Finite Group Databasebeta

← #29663375

Subgroups — #29663375

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456994C212yesmaximal
2#21456405C211yesmaximal, characteristic, complemented
3#21445401C212yesmaximal, complemented
4#21445394C212yesmaximal, complemented
5#21445223C211yesmaximal, characteristic, complemented
6#21445218C211yesmaximal, characteristic, complemented
7#21445189C211yesmaximal, characteristic, complemented
8#21445184C211yesmaximal, characteristic, complemented
9#21444567C211yesmaximal, characteristic, complemented
10#21444530C212yesmaximal, complemented
11C61×#2966C211yesmaximal, characteristic
12#16840319C2212yes—
13#16840167C2211yescharacteristic
14#16840161—22no—
15#16840142C2211yescharacteristic
16#16840116—22no—
17#16840113C2212yes—
18#16840100—22no—
19#16840083—22no—
20#16840057C2212yes—
21#16840052C2212yes—
22#16840047C2212yes—
23#16840042C2212yes—
24#16840003—21no—
25#16840003—21no—
26C4×Dic244—21no—
27C4×Dic244—21no—
28#16839815C2212yes—
29#16839515C2211yescharacteristic
30#16839515C2211yescharacteristic
31#16839513C2212yes—
32#16839510C2212yes—
33#16839510C2212yes—
34#16839509C2212yes—
35#16839509C2212yes—
36C61×#529C2211yescharacteristic
37C61×#446C2211yescharacteristic
38C8×Q8×C61C2211yescharacteristic
39C61×#436C2211yescharacteristic
40C61×#435—21no—
41C8×D4×C61—21no—
42C8×D4×C61C2211yescharacteristic
43C61×#433—21no—
44C61×#432C2211yescharacteristic
45C61×#405—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×Dic244—21no—
48C2×Dic244—21no—
49#14519777C2312yes—
50#14519777C2312yes—
51#14519769—22no—
52#14519756—22no—
53#14519756—22no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56#14519736—22no—
57#14519711—21no—
58#14519711—21no—
59#14519710—22no—
60#14519710—22no—
61#14519708—22no—
62#14519708—22no—
63C4⋊#92418C2312yes—
64C4⋊#92418C2312yes—
65#14519702—22no—
66#14519702—22no—
67#14519701—22no—
68#14519701—22no—
69#14519698—22no—
70#14519698—22no—
71C22⋊#92418—22no—
72C4×#92418—22no—
73C2×M16×C61—21no—
74C22×C8×C61—21no—
75C22×C8×C61D411yescharacteristic
76C22×C8×C61D411yescharacteristic
77#126×C61—21no—
78#125×C61D411yescharacteristic
79#124×C61D411yescharacteristic
80#123×C61—21no—
81C22⋊Q8×C61D411yescharacteristic
82#121×C61D411yescharacteristic
83C4×Q8×C61C2311yescharacteristic
84C4×D4×C61—21no—
85C4×D4×C61C2311yescharacteristic
86#117×C61—21no—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61—21no—
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2311yescharacteristic
91C4⋊C8×C61C2311yescharacteristic
92C22⋊C8×C61—21no—
93C22⋊C8×C61—21no—
94C22⋊C8×C61—21no—
95#97×C61—21no—
96#97×C61—21no—
97C4×C8×C61—21no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61C2×D411yescharacteristic
103C2×D4×C61—21no—
104C2×D4×C61C2×D411yescharacteristic
105C22×C4×C61—21no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C2×Dic122—22no—
110C2×Dic122—22no—
111M16×C61—42no—
112C2×C8×C61—22no—
113C2×C8×C61—21no—
114C2×C8×C61—21no—
115C2×C8×C61—41no—
116C2×C8×C61—42no—
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61C2×D411yescharacteristic
126C4⋊C4×C61C2×D411yescharacteristic
127C22⋊C4×C61—21no—
128C22⋊C4×C61—21no—
129C22⋊C4×C61—21no—
130C22⋊C4×C61—21no—
131C22⋊C4×C61—21no—
132C42×C61—21no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137C22⋊Dic61—42no—
138C4⋊Dic61—22no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142#11758196—42no—
143#11758196—42no—
144C4×Dic61—42no—
145C2×#92418—22no—
146C2×#92418—22no—
147Dic244—42no—
148Dic244—42no—
149C23×C61—21no—
150C23×C61#12111yescharacteristic
151Q8×C61—21no—
152Q8×C61—21no—
153D4×C61—21no—
154D4×C61—21no—
155D4×C61—42no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C2×Dic61—42no—
173C2×Dic61—42no—
174Dic122—42no—
175Dic122—42no—
176Dic122—82no—
177C8×C61—21no—
178C8×C61—21no—
179C8×C61—21no—
180C8×C61—41no—
181C8×C61—41no—
182C8×C61#13512yes—
183#92418—42no—
184#92418—42no—
185#19220—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61C2×#12111yescharacteristic
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61C2×#13511yescharacteristic
201C4×C61#57511yescharacteristic
202C4×C61#57811yescharacteristic
203C4×C61#57811yescharacteristic
204Dic61—82no—
205Dic61—82no—
206#3233—612no—
207#3170—611no—
208#2966D6111yescharacteristic
209#1704—612no—
210#1703—612no—
211#1676—611no—
212#1675—611no—
213#1670—611no—
214#1669—611no—
215#1564—611no—
216#1554—612no—
217C2×C61—41no—
218C2×C61—81no—
219C2×C61#303511yescharacteristic
220C2×C61#305911yescharacteristic
221C2×C61#306711yescharacteristic
222#533—612no—
223#529C2×D6111yescharacteristic
224#500—611no—
225#499—1222no—
226#494—611no—
227#488—1222no—
228#487—612no—
229#485—1222no—
230#482—1222no—
231#477—612no—
232#476—612no—
233#475—612no—
234#474—612no—
235#467—1221no—
236#467—1221no—
237#446C2×D6111yescharacteristic
238C8×Q8C2×D6111yescharacteristic
239C4×Q16—1221no—
240C4×Q16—1221no—
241#436C2×D6111yescharacteristic
242#435—21no—
243C8×D4—21no—
244C8×D4C2×D6111yescharacteristic
245#433—21no—
246#432C2×D6111yescharacteristic
247#431—612no—
248#405—21no—
249C2×C4×C8C2×D6111yescharacteristic
250#335—611no—
251#335—611no—
252#334—612no—
253#333—612no—
254#333—612no—
255#332—612no—
256#332—612no—
257C61#1922011yescharacteristic, complemented, Sylow
258C2×Q16—1221no—
259C2×Q16—1221no—
260C2×M16—21no—
261C22×C8—21no—
262C22×C8C22⋊D6111yescharacteristic
263C22×C8C22⋊D6111yescharacteristic
264C4⋊Q8—612no—
265C4⋊Q8—612no—
266#126—21no—
267#125—1222no—
268#125D24411yescharacteristic
269#124D24411yescharacteristic
270#123—21no—
271C22⋊Q8—1222no—
272C22⋊Q8—1222no—
273C22⋊Q8D24411yescharacteristic
274#121D24411yescharacteristic
275C4×Q8—1222no—
276C4×Q8—1222no—
277C4×Q8C22×D6111yescharacteristic
278C4×D4—21no—
279C4×D4C22×D6111yescharacteristic
280#117—21no—
281#117—1222no—
282C2×C42C22×D6111yescharacteristic
283#107—1221no—
284#107—1221no—
285#107—1222no—
286#107—1222no—
287#106—1222no—
288#106—1222no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—612no—
292C4⋊C8—612no—
293C4⋊C8C22×D6111yescharacteristic
294C4⋊C8C22×D6111yescharacteristic
295#103—1222no—
296#103—1222no—
297#103—1222no—
298#103—1222no—
299#102—1222no—
300#102—1222no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—1222no—
305#97—21no—
306#97—21no—
307C4×C8—21no—
308C4×C8—1222no—
309C4×C8C22⋊D6111yescharacteristic
310C4×C8C22⋊D6111yescharacteristic
311C4×C8C22×D6111yescharacteristic
312C4×C8C22×D6111yescharacteristic
313C2×Q8—1222no—
314C2×Q8—1222no—
315C2×Q8C2×D24411yescharacteristic
316C2×D4—21no—
317C2×D4C2×D24411yescharacteristic
318C22×C4—21no—
319C22×C4C2×D24411yescharacteristic
320C22×C4#1175822211yescharacteristic
321C22×C4C2×C22⋊D6111yescharacteristic
322Q16—2442no—
323Q16—2442no—
324M16—42no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—1222no—
331C2×C8—1222no—
332C2×C8C2×C22⋊D6111yescharacteristic
333C2×C8C2×C22⋊D6111yescharacteristic
334C2×C8C2×C22⋊D6111yescharacteristic
335C2×C8C2×C22⋊D6111yescharacteristic
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—1222no—
341C4⋊C4—1222no—
342C4⋊C4—1222no—
343C4⋊C4—1222no—
344C4⋊C4—2442no—
345C4⋊C4—2442no—
346C4⋊C4C2×D24411yescharacteristic
347C4⋊C4C2×D24411yescharacteristic
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—2442no—
354C42—21no—
355C42—2442no—
356C42C2×D24411yescharacteristic
357C42#1175822211yescharacteristic
358C42C2×C22⋊D6111yescharacteristic
359C42C23×D6111yescharacteristic
360C23—21no—
361C23#1451975111yescharacteristic
362Q8—21no—
363Q8—21no—
364Q8—4882no—
365Q8—2442no—
366Q8—2442no—
367D4—21no—
368D4—21no—
369D4—42no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4#1451975111yescharacteristic
383C2×C4#1451975111yescharacteristic
384C2×C4#1451975111yescharacteristic
385C2×C4C22×C22⋊D6111yescharacteristic
386C2×C4C22×D24411yescharacteristic
387C2×C4C2×#1175822211yescharacteristic
388C8—21no—
389C8—21no—
390C8—21no—
391C8—41no—
392C8—41no—
393C8—2442no—
394C8—2442no—
395C8#1451979712yes—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22C2×#1451975111yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—4882no—
411C4—4882no—
412C4C2×#1451979711yescharacteristic
413C4#1684056011yescharacteristic
414C4#1684057911yescharacteristic
415C4#1684057911yescharacteristic
416C2—41no—
417C2—81no—
418C2#2145527411yescharacteristic
419C2#2145547411yescharacteristic
420C2#2145553911yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2966C2
socleC22×C61C2×#121
prime radical#121×C61D4
hypercenter#2966D61
hypocenterC61#19220
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
2. #21456405C2semidirect
3. #21445401C2semidirect
4. #21445394C2semidirect
5. #21445223C2semidirect
6. #21445218C2semidirect
7. #21445189C2semidirect
8. #21445184C2semidirect
9. #21444567C2semidirect
10. #21444530C2semidirect
257. C61#19220semidirect

Maximal subgroups

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

subgroupindex
#214569942
#214564052
#214454012
#214453942
#214452232
#214452182
#214451892
#214451842
#214445672
#214445302
C61×#29662
#1922061