Finite Group Databasebeta

← #29836375

Subgroups — #29836375

Order 15616.

528 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456403C211yesmaximal, characteristic, complemented
2#21456395C211yesmaximal, characteristic
3#21447969C212yesmaximal, complemented
4#21447957C211yesmaximal, characteristic, complemented
5#21447956C211yesmaximal, characteristic, complemented
6#21447935C212yesmaximal, complemented
7#21447929C212yesmaximal, complemented
8#21447861C211yesmaximal, characteristic, complemented
9#21447857C212yesmaximal, complemented
10#21447573C211yesmaximal, characteristic, complemented
11C61×#2208C211yesmaximal, characteristic, complemented
12#16840317—21no—
13#16840309—21no—
14C2×#14519720C2212yes—
15C2×#14519716C2212yes—
16#16840167C2211yescharacteristic
17#16840166C2211yescharacteristic
18#16840137C2211yescharacteristic
19#16840112—21no—
20#16840107—21no—
21#16840000—21no—
22#16840000C2212yes—
23#16839999C2211yescharacteristic
24#16839997—21no—
25#16839995C2211yescharacteristic
26Q16×Dic61—21no—
27Q16×Dic61C2212yes—
28C4×D488C2211yescharacteristic
29C4×#11758199—21no—
30C4×C8×D61C2211yescharacteristic
31#16839588C2212yes—
32#16839587C2212yes—
33C4⋊#11758185C2211yescharacteristic
34#16839580C2212yes—
35#16839566C2212yes—
36#16839564C2212yes—
37#16839563C2212yes—
38C2×SD32×C61C2212yes—
39C61×#494C2211yescharacteristic
40C4×D8×C61C2211yescharacteristic
41C61×C4⋊C16C2211yescharacteristic
42C61×#358C2212yes—
43C2×#11758222—41no—
44C2×D488C2311yescharacteristic
45C2×#11758199—21no—
46C2×C8×D61D411yescharacteristic
47C2×C8×D61C2311yescharacteristic
48#14519777—21no—
49#14519776—21no—
50#14519773—21no—
51#14519768—41no—
52#14519763—41no—
53#14519757—41no—
54#14519753—22no—
55#14519752—21no—
56#14519750—41no—
57#14519748—21no—
58Q8×Dic61—22no—
59#14519742—41no—
60C4×D244—21no—
61C4×C22⋊D61—41no—
62#14519735—41no—
63C42×D61D411yescharacteristic
64#14519720—42no—
65#14519716—42no—
66C2×#11758185C2312yes—
67#14519711—21no—
68#14519711C2311yescharacteristic
69#14519711C2312yes—
70#14519710—21no—
71#14519701—22no—
72#14519701—22no—
73#14519699—21no—
74#14519698—21no—
75#14519697—21no—
76C4×#92418D411yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C2×Q16×C61—21no—
80C2×Q16×C61C2312yes—
81C2×D8×C61C2311yescharacteristic
82C4⋊Q8×C61—21no—
83C4×D4×C61—21no—
84SD32×C61—42no—
85C2×C16×C61C2312yes—
86#107×C61C2311yescharacteristic
87#102×C61—21no—
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—22no—
90C2×D4×C61—21no—
91C22×C4×C61—41no—
92C2×C22⋊D61—41no—
93C2×C22⋊D61—41no—
94C2×C22⋊D61—41no—
95#11758222—84no—
96C2×D244—21no—
97C2×D244—41no—
98C2×C4×D61—41no—
99C2×C4×D61D811yescharacteristic
100C2×C4×D61D811yescharacteristic
101C2×C4×D61C2×D411yescharacteristic
102C2×Dic122—41no—
103C2×Dic122—41no—
104Q16×C61—22no—
105Q16×C61—22no—
106Q16×C61—42no—
107D8×C61—21no—
108D8×C61—21no—
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2411yescharacteristic
111C4⋊C4×C61—21no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C42×C61C2×D411yescharacteristic
116C22⋊Dic61—41no—
117#11758199—42no—
118#11758198—42no—
119#11758198—41no—
120#11758198—41no—
121C4⋊Dic61—21no—
122C4⋊Dic61—22no—
123C4⋊Dic61—41no—
124C4⋊Dic61—42no—
125#11758196—41no—
126#11758196—42no—
127C4×Dic61—42no—
128C4×Dic61D811yescharacteristic
129C4×Dic61D811yescharacteristic
130C4×Dic61C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133D488—21no—
134D488—21no—
135C8×D61—21no—
136C8×D61—21no—
137C8×D61—41no—
138C16×C61—22no—
139#11758185—22no—
140C23×C61—41no—
141C22×D61—41no—
142C22×D61C2×D811yescharacteristic
143Q8×C61—42no—
144Q8×C61—42no—
145D4×C61—41no—
146D4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61—82no—
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C22⋊D61—82no—
154C22⋊D61—82no—
155C22⋊D61—82no—
156C2×Dic61—42no—
157C2×Dic61—41no—
158C2×Dic61C2×D811yescharacteristic
159C2×Dic61C2×D811yescharacteristic
160C2×Dic61C2×D811yescharacteristic
161D244—41no—
162D244—41no—
163D244—82no—
164C4×D61—21no—
165C4×D61—21no—
166C4×D61—22no—
167C4×D61—22no—
168C4×D61—82no—
169Dic122—82no—
170Dic122—82no—
171C8×C61—21no—
172C8×C61H(5,𝔽2)11yescharacteristic
173C8×C612−1+411yescharacteristic
174#92418—21no—
175#92418—21no—
176#33820—611nomaximal, Sylow
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61C22×D811yescharacteristic
181C2×D61—21no—
182C2×D61—21no—
183C2×D61—21no—
184C2×D61—81no—
185C2×D61—81no—
186C2×D61—81no—
187C4×C61—21no—
188C4×C61—81no—
189C4×C61—82no—
190C4×C61#50912yes—
191C4×C61#53711yescharacteristic
192C4×C61#57511yescharacteristic
193Dic61—21no—
194Dic61—21no—
195Dic61—81no—
196Dic61—82no—
197Dic61#51012yes—
198#3170—611no—
199#3169—611no—
200#2227—612no—
201#2224—611no—
202#2224—611no—
203#2220—612no—
204#2220—612no—
205#2208—611no—
206#2208D6111yescharacteristic, complemented
207#2207—612no—
208#2150—611no—
209C2×C61—161no—
210C2×C61#324511yescharacteristic
211C2×C61C2×#50911yescharacteristic
212C2×C61C2×#51011yescharacteristic
213D61—41no—
214D61—161no—
215#533—1221no—
216#532—1221no—
217C2×SD32—612no—
218C2×SD32C2×D6112yes—
219C2×M32—612no—
220#500—611no—
221#500—611no—
222#494C2×D6111yescharacteristic
223#493—611no—
224#487—1221no—
225#486—1221no—
226#467—612no—
227#467—1221no—
228#466—611no—
229#466—611no—
230#466—1221no—
231C4×Q16—612no—
232C4×Q16—1221no—
233C4×D8—611no—
234C4×D8—1221no—
235C4×D8C2×D6111yescharacteristic
236C2×C4×C8—611no—
237#366—612no—
238#366—612no—
239C4⋊C16—612no—
240C4⋊C16—611no—
241C4⋊C16C2×D6111yescharacteristic
242#358—612no—
243#358C2×D6112yes—
244#357—612no—
245#357—612no—
246C61#3382011yescharacteristic, complemented, Sylow
247C2×#41—2441no—
248C2×Q16—21no—
249C2×Q16C22×D6112yes—
250C2×D8—611no—
251C2×D8—1221no—
252C2×D8C22×D6111yescharacteristic
253C22×C8—611no—
254C22×C8—611no—
255C4⋊Q8—21no—
256C4⋊Q8—1221no—
257C4⋊Q8—1221no—
258#127—1221no—
259#125—2441no—
260#124—2441no—
261C22⋊Q8—1222no—
262C22⋊Q8—2441no—
263#121—1221no—
264#121—1221no—
265#121—2441no—
266C4×Q8—1222no—
267C4×Q8—2441no—
268C4×D4—21no—
269C4×D4—1221no—
270C4×D4—2441no—
271#117—2441no—
272C2×C42—611no—
273SD32—42no—
274SD32—2442no—
275M32—2442no—
276C2×C16—612no—
277C2×C16C22×D6112yes—
278#107—612no—
279#107—611no—
280#107—1221no—
281#107—1221no—
282#107C22×D6111yescharacteristic
283#103—1222no—
284#103—1222no—
285#102—21no—
286#102—1221no—
287#102—1221no—
288#102—1221no—
289C4×C8—611no—
290C4×C8—611no—
291C4×C8—611no—
292C4×C8C22×D6111yescharacteristic
293#41—4884no—
294C2×Q8—22no—
295C2×Q8—2441no—
296C2×Q8—2441no—
297C2×D4—21no—
298C2×D4—1221no—
299C2×D4—2441no—
300C2×D4—2441no—
301C2×D4—2441no—
302C2×D4—2441no—
303C22×C4—41no—
304C22×C4—611no—
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—2441no—
308Q16—22no—
309Q16—22no—
310Q16—42no—
311D8—21no—
312D8—21no—
313D8—1221no—
314D8—1221no—
315D8—2442no—
316C2×C8—611no—
317C2×C8—611no—
318C2×C8—1221no—
319C2×C8—1221no—
320C2×C8—2441no—
321C2×C8D4×D6111yescharacteristic
322C2×C8C23×D6111yescharacteristic
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—1221no—
327C4⋊C4—1222no—
328C4⋊C4—2441no—
329C4⋊C4—2442no—
330C4⋊C4—2442no—
331C4⋊C4—2441no—
332C22⋊C4—41no—
333C22⋊C4—2441no—
334C22⋊C4—2442no—
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C42—611no—
338C42—611no—
339C42—611no—
340C42—2442no—
341C42D4×D6111yescharacteristic
342C16—22no—
343C16—1222no—
344C23—41no—
345C23—611no—
346C23—2441no—
347Q8—42no—
348Q8—42no—
349Q8—4882no—
350Q8—4882no—
351D4—41no—
352D4—41no—
353D4—4882no—
354D4—4882no—
355D4—4882no—
356D4—4882no—
357D4—2441no—
358D4—2441no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—82no—
362C2×C4—611no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—1222no—
366C2×C4—1222no—
367C2×C4—1221no—
368C2×C4—1221no—
369C2×C4—4882no—
370C2×C4—2442no—
371C2×C4—2441no—
372C2×C4D8×D6111yescharacteristic
373C2×C4D8×D6111yescharacteristic
374C2×C4C2×D4×D6111yescharacteristic
375C8—21no—
376C8—1221no—
377C8—1221no—
378C8#1451982211yescharacteristic
379C8#1451982411yescharacteristic
380C22—81no—
381C22—81no—
382C22—81no—
383C22—1221no—
384C22—1221no—
385C22—1221no—
386C22—4881no—
387C22—4881no—
388C22—4881no—
389C22C2×D8×D6111yescharacteristic
390C4—21no—
391C4—82no—
392C4—81no—
393C4—612no—
394C4—1221no—
395C4—1221no—
396C4—4881no—
397C4—4882no—
398C4#1684019912yes—
399C4#1684034111yescharacteristic
400C4#1684055811yescharacteristic
401C2—161no—
402C2—9761no—
403C2—2441no—
404C2#2145712711yescharacteristic
405C2C2×#1684019911yescharacteristic
406C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2208C2
socleC22×C61C22×D8
prime radical#21456395C2
hypercenter#2208D61
hypocenterC61#33820
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2208

Complements

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

NH ≅ G/Ntype
1. #21456403C2semidirect
3. #21447969C2semidirect
4. #21447957C2semidirect
5. #21447956C2semidirect
6. #21447935C2semidirect
7. #21447929C2semidirect
8. #21447861C2semidirect
9. #21447857C2semidirect
10. #21447573C2semidirect
11. C61×#2208C2semidirect
206. #2208D61semidirect
246. C61#33820semidirect

Maximal subgroups

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

subgroupindex
#214564032
#214563952
#214479692
#214479572
#214479562
#214479352
#214479292
#214478612
#214478572
#214475732
C61×#22082
#3382061