Finite Group Databasebeta

← #29725114

Subgroups — #29725114

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455539C211yesmaximal, characteristic
2#21454619C211yesmaximal, characteristic, complemented
3#21447271C212yesmaximal, complemented
4#21447124C212yesmaximal, complemented
5#21446715C211yesmaximal, characteristic, complemented
6#21446573C211yesmaximal, characteristic, complemented
7#21446317C212yesmaximal, complemented
8#21445942C211yesmaximal, characteristic, complemented
9#21445921C212yesmaximal, complemented
10#21445661C211yesmaximal, characteristic, complemented
11C61×#1812C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519799C2211yescharacteristic
14#16840230C2212yes—
15#16840037—21no—
16#16840032—21no—
17#16840025—22no—
18#16839873—21no—
19#16839868—21no—
20#16839861—22no—
21C2×#14519713C2211yescharacteristic
22#16839789C2212yes—
23#16839780C2212yes—
24C2×#14519704C2211yescharacteristic
25#16839758C2211yescharacteristic
26#16839756C2212yes—
27#16839744C2212yes—
28#16839735C2212yes—
29#16839717C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
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—22no—
69#14519701—21no—
70#14519700—22no—
71#14519698—21no—
72#14519697—22no—
73#14519695—22no—
74#14519693—22no—
75#14519691—42no—
76#14519687—21no—
77#14519687—21no—
78#14519687—21no—
79C8×Dic61—21no—
80C2×#41×C61C2311yescharacteristic
81#131×C61—21no—
82#131×C61—21no—
83#131×C61—21no—
84#131×C61—21no—
85C2×M16×C61—21no—
86C2×M16×C61C2311yescharacteristic
87C22×C8×C61—21no—
88C22×C8×C61C2311yescharacteristic
89#117×C61C2312yes—
90#117×C61C2312yes—
91#107×C61—21no—
92#107×C61—21no—
93#106×C61—21no—
94#106×C61—21no—
95#99×C61—42no—
96#41×C61—21no—
97#41×C61—21no—
98#41×C61—21no—
99#41×C61—21no—
100C2×Q8×C61#4111yescharacteristic
101C2×D4×C61—21no—
102C2×D4×C61#4111yescharacteristic
103C22×C4×C61—21no—
104C22×C4×C61C2411yescharacteristic
105C2×C22⋊D61—22no—
106C22×Dic61—21no—
107#11758222—22no—
108#11758222—22no—
109#11758222—22no—
110#11758222—22no—
111C2×D244C2×D412yes—
112C2×C4×D61—22no—
113C2×C4×D61—42no—
114C2×Dic122C2×D412yes—
115M16×C61—21no—
116M16×C61—21no—
117M16×C61—41no—
118M16×C61—41no—
119C2×C8×C61—21no—
120C2×C8×C61—41no—
121C2×C8×C61—41no—
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61#4111yescharacteristic
125C2×C8×C61#4111yescharacteristic
126C2×C8×C61#4111yescharacteristic
127C2×C8×C61#4111yescharacteristic
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C22⋊C4×C61—22no—
131C22⋊C4×C61—22no—
132C42×C61—22no—
133C42×C61—22no—
134C22⋊Dic61—21no—
135#11758202—42no—
136#11758201—42no—
137#11758200—42no—
138#11758199—42no—
139#11758198—42no—
140C4⋊Dic61—21no—
141#11758196—42no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C4×Dic61—42no—
145C4×Dic61#4112yes—
146#11758194—22no—
147#11758194—22no—
148#11758194—21no—
149#11758194—21no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—22no—
155Dic244—21no—
156Dic244—21no—
157D488—21no—
158D488—21no—
159#11758190—22no—
160#11758190—22no—
161C23×C61—21no—
162C23×C61H(5,𝔽2)11yescharacteristic
163C22×D61—22no—
164Q8×C61—21no—
165Q8×C61—21no—
166D4×C61—21no—
167D4×C61—21no—
168D4×C61—41no—
169D4×C61—41no—
170C2×C4×C61—21no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61C22×D411yescharacteristic
176C2×C4×C61C2×#4111yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C2×C4×C61H(5,𝔽2)11yescharacteristic
181C22⋊D61—42no—
182C22⋊D61—42no—
183C2×Dic61—22no—
184C2×Dic61—22no—
185C2×Dic61—41no—
186C2×Dic61—81no—
187D244—22no—
188D244—22no—
189C4×D61—42no—
190C4×D61—42no—
191C4×D61—84no—
192Dic122—22no—
193Dic122—22no—
194C8×C61—41no—
195C8×C61#12311yescharacteristic
196C8×C61#12311yescharacteristic
197C8×C61#12311yescharacteristic
198C8×C61#12311yescharacteristic
199#92418—41no—
200#92418—42no—
201#92418—41no—
202#24986—611nomaximal, Sylow
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—81no—
206C22×C61#54011yescharacteristic
207C22×C61#54611yescharacteristic
208C22×C61#55311yescharacteristic
209C2×D61—42no—
210C2×D61—82no—
211C4×C61—41no—
212C4×C61—82no—
213C4×C61—82no—
214C4×C61C2×#12311yescharacteristic
215C4×C61#52911yescharacteristic
216C4×C61#54611yescharacteristic
217C4×C61#55511yescharacteristic
218Dic61—42no—
219Dic61—42no—
220Dic61—81no—
221#3067—611no—
222#2943—611no—
223#2080—612no—
224#2053—612no—
225#1976—611no—
226#1946—611no—
227#1906—612no—
228#1833—611no—
229#1827—612no—
230#1812D6111yescharacteristic, complemented
231#1765—611no—
232C2×C61—21no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—81no—
236C2×C61#242011yescharacteristic
237D61—82no—
238#575—612no—
239C2×#135—611no—
240C2×#131C2×D6111yescharacteristic
241C4×#41—612no—
242#473—1221no—
243#472—1221no—
244#471—1222no—
245#442—1221no—
246#441—1221no—
247#440—1222no—
248C2×#108—611no—
249#427C2×D6111yescharacteristic
250#427C2×D6111yescharacteristic
251#424—612no—
252#421—612no—
253C2×#104—611no—
254#416—611no—
255#416—612no—
256#413—612no—
257#410—612no—
258#410C2×D6112yes—
259#405—611no—
260C2×#97—611no—
261#344—612no—
262#341—611no—
263#339—611no—
264#339—612no—
265H(3,ℤ/4)—612no—
266H(3,ℤ/4)C2×D6112yes—
267C61#2498611yescharacteristic, complemented, Sylow
268C2×#41—612no—
269C2×#41C22×D6111yescharacteristic
270#137—1222no—
271#137—1222no—
272AGL(1,ℤ/8)—1222no—
273AGL(1,ℤ/8)—1222no—
274#135—1221no—
275#135—1221no—
276#135—1221no—
277#135—1221no—
278#135—2442no—
279C2×Q16—611no—
280C2×SD16—612no—
281C2×D8—611no—
282#131—21no—
283#131—21no—
284#131—21no—
285#131—21no—
286C2×M16—21no—
287C2×M16—612no—
288C2×M16—611no—
289C2×M16C22×D6111yescharacteristic
290C22×C8—21no—
291C22×C8—611no—
292C22×C8C22×D6111yescharacteristic
293C4×Q8—1222no—
294C4×D4—1222no—
295C4×D4—2442no—
296#117—611no—
297#117—1222no—
298#117C22×D6112yes—
299#117C22×D6112yes—
300C2×C42—611no—
301C2×C42—1222no—
302#108—1221no—
303#108—1221no—
304#107—21no—
305#107—21no—
306#106—21no—
307#106—21no—
308C4⋊C8—2442no—
309#104—1222no—
310#104—2442no—
311#103—1222no—
312#103—1221no—
313#102—1222no—
314#102—1221no—
315#101—1222no—
316#100—1222no—
317#99—42no—
318#99—2442no—
319#97—1221no—
320#97—1221no—
321#97—1221no—
322C4×C8—1221no—
323#41—21no—
324#41—21no—
325#41—21no—
326#41—21no—
327#41—1222no—
328#41—1222no—
329#41—1222no—
330#41—1222no—
331C2×Q8—612no—
332C2×Q8#1175822611yescharacteristic
333C2×D4—21no—
334C2×D4—612no—
335C2×D4—1222no—
336C2×D4#1175822611yescharacteristic
337C22×C4—21no—
338C22×C4—1221no—
339C22×C4—1222no—
340C22×C4—2442no—
341C22×C4C23×D6111yescharacteristic
342Q16—1221no—
343Q16—1221no—
344Q16—2442no—
345SD16—1222no—
346SD16—1222no—
347SD16—2442no—
348SD16—2442no—
349D8—1221no—
350D8—1221no—
351D8—2442no—
352M16—21no—
353M16—21no—
354M16—41no—
355M16—41no—
356M16—1222no—
357M16—1221no—
358M16—1222no—
359M16—1221no—
360C2×C8—21no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—1221no—
364C2×C8—1221no—
365C2×C8—1221no—
366C2×C8—1221no—
367C2×C8—1222no—
368C2×C8#1175822211yescharacteristic
369C2×C8#1175822211yescharacteristic
370C2×C8D4×D6111yescharacteristic
371C2×C8D4×D6111yescharacteristic
372C2×C8#1175822411yescharacteristic
373C2×C8#1175822411yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—1221no—
377C4⋊C4—2442no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—1221no—
381C22⋊C4—2442no—
382C42—22no—
383C42—22no—
384C42—612no—
385C42—1221no—
386C42—1221no—
387C42—2442no—
388C23—21no—
389C23—1222no—
390C23#1451982211yescharacteristic
391Q8—21no—
392Q8—21no—
393Q8—1222no—
394Q8—1222no—
395D4—21no—
396D4—21no—
397D4—41no—
398D4—41no—
399D4—1222no—
400D4—1222no—
401D4—2442no—
402D4—2442no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—1222no—
409C2×C4—1222no—
410C2×C4—4884no—
411C2×C4—4881no—
412C2×C4—2441no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4#41×D6111yescharacteristic
417C2×C4C2×#1175822411yescharacteristic
418C2×C4C2×#1175822211yescharacteristic
419C2×C4C2×#1175822611yescharacteristic
420C2×C4#1451982311yescharacteristic
421C8—41no—
422C8—2441no—
423C8—2441no—
424C8—2442no—
425C8#1451975911yescharacteristic
426C8#1451975911yescharacteristic
427C8#1451975911yescharacteristic
428C8#1451975911yescharacteristic
429C22—41no—
430C22—41no—
431C22—81no—
432C22—4882no—
433C22—2442no—
434C22#1684036511yescharacteristic
435C22#1684040911yescharacteristic
436C22#1684046011yescharacteristic
437C4—41no—
438C4—82no—
439C4—82no—
440C4—4881no—
441C4—2442no—
442C4—2442no—
443C4C2×#1451975911yescharacteristic
444C4#1684029811yescharacteristic
445C4#1684040311yescharacteristic
446C4#1684047511yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—81no—
451C2—4882no—
452C2#2144931311yescharacteristic
subgroupG/H
centerC4#16840475
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1812C2
socleC2×C61#2420
prime radical#21455539C2
hypercenter#1812D61
hypocenterC61#24986
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. #21454619C2semidirect
3. #21447271C2semidirect
4. #21447124C2semidirect
5. #21446715C2semidirect
6. #21446573C2semidirect
7. #21446317C2semidirect
8. #21445942C2semidirect
9. #21445921C2semidirect
10. #21445661C2semidirect
11. C61×#1812C2semidirect
230. #1812D61semidirect
267. C61#24986semidirect

Maximal subgroups

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

subgroupindex
#214555392
#214546192
#214472712
#214471242
#214467152
#214465732
#214463172
#214459422
#214459212
#214456612
C61×#18122
#2498661