Finite Group Databasebeta

← #29575913

Subgroups — #29575913

Order 15616.

516 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455299C211yesmaximal, characteristic
2#21444488C211yesmaximal, characteristic, complemented
3C2×#16839505C211yesmaximal, characteristic, complemented
4#21444206C211yesmaximal, characteristic, complemented
5#21443716C211yesmaximal, characteristic, complemented
6#21443693C211yesmaximal, characteristic, complemented
7C61×#2925C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×#14519774C2211yescharacteristic
10#16839962—21no—
11#16839962C412yes—
12#16839957—21no—
13#16839957C412yes—
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16C2×C4⋊#92418C2211yescharacteristic
17#16839766—21no—
18#16839505—21no—
19#16839505—21no—
20#16839505—22no—
21Q8⋊#92418—21no—
22D4⋊#92418—21no—
23C4×#41×C61C2211yescharacteristic
24C61×#433—22no—
25C61×#432—21no—
26C61×#431—21no—
27C61×#423C2211yescharacteristic
28C2×C4⋊C8×C61C2211yescharacteristic
29C61×#408—21no—
30C61×#405—21no—
31C22×D244—21no—
32C22×D244—21no—
33C22×D244—21no—
34#14519802—42no—
35#14519802—42no—
36C2×#11758201—22no—
37C2×#11758199—22no—
38C2×#11758194—21no—
39C22×#92418—21no—
40#14519774—21no—
41#14519774—21no—
42#14519774—21no—
43#14519774C2×C412yes—
44C4⋊#92418—21no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47C4⋊#92418—21no—
48C4⋊#92418C2×C412yes—
49#14519697—22no—
50#14519697—22no—
51C22⋊#92418—21no—
52C4×#92418—21no—
53C2×#41×C61D411yescharacteristic
54C2×M16×C61—21no—
55C22×C8×C61—21no—
56C4×Q8×C61C2×C411yescharacteristic
57C4×Q8×C61D411yescharacteristic
58C4×D4×C61—21no—
59C4×D4×C61—21no—
60C4×D4×C61C2×C411yescharacteristic
61C4×D4×C61D411yescharacteristic
62#117×C61—21no—
63#117×C61D411yescharacteristic
64C2×C42×C61—21no—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C4⋊C8×C61D411yescharacteristic
69C4⋊C8×C61D411yescharacteristic
70C4⋊C8×C61D412yes—
71C22⋊C8×C61—22no—
72C22⋊C8×C61—22no—
73#97×C61—22no—
74C4×C8×C61—22no—
75C23×D61—41no—
76#41×C61—21no—
77#41×C61—21no—
78#41×C61—21no—
79#41×C61—21no—
80C2×Q8×C61C22⋊C411yescharacteristic
81C2×D4×C61—21no—
82C2×D4×C61C22⋊C411yescharacteristic
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C2×D244—22no—
90C2×D244—22no—
91C2×D244—22no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—41no—
98C2×D244—41no—
99C2×D244—82no—
100C2×D244—81no—
101C2×D244—81no—
102M16×C61—41no—
103M16×C61—41no—
104C2×C8×C61—21no—
105C2×C8×C61—22no—
106C2×C8×C61—21no—
107C2×C8×C61—41no—
108C2×C8×C61—41no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61C22⋊C411yescharacteristic
112C4⋊C4×C61C22⋊C411yescharacteristic
113C22⋊C4×C61—21no—
114C22⋊C4×C61—21no—
115C22⋊C4×C61—21no—
116C42×C61—21no—
117C42×C61—21no—
118C42×C61—21no—
119C42×C61C22×C411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123#11758201—44no—
124#11758199—44no—
125#11758194—41no—
126#11758194—41no—
127C2×#92418—22no—
128C2×#92418—41no—
129C2×#92418—41no—
130C2×#92418—81no—
131C23×C61—21no—
132C23×C61#12311yescharacteristic
133C22×D61—42no—
134C22×D61—82no—
135C22×D61—82no—
136C22×D61—81no—
137C22×D61—81no—
138Q8×C61—21no—
139Q8×C61—21no—
140D4×C61—21no—
141D4×C61—21no—
142D4×C61—41no—
143D4×C61—41no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—81no—
155C2×C4×C61C2×C22⋊C411yescharacteristic
156C2×C4×C61C4×D411yescharacteristic
157C2×C4×C61C4×D411yescharacteristic
158C2×C4×C61#12011yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#13511yescharacteristic
162C2×C4×C61#13511yescharacteristic
163C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165D244—42no—
166D244—42no—
167D244—82no—
168D244—82no—
169D244—82no—
170D244—81no—
171D244—81no—
172D244—81no—
173D244—81no—
174C8×C61—41no—
175C8×C61—41no—
176C8×C61—42no—
177#92418—42no—
178#92418—81no—
179#92418—81no—
180#9174—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61#38611yescharacteristic
187C22×C61#48211yescharacteristic
188C22×C61#48211yescharacteristic
189C2×D61—82no—
190C2×D61—82no—
191C2×D61—82no—
192C2×D61—162no—
193C2×D61—162no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61#41611yescharacteristic
201C4×C61#41811yescharacteristic
202C4×C61#44911yescharacteristic
203C4×C61#44911yescharacteristic
204C4×C61#47111yescharacteristic
205C4×C61#47211yescharacteristic
206#3038—611no—
207#2925D6111yescharacteristic, complemented
208#1545—611no—
209C2×#331—611no—
210#1477—611no—
211#1301—611no—
212#1292—611no—
213C2×C61—21no—
214C2×C61—21no—
215C2×C61—81no—
216C2×C61#187511yescharacteristic
217C2×C61#190311yescharacteristic
218C2×C61#190711yescharacteristic
219D61—162no—
220C2×AGL(1,ℤ/8)—1221no—
221C2×#127—611no—
222C4×#41C2×D6111yescharacteristic
223#460—612no—
224#460—1221no—
225#459—612no—
226#459—1221no—
227#433—22no—
228#432—21no—
229#431—21no—
230#431—611no—
231#423—611no—
232#423C2×D6111yescharacteristic
233C2×C4⋊C8—611no—
234C2×C4⋊C8C2×D6111yescharacteristic
235#418—1221no—
236#408—21no—
237#405—21no—
238#331—1221no—
239#331—1221no—
240#331—1222no—
241Q8⋊C8—1221no—
242D4⋊C8—1221no—
243C61#917411yescharacteristic, complemented, Sylow
244C2×#41C22⋊D6111yescharacteristic
245C22×D4—1221no—
246C22×D4—1221no—
247C22×D4—1221no—
248AGL(1,ℤ/8)—2442no—
249AGL(1,ℤ/8)—2442no—
250C2×SD16—1222no—
251C2×D8—1222no—
252C2×M16—21no—
253C2×M16—1221no—
254C22×C8—21no—
255C22×C8—1221no—
256#127—612no—
257#127—1221no—
258#127—1221no—
259#127—1221no—
260C4×Q8C4×D6111yescharacteristic
261C4×Q8D24411yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264C4×D4C4×D6111yescharacteristic
265C4×D4D24411yescharacteristic
266#117—21no—
267#117C22⋊D6111yescharacteristic
268C2×C42—21no—
269C2×C42C22×D6111yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—612no—
273C4⋊C8—1221no—
274C4⋊C8—1221no—
275C4⋊C8—1221no—
276C4⋊C8—1221no—
277C4⋊C8C22⋊D6111yescharacteristic
278C4⋊C8C22⋊D6111yescharacteristic
279C4⋊C8C22⋊D6112yes—
280#102—1222no—
281#102—1222no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—1221no—
285#97—22no—
286C4×C8—22no—
287C4×C8—1221no—
288C24—2441no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—21no—
293C2×Q8#1175819811yescharacteristic
294C2×D4—21no—
295C2×D4—1222no—
296C2×D4—1222no—
297C2×D4—1222no—
298C2×D4—4881no—
299C2×D4—4881no—
300C2×D4—4882no—
301C2×D4—2441no—
302C2×D4—2441no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2441no—
306C2×D4—2441no—
307C2×D4—2441no—
308C2×D4#1175819811yescharacteristic
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4#1175822211yescharacteristic
313C22×C4#1175822211yescharacteristic
314C22×C4C2×C22⋊D6111yescharacteristic
315SD16—2444no—
316D8—2444no—
317M16—41no—
318M16—41no—
319M16—2441no—
320M16—2441no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—1222no—
327C2×C8—4881no—
328C2×C8—2441no—
329C2×C8—2441no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4#1175819811yescharacteristic
333C4⋊C4#1175819811yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42C2×C4×D6111yescharacteristic
341C42C2×D24411yescharacteristic
342C42C2×C22⋊D6111yescharacteristic
343C42C2×C22⋊D6111yescharacteristic
344C23—21no—
345C23—4882no—
346C23—4882no—
347C23—4881no—
348C23—4881no—
349C23—2442no—
350C23#1451976011yescharacteristic
351Q8—21no—
352Q8—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—41no—
357D4—4882no—
358D4—4882no—
359D4—4882no—
360D4—4881no—
361D4—4881no—
362D4—4881no—
363D4—4881no—
364D4—2442no—
365D4—2442no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4C2×#1175819811yescharacteristic
378C2×C4C4×C22⋊D6111yescharacteristic
379C2×C4C4×C22⋊D6111yescharacteristic
380C2×C4#1451974611yescharacteristic
381C2×C4#1451975111yescharacteristic
382C2×C4#1451975111yescharacteristic
383C2×C4#1451979711yescharacteristic
384C2×C4#1451979711yescharacteristic
385C2×C4#1451980211yescharacteristic
386C2×C4#1451980211yescharacteristic
387C8—41no—
388C8—41no—
389C8—42no—
390C8—4881no—
391C8—4881no—
392C8—2442no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—9762no—
399C22—9762no—
400C22—4882no—
401C22—4882no—
402C22—4882no—
403C22#1683964111yescharacteristic
404C22#1684008611yescharacteristic
405C22#1684008611yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4#1683975611yescharacteristic
413C4#1683976611yescharacteristic
414C4#1683990711yescharacteristic
415C4#1683990711yescharacteristic
416C4#1684002511yescharacteristic
417C4#1684003211yescharacteristic
418C2—21no—
419C2—21no—
420C2—81no—
421C2—9762no—
422C2#2144614111yescharacteristic
423C2#2144629911yescharacteristic
424C2#2144632711yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2925C2
socleC22×C61#386
prime radical#21455299C2
cosocleC2×C42×C61C23
hypercenter#2925D61
hypocenterC61#9174
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #21444488C2semidirect
3. C2×#16839505C2semidirect
4. #21444206C2semidirect
5. #21443716C2semidirect
6. #21443693C2semidirect
7. C61×#2925C2semidirect
207. #2925D61semidirect
243. C61#9174semidirect

Maximal subgroups

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

subgroupindex
#214552992
#214444882
C2×#168395052
#214442062
#214437162
#214436932
C61×#29252
#917461