Finite Group Databasebeta

← #29836383

Subgroups — #29836383

Order 15616.

484 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456403C211yesmaximal, characteristic
2#21456402C211yesmaximal, characteristic
3#21447962C211yesmaximal, characteristic, complemented
4#21447961C211yesmaximal, characteristic, complemented
5#21447959C212yesmaximal, complemented
6#21447926C212yesmaximal
7#21447924C212yesmaximal
8#21447867C212yesmaximal, complemented
9#21447866C211yesmaximal, characteristic, complemented
10#21447573C211yesmaximal, characteristic
11C61×#2209C211yesmaximal, characteristic, complemented
12#16840317—21no—
13#16840316—21no—
14C2×#14519723C2212yes—
15C2×#14519716C2212yes—
16#16840167C2211yescharacteristic
17#16840166C2211yescharacteristic
18#16840140C2211yescharacteristic
19#16840112—21no—
20#16840111—21no—
21#16840004C2211yescharacteristic
22#16840002—21no—
23#16840000—21no—
24#16840000C2211yescharacteristic
25#16840000C2212yes—
26C4×Dic244C2211yescharacteristic
27Q16×Dic61—21no—
28Q16×Dic61C2212yes—
29C4×#11758202—21no—
30C4×C8×D61C2211yescharacteristic
31#16839590C2212yes—
32#16839589C2212yes—
33C4⋊#11758185C2211yescharacteristic
34#16839580C2212yes—
35#16839567C2212yes—
36#16839566C2212yes—
37#16839566C2212yes—
38C2×Q32×C61C2212yes—
39C61×#494C2211yescharacteristic
40C4×Q16×C61C2211yescharacteristic
41C61×C4⋊C16C2211yescharacteristic
42C61×#358C2212yes—
43C2×Dic244C2311yescharacteristic
44C2×#11758202—21no—
45C2×C8×D61D411yescharacteristic
46C2×C8×D61C2311yescharacteristic
47#14519777—21no—
48#14519776—21no—
49#14519775—21no—
50#14519768—41no—
51#14519767—41no—
52#14519757—21no—
53#14519757—41no—
54#14519755—41no—
55#14519753—22no—
56#14519753—21no—
57Q8×Dic61—22no—
58C4×Dic122—21no—
59C4×Dic122—41no—
60#14519742—41no—
61#14519735—41no—
62#14519734—41no—
63C42×D61D411yescharacteristic
64#14519723—42no—
65#14519716—42no—
66C2×#11758185C2312yes—
67#14519711—21no—
68#14519711C2311yescharacteristic
69#14519711C2312yes—
70#14519710—21no—
71#14519702—21no—
72#14519701—22no—
73#14519701—21no—
74#14519701—22no—
75#14519700—21no—
76C4×#92418D411yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C2×Q16×C61—21no—
80C2×Q16×C61C2312yes—
81C2×Q16×C61C2311yescharacteristic
82C4⋊Q8×C61—21no—
83C4×Q8×C61—21no—
84Q32×C61—42no—
85C2×C16×C61C2312yes—
86#107×C61C2311yescharacteristic
87#103×C61—21no—
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—21no—
90C2×Q8×C61—22no—
91C2×C4×D61D811yescharacteristic
92C2×C4×D61D811yescharacteristic
93C2×C4×D61C2×D411yescharacteristic
94C2×Dic122—21no—
95C2×Dic122—41no—
96C2×Dic122—41no—
97Q16×C61—21no—
98Q16×C61—22no—
99Q16×C61—22no—
100Q16×C61—21no—
101Q16×C61—42no—
102C2×C8×C61C2×D411yescharacteristic
103C2×C8×C61C2411yescharacteristic
104C4⋊C4×C61—21no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C42×C61—41no—
109C42×C61C2×D411yescharacteristic
110#11758202—42no—
111#11758198—41no—
112#11758198—41no—
113#11758198—42no—
114C4⋊Dic61—22no—
115C4⋊Dic61—21no—
116C4⋊Dic61—41no—
117C4⋊Dic61—41no—
118C4⋊Dic61—42no—
119#11758196—41no—
120#11758196—41no—
121#11758196—41no—
122#11758196—42no—
123#11758196—41no—
124#11758196—41no—
125C4×Dic61—41no—
126C4×Dic61—42no—
127C4×Dic61D811yescharacteristic
128C4×Dic61D811yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132Dic244—21no—
133Dic244—21no—
134C8×D61—21no—
135C8×D61—21no—
136C8×D61—41no—
137C16×C61—22no—
138#11758185—22no—
139C22×D61C2×D811yescharacteristic
140Q8×C61—41no—
141Q8×C61—42no—
142Q8×C61—41no—
143Q8×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C22×D411yescharacteristic
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—42no—
153C2×Dic61C2×D811yescharacteristic
154C2×Dic61C2×D811yescharacteristic
155C2×Dic61C2×D811yescharacteristic
156C4×D61—21no—
157C4×D61—21no—
158C4×D61—22no—
159C4×D61—22no—
160Dic122—41no—
161Dic122—41no—
162Dic122—82no—
163Dic122—82no—
164C8×C61—21no—
165C8×C61H(5,𝔽2)11yescharacteristic
166C8×C612−1+411yescharacteristic
167#92418—21no—
168#92418—21no—
169#33821—611nomaximal, Sylow
170C22×C61C22×D811yescharacteristic
171C2×D61—21no—
172C2×D61—21no—
173C2×D61—21no—
174C4×C61—21no—
175C4×C61—81no—
176C4×C61—82no—
177C4×C61—81no—
178C4×C61#51012yes—
179C4×C61#53711yescharacteristic
180C4×C61#57511yescharacteristic
181Dic61—21no—
182Dic61—21no—
183Dic61—81no—
184Dic61—82no—
185Dic61—81no—
186Dic61#51012yes—
187#3170—611no—
188#3170—611no—
189#2225—611no—
190#2225—611no—
191#2225—612no—
192#2219—612no—
193#2219—612no—
194#2209—612no—
195#2209—611no—
196#2209D6111yescharacteristic, complemented
197#2150—611no—
198C2×C61#324511yescharacteristic
199C2×C61C2×#51011yescharacteristic
200C2×C61C2×#51011yescharacteristic
201D61—41no—
202#533—1221no—
203#533—1221no—
204C2×Q32—612no—
205C2×Q32C2×D6112yes—
206C2×M32—612no—
207#500—611no—
208#500—611no—
209#494—611no—
210#494C2×D6111yescharacteristic
211#487—1221no—
212#487—1221no—
213#467—611no—
214#467—612no—
215#467—611no—
216#467—1221no—
217#467—1221no—
218C4×Q16—612no—
219C4×Q16—611no—
220C4×Q16—1221no—
221C4×Q16—1221no—
222C4×Q16C2×D6111yescharacteristic
223C2×C4×C8—611no—
224#367—612no—
225#367—612no—
226C4⋊C16—612no—
227C4⋊C16—611no—
228C4⋊C16C2×D6111yescharacteristic
229#358—612no—
230#358—612no—
231#358—612no—
232#358C2×D6112yes—
233C61#3382111yescharacteristic, complemented, Sylow
234C2×Q16—21no—
235C2×Q16—611no—
236C2×Q16—1221no—
237C2×Q16C22×D6111yescharacteristic
238C2×Q16C22×D6112yes—
239C22×C8—611no—
240C22×C8—611no—
241C4⋊Q8—21no—
242C4⋊Q8—1221no—
243C4⋊Q8—1221no—
244C4⋊Q8—1221no—
245#125—2441no—
246#125—2441no—
247C22⋊Q8—1221no—
248C22⋊Q8—1221no—
249C22⋊Q8—1222no—
250C22⋊Q8—2441no—
251C22⋊Q8—2441no—
252C4×Q8—21no—
253C4×Q8—1222no—
254C4×Q8—1221no—
255C4×Q8—2441no—
256C4×Q8—2441no—
257#117—2441no—
258#117—2441no—
259C2×C42—611no—
260Q32—42no—
261Q32—2442no—
262M32—2442no—
263C2×C16—612no—
264C2×C16C22×D6112yes—
265#107—611no—
266#107—612no—
267#107—1221no—
268#107—1221no—
269#107C22×D6111yescharacteristic
270#103—21no—
271#103—1221no—
272#103—1222no—
273#103—1221no—
274#103—1222no—
275#103—1221no—
276C4×C8—611no—
277C4×C8—611no—
278C4×C8—611no—
279C4×C8C22×D6111yescharacteristic
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—1221no—
283C2×Q8—2441no—
284C2×Q8—2441no—
285C22×C4—611no—
286C22×C4—611no—
287C22×C4—611no—
288Q16—21no—
289Q16—22no—
290Q16—21no—
291Q16—22no—
292Q16—42no—
293Q16—1221no—
294Q16—1221no—
295Q16—2442no—
296C2×C8—611no—
297C2×C8—611no—
298C2×C8—1221no—
299C2×C8—1221no—
300C2×C8—2441no—
301C2×C8D4×D6111yescharacteristic
302C2×C8C23×D6111yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1222no—
308C4⋊C4—1221no—
309C4⋊C4—2441no—
310C4⋊C4—2441no—
311C4⋊C4—2441no—
312C4⋊C4—2442no—
313C4⋊C4—2441no—
314C4⋊C4—2442no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4—2441no—
318C22⋊C4—2441no—
319C22⋊C4—2441no—
320C22⋊C4—2442no—
321C42—41no—
322C42—611no—
323C42—611no—
324C42—611no—
325C42—2442no—
326C42—2441no—
327C42D4×D6111yescharacteristic
328C16—22no—
329C16—1222no—
330C23—611no—
331Q8—42no—
332Q8—41no—
333Q8—41no—
334Q8—42no—
335Q8—4882no—
336Q8—4882no—
337Q8—2441no—
338Q8—2441no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—611no—
343C2×C4—611no—
344C2×C4—611no—
345C2×C4—1222no—
346C2×C4—1222no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—2441no—
350C2×C4—2442no—
351C2×C4—2441no—
352C2×C4D8×D6111yescharacteristic
353C2×C4D8×D6111yescharacteristic
354C2×C4C2×D4×D6111yescharacteristic
355C8—21no—
356C8—1221no—
357C8—1221no—
358C8#1451982211yescharacteristic
359C8#1451982411yescharacteristic
360C22—1221no—
361C22—1221no—
362C22—1221no—
363C22C2×D8×D6111yescharacteristic
364C4—21no—
365C4—81no—
366C4—81no—
367C4—82no—
368C4—612no—
369C4—1221no—
370C4—1221no—
371C4—4881no—
372C4—4882no—
373C4—4881no—
374C4#1684020412yes—
375C4#1684034111yescharacteristic
376C4#1684055811yescharacteristic
377C2—2441no—
378C2#2145712711yescharacteristic
379C2#510×D6111yescharacteristic
380C2C2×#1684020411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2209C2
socleC22×C61C22×D8
prime radicalC22×D61C2×D8
hypercenter#2209D61
hypocenterC61#33821
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2209

Complements

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

NH ≅ G/Ntype
3. #21447962C2semidirect
4. #21447961C2semidirect
5. #21447959C2semidirect
8. #21447867C2semidirect
9. #21447866C2semidirect
11. C61×#2209C2semidirect
196. #2209D61semidirect
233. C61#33821semidirect

Maximal subgroups

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

subgroupindex
#214564032
#214564022
#214479622
#214479612
#214479592
#214479262
#214479242
#214478672
#214478662
#214475732
C61×#22092
#3382161