168401582
#214545302
#214453172
#214451932
#214445652
#214445012
C61×#15492
#1919161