Finite Group Databasebeta

← #29576901

Subgroups — #29576901

Order 15616.

558 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455299C211yesmaximal, characteristic
2#21446745C211yesmaximal, characteristic, complemented
3#21445839C211yesmaximal, characteristic, complemented
4#21444486C211yesmaximal, characteristic, complemented
5#21443721C211yesmaximal, characteristic, complemented
6#21443694C211yesmaximal, characteristic, complemented
7C61×#2989C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×#14519774C2211yescharacteristic
10#16839962—21no—
11#16839962C412yescomplemented
12#16839957—21no—
13#16839957C412yescomplemented
14#16839786C2211yescharacteristic
15#16839786C2211yescharacteristic
16C2×#14519704—21no—
17#16839766—21no—
18C2×#14519699—22no—
19C4×C4⋊Dic61C2211yescharacteristic
20#16839521—21no—
21#16839505—21no—
22#16839505—21no—
23C4×#41×C61C2211yescharacteristic
24C61×#446—21no—
25C61×#436—21no—
26C61×#435—22no—
27C61×#432—21no—
28C61×#431C2211yescharacteristic
29C61×#423—21no—
30C2×C4⋊C8×C61C2211yescharacteristic
31C22×D244—21no—
32C22×D244—21no—
33C22×D244—21no—
34#14519802—42no—
35#14519802—42no—
36C2×#11758201—22no—
37C2×#11758199—22no—
38C2×#11758194—21no—
39C2×#11758194—21no—
40#14519774—21no—
41#14519774—21no—
42#14519774—21no—
43#14519774C2×C412yes—
44C2×C4⋊Dic61—21no—
45C2×C4×Dic61—21no—
46C4⋊#92418—21no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418C2×C412yes—
50#14519704—42no—
51#14519699—44no—
52#14519697—22no—
53#14519697—22no—
54#14519684—21no—
55C2×#41×C61D411yescharacteristic
56C2×M16×C61—22no—
57C22×C8×C61D412yes—
58C4×Q8×C61C2×C411yescharacteristic
59C4×Q8×C61D411yescharacteristic
60C4×D4×C61—21no—
61C4×D4×C61—21no—
62C4×D4×C61C2×C411yescharacteristic
63C4×D4×C61D411yescharacteristic
64#117×C61—21no—
65#117×C61D411yescharacteristic
66C2×C42×C61—21no—
67C2×C42×C61C2311yescharacteristic
68C4⋊C8×C61—22no—
69C4⋊C8×C61—21no—
70C4⋊C8×C61—21no—
71C4⋊C8×C61D411yescharacteristic
72C4⋊C8×C61D411yescharacteristic
73C22⋊C8×C61—22no—
74C22⋊C8×C61—21no—
75#97×C61—22no—
76C4×C8×C61—21no—
77C23×D61—41no—
78#41×C61—21no—
79#41×C61—21no—
80#41×C61—21no—
81#41×C61—21no—
82C2×Q8×C61C22⋊C411yescharacteristic
83C2×D4×C61—21no—
84C2×D4×C61C22⋊C411yescharacteristic
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C22×Dic61—41no—
92C2×D244—22no—
93C2×D244—22no—
94C2×D244—22no—
95C2×D244—42no—
96C2×D244—42no—
97C2×D244—41no—
98C2×D244—41no—
99C2×D244—41no—
100C2×D244—41no—
101C2×D244—41no—
102C2×D244—41no—
103C2×D244—41no—
104C2×D244—82no—
105M16×C61—82no—
106C2×C8×C61—21no—
107C2×C8×C61—21no—
108C2×C8×C61—22no—
109C2×C8×C61—42no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61C22⋊C411yescharacteristic
113C4⋊C4×C61C22⋊C411yescharacteristic
114C22⋊C4×C61—21no—
115C22⋊C4×C61—21no—
116C22⋊C4×C61—21no—
117C42×C61—21no—
118C42×C61—21no—
119C42×C61—21no—
120C42×C61C22×C411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124#11758201—44no—
125#11758199—44no—
126C4⋊Dic61—22no—
127C4⋊Dic61—41no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130#11758194—41no—
131#11758194—41no—
132#11758194—41no—
133#11758194—41no—
134C2×#92418—22no—
135C2×#92418—41no—
136C23×C61—21no—
137C23×C61#12111yescharacteristic
138C22×D61—42no—
139C22×D61—82no—
140C22×D61—82no—
141C22×D61—81no—
142C22×D61—81no—
143Q8×C61—21no—
144Q8×C61—21no—
145D4×C61—21no—
146D4×C61—21no—
147D4×C61—41no—
148D4×C61—41no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—81no—
160C2×C4×C61C2×C22⋊C411yescharacteristic
161C2×C4×C61C4×D411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61#12011yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12311yescharacteristic
166C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C61#13711yescharacteristic
170C2×Dic61—42no—
171C2×Dic61—81no—
172C2×Dic61—81no—
173D244—44no—
174D244—42no—
175D244—42no—
176D244—82no—
177D244—82no—
178D244—81no—
179D244—81no—
180D244—81no—
181D244—81no—
182C8×C61—42no—
183C8×C61—41no—
184C8×C61—41no—
185#92418—42no—
186#92418—81no—
187#9322—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61#38611yescharacteristic
194C22×C61#46811yescharacteristic
195C22×C61#46911yescharacteristic
196C2×D61—82no—
197C2×D61—82no—
198C2×D61—82no—
199C2×D61—162no—
200C2×D61—162no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—41no—
205C4×C61—41no—
206C4×C61—41no—
207C4×C61#41711yescharacteristic
208C4×C61#41811yescharacteristic
209C4×C61#45311yescharacteristic
210C4×C61#45411yescharacteristic
211C4×C61#46811yescharacteristic
212C4×C61#46911yescharacteristic
213Dic61—82no—
214#3038—611no—
215#2989D6111yescharacteristic, complemented
216#1982—611no—
217#1806—611no—
218#1545—611no—
219#1303—611no—
220#1292—611no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61—81no—
224C2×C61#188311yescharacteristic
225C2×C61#189511yescharacteristic
226C2×C61#190311yescharacteristic
227D61—162no—
228C2×AGL(1,ℤ/8)—1221no—
229C2×#127—611no—
230C4×#41C2×D6111yescharacteristic
231#460—612no—
232#460—1221no—
233#459—612no—
234#459—1221no—
235#446—21no—
236#436—21no—
237#435—22no—
238#432—21no—
239#431C2×D6111yescharacteristic
240#423—21no—
241#423—611no—
242#423—611no—
243C2×C4⋊C8C2×D6111yescharacteristic
244C2×#104—1221no—
245#418—1221no—
246C2×#102—1222no—
247C4×C4⋊C4—611no—
248#339—1221no—
249#331—1221no—
250#331—1221no—
251C61#932211yescharacteristic, complemented, Sylow
252C2×#41D24411yescharacteristic
253C22×D4—1221no—
254C22×D4—1221no—
255C22×D4—1221no—
256AGL(1,ℤ/8)—2442no—
257AGL(1,ℤ/8)—2442no—
258C2×SD16—1222no—
259C2×D8—1222no—
260C2×M16—22no—
261C2×M16—1221no—
262C2×M16—1221no—
263C22×C8C22⋊D6112yes—
264#127—612no—
265#127—1221no—
266#127—1221no—
267#127—1221no—
268C4×Q8C4×D6111yescharacteristic
269C4×Q8C22⋊D6111yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C4×D6111yescharacteristic
273C4×D4C22⋊D6111yescharacteristic
274#117—21no—
275#117D24411yescharacteristic
276C2×C4⋊C4—1221no—
277C2×C42—21no—
278C2×C42—1221no—
279C2×C42C22×D6111yescharacteristic
280C4⋊C8—22no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—612no—
284C4⋊C8—1221no—
285C4⋊C8—1221no—
286C4⋊C8—1221no—
287C4⋊C8C22⋊D6111yescharacteristic
288C4⋊C8C22⋊D6111yescharacteristic
289#104—2442no—
290#102—1222no—
291#102—1222no—
292#102—2444no—
293C22⋊C8—22no—
294C22⋊C8—21no—
295#97—22no—
296C4×C8—21no—
297#95—1221no—
298C24—2441no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—21no—
303C2×Q8#1175819811yescharacteristic
304C2×D4—21no—
305C2×D4—1222no—
306C2×D4—1222no—
307C2×D4—1222no—
308C2×D4—4882no—
309C2×D4—2442no—
310C2×D4—2442no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4—2441no—
315C2×D4—2441no—
316C2×D4—2441no—
317C2×D4—2441no—
318C2×D4#1175819811yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—2441no—
323C22×C4C2×D24411yescharacteristic
324C22×C4#1175822211yescharacteristic
325C22×C4C2×C22⋊D6111yescharacteristic
326SD16—2444no—
327D8—2444no—
328M16—82no—
329M16—2441no—
330M16—2441no—
331M16—2441no—
332M16—2441no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—22no—
336C2×C8—42no—
337C2×C8—1222no—
338C2×C8—2441no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—1222no—
342C4⋊C4—2441no—
343C4⋊C4#1175819811yescharacteristic
344C4⋊C4#1175819811yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—2441no—
352C42—2441no—
353C42C2×C4×D6111yescharacteristic
354C42#1175822211yescharacteristic
355C42C2×C22⋊D6111yescharacteristic
356C42C2×C22⋊D6111yescharacteristic
357C23—21no—
358C23—4882no—
359C23—4882no—
360C23—4881no—
361C23—4881no—
362C23—2442no—
363C23#1451975111yescharacteristic
364Q8—21no—
365Q8—21no—
366D4—21no—
367D4—21no—
368D4—41no—
369D4—41no—
370D4—4882no—
371D4—4882no—
372D4—4881no—
373D4—4881no—
374D4—4881no—
375D4—4881no—
376D4—2444no—
377D4—2442no—
378D4—2442no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—4881no—
391C2×C4—4881no—
392C2×C4—2442no—
393C2×C4C2×#1175819811yescharacteristic
394C2×C4C4×C22⋊D6111yescharacteristic
395C2×C4C4×C22⋊D6111yescharacteristic
396C2×C4#1451974611yescharacteristic
397C2×C4#1451975111yescharacteristic
398C2×C4#1451976011yescharacteristic
399C2×C4#1451980511yescharacteristic
400C2×C4#1451980511yescharacteristic
401C2×C4#1451980511yescharacteristic
402C2×C4#1451981011yescharacteristic
403C8—42no—
404C8—41no—
405C8—41no—
406C8—4881no—
407C8—2442no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—9762no—
414C22—9762no—
415C22—4882no—
416C22—4882no—
417C22—4882no—
418C22#1683964111yescharacteristic
419C22#1684000811yescharacteristic
420C22#1684001511yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—4882no—
428C4#1683976411yescharacteristic
429C4#1683976811yescharacteristic
430C4#1683993011yescharacteristic
431C4#1683993511yescharacteristic
432C4#1684000811yescharacteristic
433C4#1684001511yescharacteristic
434C2—21no—
435C2—21no—
436C2—81no—
437C2—9762no—
438C2#2144619511yescharacteristic
439C2#2144625711yescharacteristic
440C2#2144629911yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2989C2
socleC22×C61#386
prime radical#21455299C2
cosocleC2×C42×C61C23
hypercenter#2989D61
hypocenterC61#9322
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2989

Complements

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

NH ≅ G/Ntype
2. #21446745C2semidirect
3. #21445839C2semidirect
4. #21444486C2semidirect
5. #21443721C2semidirect
6. #21443694C2semidirect
7. C61×#2989C2semidirect
11. #16839962C4semidirect
13. #16839957C4semidirect
215. #2989D61semidirect
251. C61#9322semidirect

Maximal subgroups

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

subgroupindex
#214552992
#214467452
#214458392
#214444862
#214437212
#214436942
C61×#29892
#932261