Finite Group Databasebeta

← #29725103

Subgroups — #29725103

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455539C211yesmaximal, characteristic
2#21454590C211yesmaximal, characteristic, complemented
3#21447271C212yesmaximal, complemented
4#21447124C212yesmaximal, complemented
5#21446715C211yesmaximal, characteristic, complemented
6#21446568C211yesmaximal, characteristic, complemented
7#21446317C212yesmaximal, complemented
8#21445942C211yesmaximal, characteristic, complemented
9#21445921C212yesmaximal, complemented
10#21445654C211yesmaximal, characteristic, complemented
11C61×#1812C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519799C2211yescharacteristic
14#16840230C2212yes—
15#16840037—21no—
16#16840032—21no—
17#16840025—22no—
18#16839856—21no—
19#16839849—22no—
20#16839844—21no—
21C2×#14519713C2211yescharacteristic
22#16839789C2212yes—
23#16839780C2212yes—
24C2×#14519704C2211yescharacteristic
25#16839758C2211yescharacteristic
26#16839756C2212yes—
27#16839744C2212yes—
28#16839735C2212yes—
29#16839717C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839532C2212yes—
32#16839526C2211yescharacteristic
33#16839523C2211yescharacteristic
34#16839521C2212yes—
35#16839518C2212yes—
36C2×#131×C61C2211yescharacteristic
37C61×#427C2211yescharacteristic
38C61×#427C2211yescharacteristic
39C61×#410C2212yes—
40C61×H(3,ℤ/4)C2212yes—
41C2×#11758222C2312yes—
42#14519808—22no—
43#14519807—22no—
44#14519803—22no—
45#14519802—22no—
46#14519799—21no—
47#14519799—21no—
48#14519799—21no—
49#14519799—21no—
50#14519797—42no—
51C2×Dic244D411yescharacteristic
52C2×#11758190D412yes—
53C2×D488D411yescharacteristic
54C2×#11758194C2311yescharacteristic
55C2×#11758194C2312yes—
56C22×#92418C2311yescharacteristic
57#14519742—22no—
58#14519739—22no—
59#14519737—42no—
60#14519736C2311yescharacteristic
61#14519734—22no—
62C2×C4×Dic61C2311yescharacteristic
63C42×D61—22no—
64#14519713—21no—
65#14519713—21no—
66C4⋊#92418—42no—
67#14519705—42no—
68#14519704—21no—
69#14519704—21no—
70#14519701—21no—
71#14519700—22no—
72#14519698—21no—
73#14519697—22no—
74#14519695—22no—
75#14519693—22no—
76#14519691—42no—
77#14519687—21no—
78C8×Dic61—21no—
79C8×Dic61—21no—
80C8×Dic61—21no—
81C2×#41×C61C2311yescharacteristic
82#131×C61—21no—
83#131×C61—21no—
84#131×C61—21no—
85#131×C61—21no—
86C2×M16×C61—21no—
87C2×M16×C61C2311yescharacteristic
88C22×C8×C61—21no—
89C22×C8×C61C2311yescharacteristic
90#117×C61C2312yes—
91#117×C61C2312yes—
92#107×C61—21no—
93#107×C61—21no—
94#106×C61—21no—
95#106×C61—21no—
96#99×C61—42no—
97#41×C61—21no—
98#41×C61—21no—
99#41×C61—21no—
100#41×C61—21no—
101C2×Q8×C61#4111yescharacteristic
102C2×D4×C61—21no—
103C2×D4×C61#4111yescharacteristic
104C22×C4×C61—21no—
105C22×C4×C61C2411yescharacteristic
106C2×C22⋊D61—22no—
107C22×Dic61—21no—
108#11758222—22no—
109#11758222—22no—
110#11758222—22no—
111#11758222—22no—
112C2×D244C2×D412yes—
113C2×C4×D61—22no—
114C2×C4×D61—42no—
115C2×Dic122C2×D412yes—
116M16×C61—21no—
117M16×C61—21no—
118M16×C61—41no—
119M16×C61—41no—
120C2×C8×C61—21no—
121C2×C8×C61—41no—
122C2×C8×C61—41no—
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61#4111yescharacteristic
126C2×C8×C61#4111yescharacteristic
127C2×C8×C61#4111yescharacteristic
128C2×C8×C61#4111yescharacteristic
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C42×C61—22no—
134C42×C61—22no—
135C22⋊Dic61—21no—
136#11758202—42no—
137#11758201—42no—
138#11758200—42no—
139#11758199—42no—
140#11758198—42no—
141C4⋊Dic61—21no—
142#11758196—42no—
143C4×Dic61—21no—
144C4×Dic61—21no—
145C4×Dic61—42no—
146C4×Dic61#4111yescharacteristic
147C4×Dic61#4111yescharacteristic
148#11758194—22no—
149#11758194—22no—
150#11758194—21no—
151#11758194—21no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—22no—
157Dic244—21no—
158Dic244—21no—
159D488—21no—
160D488—21no—
161#11758190—22no—
162#11758190—22no—
163C23×C61—21no—
164C23×C61H(5,𝔽2)11yescharacteristic
165C22×D61—22no—
166Q8×C61—21no—
167Q8×C61—21no—
168D4×C61—21no—
169D4×C61—21no—
170D4×C61—41no—
171D4×C61—41no—
172C2×C4×C61—21no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C2×C4×C61C2×#4111yescharacteristic
182C2×C4×C61H(5,𝔽2)11yescharacteristic
183C22⋊D61—42no—
184C22⋊D61—42no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—22no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190D244—22no—
191D244—22no—
192C4×D61—42no—
193C4×D61—42no—
194C4×D61—84no—
195Dic122—22no—
196Dic122—22no—
197C8×C61—41no—
198C8×C61#12311yescharacteristic
199C8×C61#12311yescharacteristic
200C8×C61#12311yescharacteristic
201C8×C61#12311yescharacteristic
202#92418—41no—
203#92418—42no—
204#92418—41no—
205#24985—611nomaximal, Sylow
206C22×C61—41no—
207C22×C61—41no—
208C22×C61—81no—
209C22×C61#54011yescharacteristic
210C22×C61#54611yescharacteristic
211C22×C61#55311yescharacteristic
212C2×D61—42no—
213C2×D61—82no—
214C4×C61—41no—
215C4×C61—82no—
216C4×C61—82no—
217C4×C61C2×#12311yescharacteristic
218C4×C61#52911yescharacteristic
219C4×C61#54611yescharacteristic
220C4×C61#55511yescharacteristic
221Dic61—22no—
222Dic61—41no—
223Dic61—42no—
224Dic61—81no—
225#3067—611no—
226C4×#135—611no—
227#2080—612no—
228#2053—612no—
229#1976—611no—
230#1945—611no—
231#1906—612no—
232#1833—611no—
233#1827—612no—
234#1812D6111yescharacteristic, complemented
235#1764—611no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—81no—
240C2×C61#242011yescharacteristic
241D61—82no—
242#575—612no—
243C2×#135—611no—
244C2×#131C2×D6111yescharacteristic
245C4×#41—612no—
246#473—1221no—
247#472—1221no—
248#471—1222no—
249C4×Q16—1221no—
250C4×SD16—1222no—
251C4×D8—1221no—
252C2×#108—611no—
253#427C2×D6111yescharacteristic
254#427C2×D6111yescharacteristic
255#424—612no—
256#421—612no—
257C2×#104—611no—
258#416—611no—
259#416—612no—
260#413—612no—
261#410—612no—
262#410C2×D6112yes—
263#405—611no—
264C2×C4×C8—611no—
265#344—612no—
266#341—611no—
267#339—611no—
268#339—612no—
269H(3,ℤ/4)—612no—
270H(3,ℤ/4)C2×D6112yes—
271C61#2498511yescharacteristic, complemented, Sylow
272C2×#41—612no—
273C2×#41C22×D6111yescharacteristic
274#137—1222no—
275#137—1222no—
276AGL(1,ℤ/8)—1222no—
277AGL(1,ℤ/8)—1222no—
278#135—1221no—
279#135—1221no—
280#135—1221no—
281#135—1221no—
282#135—2442no—
283C2×Q16—611no—
284C2×SD16—612no—
285C2×D8—611no—
286#131—21no—
287#131—21no—
288#131—21no—
289#131—21no—
290C2×M16—21no—
291C2×M16—612no—
292C2×M16—611no—
293C2×M16C22×D6111yescharacteristic
294C22×C8—21no—
295C22×C8—611no—
296C22×C8C22×D6111yescharacteristic
297C4×Q8—1222no—
298C4×D4—1222no—
299C4×D4—2442no—
300#117—611no—
301#117—1222no—
302#117C22×D6112yes—
303#117C22×D6112yes—
304C2×C42—611no—
305C2×C42—1222no—
306#108—1221no—
307#108—1221no—
308#107—21no—
309#107—21no—
310#106—21no—
311#106—21no—
312C4⋊C8—2442no—
313#104—1221no—
314#104—1221no—
315#104—2442no—
316#103—1222no—
317#103—1221no—
318#102—1222no—
319#102—1221no—
320#101—1222no—
321#100—1222no—
322#99—42no—
323#99—2442no—
324#97—1221no—
325C4×C8—1221no—
326C4×C8—1221no—
327C4×C8—1221no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—21no—
332#41—1222no—
333#41—1222no—
334#41—1222no—
335#41—1222no—
336C2×Q8—612no—
337C2×Q8#1175822611yescharacteristic
338C2×D4—21no—
339C2×D4—612no—
340C2×D4—1222no—
341C2×D4#1175822611yescharacteristic
342C22×C4—21no—
343C22×C4—1221no—
344C22×C4—1222no—
345C22×C4—2442no—
346C22×C4C23×D6111yescharacteristic
347Q16—1221no—
348Q16—1221no—
349Q16—2442no—
350SD16—1222no—
351SD16—1222no—
352SD16—2442no—
353SD16—2442no—
354D8—1221no—
355D8—1221no—
356D8—2442no—
357M16—21no—
358M16—21no—
359M16—41no—
360M16—41no—
361M16—1222no—
362M16—1221no—
363M16—1222no—
364M16—1221no—
365C2×C8—21no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—1221no—
369C2×C8—1221no—
370C2×C8—1221no—
371C2×C8—1221no—
372C2×C8—1222no—
373C2×C8#1175822211yescharacteristic
374C2×C8#1175822211yescharacteristic
375C2×C8D4×D6111yescharacteristic
376C2×C8D4×D6111yescharacteristic
377C2×C8#1175822411yescharacteristic
378C2×C8#1175822411yescharacteristic
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—1221no—
382C4⋊C4—2442no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—1221no—
386C22⋊C4—2442no—
387C42—22no—
388C42—22no—
389C42—611no—
390C42—611no—
391C42—1221no—
392C42—1221no—
393C42—2442no—
394C23—21no—
395C23—1222no—
396C23#1451982211yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—1222no—
400Q8—1222no—
401D4—21no—
402D4—21no—
403D4—41no—
404D4—41no—
405D4—1222no—
406D4—1222no—
407D4—2442no—
408D4—2442no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1222no—
417C2×C4—4884no—
418C2×C4—2442no—
419C2×C4—2441no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4C2×D4×D6111yescharacteristic
423C2×C4#41×D6111yescharacteristic
424C2×C4C2×#1175822411yescharacteristic
425C2×C4C2×#1175822211yescharacteristic
426C2×C4C2×#1175822611yescharacteristic
427C2×C4#1451982311yescharacteristic
428C8—41no—
429C8—2441no—
430C8—2441no—
431C8—2442no—
432C8#1451975911yescharacteristic
433C8#1451975911yescharacteristic
434C8#1451975911yescharacteristic
435C8#1451975911yescharacteristic
436C22—41no—
437C22—41no—
438C22—81no—
439C22—4882no—
440C22—2442no—
441C22#1684036511yescharacteristic
442C22#1684040911yescharacteristic
443C22#1684046011yescharacteristic
444C4—41no—
445C4—82no—
446C4—82no—
447C4—1222no—
448C4—4881no—
449C4—2441no—
450C4—2442no—
451C4C2×#1451975911yescharacteristic
452C4#1684029811yescharacteristic
453C4#1684040311yescharacteristic
454C4#1684047511yescharacteristic
455C2—21no—
456C2—21no—
457C2—21no—
458C2—81no—
459C2—4882no—
460C2#2144931311yescharacteristic
subgroupG/H
centerC4#16840475
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1812C2
socleC2×C61#2420
prime radical#21455539C2
hypercenter#1812D61
hypocenterC61#24985
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1812

Complements

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

NH ≅ G/Ntype
2. #21454590C2semidirect
3. #21447271C2semidirect
4. #21447124C2semidirect
5. #21446715C2semidirect
6. #21446568C2semidirect
7. #21446317C2semidirect
8. #21445942C2semidirect
9. #21445921C2semidirect
10. #21445654C2semidirect
11. C61×#1812C2semidirect
234. #1812D61semidirect
271. C61#24985semidirect

Maximal subgroups

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

subgroupindex
#214555392
#214545902
#214472712
#214471242
#214467152
#214465682
#214463172
#214459422
#214459212
#214456542
C61×#18122
#2498561