Finite Group Databasebeta

← #29663449

Subgroups — #29663449

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456994C212yesmaximal
2#21456405C211yesmaximal, characteristic, complemented
3#21445347C211yesmaximal, characteristic, complemented
4#21445342C211yesmaximal, characteristic, complemented
5#21445313C211yesmaximal, characteristic, complemented
6#21445308C211yesmaximal, characteristic, complemented
7#21445277C212yesmaximal, complemented
8#21445270C212yesmaximal, 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#16839515C2212yes—
30#16839513C2211yescharacteristic
31#16839513C2211yescharacteristic
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—22no—
152D4×C61—22no—
153D4×C61—42no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C2×Dic61—42no—
170C2×Dic61—42no—
171Dic122—42no—
172Dic122—42no—
173Dic122—82no—
174C8×C61—21no—
175C8×C61—21no—
176C8×C61—21no—
177C8×C61—41no—
178C8×C61—41no—
179C8×C61#13512yes—
180#92418—42no—
181#92418—42no—
182#19226—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61C2×#12111yescharacteristic
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—41no—
195C4×C61—41no—
196C4×C61—41no—
197C4×C61C2×#13511yescharacteristic
198C4×C61#57511yescharacteristic
199C4×C61#57812yes—
200Dic61—82no—
201Dic61—82no—
202#3233—612no—
203#3170—611no—
204#2966D6111yescharacteristic
205#1696—611no—
206#1695—611no—
207#1690—611no—
208#1689—611no—
209#1684—612no—
210#1683—612no—
211#1564—611no—
212#1554—612no—
213C2×C61—41no—
214C2×C61—81no—
215C2×C61#303511yescharacteristic
216C2×C61#305911yescharacteristic
217C2×C61#306711yescharacteristic
218#533—612no—
219#529C2×D6111yescharacteristic
220#500—611no—
221#499—1222no—
222#494—611no—
223#488—1222no—
224#487—612no—
225#485—1222no—
226#482—1222no—
227#477—612no—
228#476—612no—
229#475—612no—
230#474—612no—
231#467—1221no—
232#467—1221no—
233#446C2×D6111yescharacteristic
234C8×Q8C2×D6111yescharacteristic
235C4×Q16—1221no—
236C4×Q16—1221no—
237#436C2×D6111yescharacteristic
238#435—21no—
239C8×D4—21no—
240C8×D4C2×D6111yescharacteristic
241#433—21no—
242#432C2×D6111yescharacteristic
243#431—612no—
244#405—21no—
245C2×C4×C8C2×D6111yescharacteristic
246#335—612no—
247#334—611no—
248#334—611no—
249#333—612no—
250#333—612no—
251#332—612no—
252#332—612no—
253C61#1922611yescharacteristic, complemented, Sylow
254C2×Q16—1221no—
255C2×Q16—1221no—
256C2×M16—21no—
257C22×C8—21no—
258C22×C8C22⋊D6111yescharacteristic
259C22×C8C22⋊D6111yescharacteristic
260C4⋊Q8—612no—
261C4⋊Q8—612no—
262#126—21no—
263#125—1222no—
264#125D24411yescharacteristic
265#124D24411yescharacteristic
266#123—21no—
267C22⋊Q8—1222no—
268C22⋊Q8—1222no—
269C22⋊Q8D24411yescharacteristic
270#121D24411yescharacteristic
271C4×Q8—1222no—
272C4×Q8—1222no—
273C4×Q8C22×D6111yescharacteristic
274C4×D4—21no—
275C4×D4C22×D6111yescharacteristic
276#117—21no—
277#117—1222no—
278C2×C42C22×D6111yescharacteristic
279#107—1221no—
280#107—1221no—
281#107—1222no—
282#107—1222no—
283#106—1222no—
284#106—1222no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—612no—
288C4⋊C8—612no—
289C4⋊C8C22×D6111yescharacteristic
290C4⋊C8C22×D6111yescharacteristic
291#103—1222no—
292#103—1222no—
293#103—1222no—
294#103—1222no—
295#102—1222no—
296#102—1222no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—1222no—
301#97—21no—
302#97—21no—
303C4×C8—21no—
304C4×C8—1222no—
305C4×C8C22⋊D6111yescharacteristic
306C4×C8C22⋊D6111yescharacteristic
307C4×C8C22×D6111yescharacteristic
308C4×C8C22×D6111yescharacteristic
309C2×Q8—1222no—
310C2×Q8—1222no—
311C2×Q8C2×D24411yescharacteristic
312C2×D4—21no—
313C2×D4C2×D24411yescharacteristic
314C22×C4—21no—
315C22×C4C2×D24411yescharacteristic
316C22×C4#1175822211yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318Q16—2442no—
319Q16—2442no—
320M16—42no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—1222no—
327C2×C8—1222no—
328C2×C8C2×C22⋊D6111yescharacteristic
329C2×C8C2×C22⋊D6111yescharacteristic
330C2×C8C2×C22⋊D6111yescharacteristic
331C2×C8C2×C22⋊D6111yescharacteristic
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—1222no—
337C4⋊C4—1222no—
338C4⋊C4—1222no—
339C4⋊C4—1222no—
340C4⋊C4—2442no—
341C4⋊C4—2442no—
342C4⋊C4C2×D24411yescharacteristic
343C4⋊C4C2×D24411yescharacteristic
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—2442no—
350C42—21no—
351C42—2442no—
352C42C2×D24411yescharacteristic
353C42#1175822211yescharacteristic
354C42C2×C22⋊D6111yescharacteristic
355C42C23×D6111yescharacteristic
356C23—21no—
357C23#1451975111yescharacteristic
358Q8—22no—
359Q8—4882no—
360Q8—2442no—
361Q8—2442no—
362D4—22no—
363D4—42no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4#1451975111yescharacteristic
376C2×C4#1451975111yescharacteristic
377C2×C4#1451975111yescharacteristic
378C2×C4C22×C22⋊D6111yescharacteristic
379C2×C4C22×D24411yescharacteristic
380C2×C4C2×#1175822211yescharacteristic
381C8—21no—
382C8—21no—
383C8—21no—
384C8—41no—
385C8—41no—
386C8—2442no—
387C8—2442no—
388C8#1451979712yes—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22C2×#1451975111yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—4882no—
404C4—4882no—
405C4C2×#1451979711yescharacteristic
406C4#1684056011yescharacteristic
407C4#1684057912yes—
408C2—41no—
409C2—81no—
410C2#2145527411yescharacteristic
411C2#2145547411yescharacteristic
412C2#2145553911yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2966C2
socleC22×C61C2×#121
prime radical#121×C61D4
hypercenter#2966D61
hypocenterC61#19226
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. #21445347C2semidirect
4. #21445342C2semidirect
5. #21445313C2semidirect
6. #21445308C2semidirect
7. #21445277C2semidirect
8. #21445270C2semidirect
9. #21444567C2semidirect
10. #21444530C2semidirect
253. C61#19226semidirect

Maximal subgroups

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

subgroupindex
#214569942
#214564052
#214453472
#214453422
#214453132
#214453082
#214452772
#214452702
#214445672
#214445302
C61×#29662
#1922661