Finite Group Databasebeta

← #29834849

Subgroups — #29834849

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840004C211yesmaximal, characteristic
2#427×D61C211yesmaximal, characteristic
3#21448030C212yesmaximal, complemented
4#21448029C212yesmaximal, complemented
5#21448023C212yesmaximal, complemented
6#21448018C212yesmaximal, complemented
7#21447605C211yesmaximal, characteristic, complemented
8#21447540C211yesmaximal, characteristic, complemented
9#21447538C211yesmaximal, characteristic, complemented
10#21447429C211yesmaximal, characteristic
11C61×#2159C211yesmaximal, characteristic, complemented
12C22×Dic244C2211yescharacteristic
13C22×C8×D61C2211yescharacteristic
14C2×#14519757—21no—
15#117×D61—21no—
16C2×#14519717C2211yescharacteristic
17#16840004—21no—
18#16840004—21no—
19#16840004C2212yes—
20#16840004C2212yes—
21#16839800C2211yescharacteristic
22#16839799C2211yescharacteristic
23#16839799C2211yescharacteristic
24#16839798—21no—
25#16839797—21no—
26#107×D61C2211yescharacteristic
27#107×D61C2211yescharacteristic
28#106×D61—21no—
29C2×#14519700—21no—
30#16839589C2212yescomplemented
31#16839587C2212yescomplemented
32#16839567C2212yescomplemented
33#16839567C2212yescomplemented
34#16839565C2212yescomplemented
35#16839565C2212yescomplemented
36#16839541C2212yes—
37#16839541C2212yes—
38C2×M32×C61C2211yescharacteristic
39C61×#427C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#367C2212yescomplemented
42C61×#366C2212yescomplemented
43C22×Dic122—21no—
44C22×C4×D61D411yescharacteristic
45C2×Dic244—22no—
46C2×Dic244—22no—
47C2×Dic244—21no—
48C2×Dic244C2312yes—
49C22×#92418D411yescharacteristic
50C2×C8×D61—22no—
51C2×C8×D61—22no—
52C2×C8×D61D411yescharacteristic
53C2×C8×D61D411yescharacteristic
54C2×C8×D61C2311yescharacteristic
55C2×C8×D61C2311yescharacteristic
56#14519757—22no—
57#14519757—22no—
58#14519757—41no—
59#14519757—41no—
60#14519736—21no—
61#14519735—21no—
62#14519735—21no—
63#14519734—41no—
64#14519733—41no—
65C2×C4⋊Dic61—41no—
66C4⋊C4×D61—21no—
67C4⋊C4×D61—21no—
68C2×#11758198—41no—
69C22⋊C4×D61—41no—
70C42×D61—41no—
71#14519717—22no—
72C2×#11758185C2312yes—
73#14519711C2311yescharacteristic
74#14519711C2311yescharacteristic
75#14519710—21no—
76#14519710—21no—
77#14519709—21no—
78#14519708—21no—
79#14519708—21no—
80#14519700—22no—
81#14519700—22no—
82C22×C8×C61C2311yescharacteristic
83#117×C61—21no—
84C2×C4⋊C4×C61—21no—
85M32×C61—22no—
86C2×C16×C61C2312yes—
87#107×C61—21no—
88#107×C61C2312yes—
89#107×C61C2311yescharacteristic
90#107×C61C2311yescharacteristic
91#106×C61—21no—
92C23×D61D811yescharacteristic
93C22×C4×C61—41no—
94C22×C4×C61C2×D411yescharacteristic
95C22×Dic61—41no—
96C22×Dic61D811yescharacteristic
97C2×C4×D61—22no—
98C2×C4×D61—22no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61D811yescharacteristic
102C2×C4×D61D811yescharacteristic
103C2×C4×D61C2×D411yescharacteristic
104C2×C4×D61C2×D411yescharacteristic
105C2×Dic122—22no—
106C2×Dic122—42no—
107C2×Dic122—42no—
108C2×Dic122—41no—
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61#4112yes—
111C2×C8×C61#4112yes—
112C2×C8×C61C2411yescharacteristic
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C42×C61—41no—
119C22⋊Dic61—41no—
120#11758198—41no—
121#11758198—41no—
122#11758198—42no—
123#11758198—42no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—42no—
127C4⋊Dic61—42no—
128#11758196—41no—
129#11758196—41no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132C4×Dic61—41no—
133C2×#92418—22no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136Dic244—22no—
137Dic244—22no—
138Dic244—42no—
139C8×D61—22no—
140C8×D61—22no—
141C8×D61—21no—
142C8×D61—21no—
143C8×D61—21no—
144C8×D61—21no—
145C16×C61—22no—
146#11758185—22no—
147C23×C61C2×D811yescharacteristic
148C22×D61—22no—
149C22×D61—22no—
150C22×D61—41no—
151C22×D61C2×D811yescharacteristic
152C22×D61C2×D811yescharacteristic
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—82no—
157C2×C4×C61#12312yes—
158C2×C4×C61C2×D811yescharacteristic
159C2×C4×C61#13712yes—
160C2×C4×C61C22×D411yescharacteristic
161C2×Dic61—22no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—42no—
165C2×Dic61—82no—
166C2×Dic61C2×D811yescharacteristic
167C2×Dic61C2×D811yescharacteristic
168C4×D61—21no—
169C4×D61—21no—
170C4×D61—21no—
171C4×D61—21no—
172C4×D61—22no—
173C4×D61—22no—
174C4×D61—82no—
175C4×D61—82no—
176Dic122—42no—
177Dic122—42no—
178Dic122—82no—
179C8×C61#12312yes—
180C8×C61C2×#4111yescharacteristic
181C8×C61C2×#4111yescharacteristic
182#92418—21no—
183#92418—21no—
184#33649—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61#48012yes—
188C22×C61C22×D811yescharacteristic
189C2×D61—21no—
190C2×D61—21no—
191C2×D61—21no—
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—44no—
195C2×D61#51012yes—
196C4×C61—81no—
197C4×C61—81no—
198C4×C61—82no—
199C4×C61#48012yes—
200C4×C61C2×#12311yescharacteristic
201C4×C61C2×#13711yescharacteristic
202Dic61—21no—
203Dic61—21no—
204Dic61—82no—
205Dic61—81no—
206Dic61—81no—
207C2×#467—611no—
208C2×#427—611no—
209#2237—612no—
210#2237—612no—
211#2236—612no—
212#2235—612no—
213#2159—611no—
214#2159D6111yescharacteristic, complemented
215#2141—611no—
216#2141—611no—
217#2112—611no—
218C2×C61—22no—
219C2×C61C2×#48011yescharacteristic
220C2×C61C2×#51011yescharacteristic
221C2×C61#341411yescharacteristic
222D61—22no—
223D61—41no—
224C22×Q16—611no—
225C23×C8—611no—
226C2×C22⋊Q8—1221no—
227C2×#117—1221no—
228C2×M32—611no—
229C2×M32C2×D6111yescharacteristic
230#467—612no—
231#467—612no—
232#467—1221no—
233#467—1221no—
234#427—611no—
235#427—611no—
236#427—611no—
237#427—1221no—
238#427—1221no—
239#427C2×D6111yescharacteristic
240C2×#107—611no—
241C2×#107—611no—
242C2×#107C2×D6111yescharacteristic
243C2×#106—1221no—
244C2×#103—1221no—
245#367—612no—
246#367C2×D6112yescomplemented
247#366—612no—
248#366C2×D6112yescomplemented
249#358—612no—
250#358—612no—
251#358—612no—
252#358—612no—
253C22⋊C16—612no—
254C22⋊C16—612no—
255C61#3364911yescharacteristic, complemented, Sylow
256C22×Q8—1221no—
257C23×C4—611no—
258C2×Q16—612no—
259C2×Q16—1222no—
260C2×Q16—1221no—
261C2×Q16—1222no—
262C22×C8—611no—
263C22×C8—611no—
264C22×C8—611no—
265C22×C8—611no—
266C22×C8—611no—
267C22×C8—1222no—
268C22×C8—1222no—
269C22×C8C22×D6111yescharacteristic
270C22⋊Q8—1222no—
271C22⋊Q8—1222no—
272C22⋊Q8—2441no—
273C22⋊Q8—2441no—
274#117—21no—
275#117—1221no—
276#117—1221no—
277#117—1221no—
278#117—2441no—
279#117—2441no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—1221no—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—2441no—
284C2×C22⋊C4—2441no—
285C2×C22⋊C4—2441no—
286C2×C42—2441no—
287M32—22no—
288M32—1222no—
289C2×C16—612no—
290C2×C16C22×D6112yes—
291#107—21no—
292#107—611no—
293#107—611no—
294#107—1221no—
295#107—1221no—
296#107C22×D6111yescharacteristic
297#107C22×D6111yescharacteristic
298#107C22×D6112yes—
299#106—21no—
300#106—1221no—
301#106—1221no—
302#106—1221no—
303#103—1222no—
304#103—1222no—
305C24—611no—
306C2×Q8—1222no—
307C2×Q8—2442no—
308C2×Q8—2442no—
309C2×Q8—2441no—
310C22×C4—41no—
311C22×C4—611no—
312C22×C4—611no—
313C22×C4—611no—
314C22×C4—611no—
315C22×C4—611no—
316C22×C4—1222no—
317C22×C4—1222no—
318C22×C4—2441no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4D4×D6111yescharacteristic
322Q16—1222no—
323Q16—1222no—
324Q16—2442no—
325C2×C8—611no—
326C2×C8—611no—
327C2×C8—1222no—
328C2×C8—1221no—
329C2×C8—1221no—
330C2×C8—1221no—
331C2×C8—1221no—
332C2×C8—1222no—
333C2×C8—1222no—
334C2×C8#1175822212yes—
335C2×C8D4×D6111yescharacteristic
336C2×C8#1175822612yes—
337C2×C8C23×D6111yescharacteristic
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C4⋊C4—2442no—
347C4⋊C4—2442no—
348C22⋊C4—41no—
349C22⋊C4—2441no—
350C22⋊C4—2441no—
351C22⋊C4—2442no—
352C22⋊C4—2442no—
353C22⋊C4—2441no—
354C42—41no—
355C42—2441no—
356C42—2441no—
357C42—2441no—
358C16—22no—
359C16—1222no—
360C23—611no—
361C23—611no—
362C23—1222no—
363C23—1222no—
364C23—2441no—
365C23D8×D6111yescharacteristic
366Q8—4882no—
367Q8—2442no—
368Q8—2442no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—611no—
374C2×C4—611no—
375C2×C4—1222no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1222no—
381C2×C4—1222no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4—4882no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2442no—
388C2×C4#1451976212yes—
389C2×C4D8×D6111yescharacteristic
390C2×C4#1451980912yes—
391C2×C4C2×D4×D6111yescharacteristic
392C8—1221no—
393C8—1221no—
394C8#1451976212yes—
395C8C2×#1175822211yescharacteristic
396C8C2×#1175822611yescharacteristic
397C22—21no—
398C22—21no—
399C22—612no—
400C22—1221no—
401C22—1221no—
402C22—1221no—
403C22—1221no—
404C22—1221no—
405C22—2444no—
406C22#1684007712yes—
407C22C2×D8×D6111yescharacteristic
408C4—81no—
409C4—82no—
410C4—81no—
411C4—1221no—
412C4—1221no—
413C4—4881no—
414C4—4881no—
415C4—4882no—
416C4#1684007712yes—
417C4C2×#1451976211yescharacteristic
418C4C2×#1451980911yescharacteristic
419C2—22no—
420C2—1222no—
421C2—2441no—
422C2C2×#1684007711yescharacteristic
423C2#510×D6111yescharacteristic
424C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
prime radicalC23×D61D8
hypercenter#2159D61
hypocenterC61#33649
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
3. #21448030C2semidirect
4. #21448029C2semidirect
5. #21448023C2semidirect
6. #21448018C2semidirect
7. #21447605C2semidirect
8. #21447540C2semidirect
9. #21447538C2semidirect
11. C61×#2159C2semidirect
30. #16839589C22semidirect
31. #16839587C22semidirect
32. #16839567C22semidirect
33. #16839567C22semidirect
34. #16839565C22semidirect
35. #16839565C22semidirect
41. C61×#367C22semidirect
42. C61×#366C22semidirect
214. #2159D61semidirect
246. #367C2×D61semidirect
248. #366C2×D61semidirect
255. C61#33649semidirect

Maximal subgroups

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

subgroupindex
C2×#168400042
#427×D612
#214480302
#214480292
#214480232
#214480182
#214476052
#214475402
#214475382
#214474292
C61×#21592
#3364961