Finite Group Databasebeta

← #29834419

Subgroups — #29834419

Order 15616.

674 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#21458953C211yesmaximal, characteristic
2C2×#16840203C211yesmaximal, characteristic, complemented
3#21454737C211yesmaximal, characteristic, complemented
4#21447897C214yesmaximal, complemented
5#21447885C214yesmaximal, complemented
6#21447585C211yesmaximal, characteristic, complemented
7#21447568C211yesmaximal, characteristic, complemented
8C2×#16839578C211yesmaximal, characteristic, complemented
9C61×#3156C211yesmaximal, characteristic, complemented
10C2×#14519799C2211yescharacteristic
11#16840528—21no—
12#16840203—22no—
13#16840203C2214yes—
14#16840203C2212yes—
15#16840202—21no—
16#16840198C2212yes—
17#16840197—21no—
18#16840192—22no—
19C2×#14519723—21no—
20C2×#14519720—21no—
21C2×#14519717C2211yescharacteristic
22C2×#14519717C2211yescharacteristic
23#16839885—22no—
24#16839880C2214yes—
25C2×#14519713C2211yescharacteristic
26#16839780—21no—
27#16839583C2214yescomplemented
28#16839578C2214yescomplemented
29#16839578C2212yescomplemented
30#16839575C2212yescomplemented
31C2×#137×C61—21no—
32C2×#135×C61C2211yescharacteristic
33C22×Q16×C61C2211yescharacteristic
34C61×#536—21no—
35C61×#497—22no—
36C61×#495C2212yescomplemented
37C61×#494C2212yescomplemented
38C61×#405C2211yescharacteristic
39C2×#11758222—21no—
40#14519799—21no—
41#14519799—21no—
42#14519799D412yes—
43#14519799C2312yes—
44C2×Dic244D411yescharacteristic
45C2×#11758190—21no—
46C2×D488D411yescharacteristic
47#14519785—42no—
48#14519783—24no—
49C2×#11758194—21no—
50M16×D61—42no—
51#14519723—22no—
52#14519723—24no—
53#14519721—22no—
54#14519720—22no—
55#14519720—24no—
56#14519718—22no—
57#14519717—21no—
58#14519717—21no—
59#14519717C2312yes—
60#14519717C2312yes—
61C2×#11758185—21no—
62C2×#11758185—21no—
63#14519713—21no—
64#14519713C2312yes—
65#14519705—24no—
66C2×#41×C61—21no—
67C22×Q8×C61—21no—
68#137×C61—42no—
69#137×C61—42no—
70#135×C61—21no—
71#135×C61—21no—
72#135×C61—21no—
73#135×C61—21no—
74C2×Q16×C61—21no—
75C2×Q16×C61—21no—
76C2×Q16×C61—21no—
77C2×Q16×C61—22no—
78C2×Q16×C61—21no—
79C2×Q16×C61—21no—
80C2×Q16×C61C2312yes—
81C2×Q16×C61C2311yescharacteristic
82C2×SD16×C61—21no—
83C2×SD16×C61—22no—
84C2×D8×C61C2311yescharacteristic
85C2×M16×C61D411yescharacteristic
86C22×C8×C61C2311yescharacteristic
87C4⋊Q8×C61—22no—
88#124×C61—22no—
89#123×C61—42no—
90C22⋊Q8×C61—42no—
91#117×C61D411yescharacteristic
92#97×C61D412yes—
93C4×C8×C61C2312yes—
94#41×C61—41no—
95#41×C61—41no—
96#41×C61—41no—
97#41×C61—41no—
98C2×Q8×C61—22no—
99C2×Q8×C61—21no—
100C2×Q8×C61—41no—
101C2×Q8×C61—41no—
102C2×Q8×C61—41no—
103C2×Q8×C61—41no—
104C2×Q8×C61—41no—
105C2×D4×C61—21no—
106C2×D4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C2×C22⋊D61—41no—
111#11758222—22no—
112#11758222—22no—
113#11758222—41no—
114#11758222—41no—
115C2×D244—21no—
116C2×C4×D61—41no—
117C2×Dic122—21no—
118Q16×C61—22no—
119Q16×C61—21no—
120Q16×C61—22no—
121Q16×C61—21no—
122Q16×C61—41no—
123Q16×C61—41no—
124Q16×C61—44no—
125SD16×C61—41no—
126SD16×C61—41no—
127SD16×C61—44no—
128D8×C61—21no—
129D8×C61—21no—
130M16×C61D812yes—
131M16×C61D812yes—
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2×D412yes—
136C2×C8×C61#4111yescharacteristic
137C2×C8×C61#4111yescharacteristic
138C2×C8×C61C2411yescharacteristic
139C4⋊C4×C61—21no—
140C4⋊C4×C61—44no—
141C22⋊C4×C61—21no—
142C22⋊C4×C61—44no—
143C42×C61C2×D412yes—
144#11758194—22no—
145#11758194—41no—
146C2×#92418—41no—
147Dic244—21no—
148Dic244C2×D412yes—
149D488—21no—
150D488C2×D412yes—
151#11758190—22no—
152#11758190—22no—
153#11758189—44no—
154C8×D61—44no—
155#11758185—22no—
156#11758185—22no—
157C23×C61—41no—
158C23×C61AGL(1,ℤ/8)11yescharacteristic
159C22×D61—41no—
160Q8×C61—41no—
161Q8×C61—42no—
162Q8×C61—42no—
163Q8×C61—41no—
164Q8×C61—81no—
165Q8×C61—81no—
166D4×C61—41no—
167D4×C61—41no—
168D4×C61—81no—
169D4×C61—81no—
170C2×C4×C61—22no—
171C2×C4×C61—42no—
172C2×C4×C61—41no—
173C2×C4×C61—81no—
174C2×C4×C61—81no—
175C2×C4×C61—81no—
176C2×C4×C61—81no—
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61C2×D811yescharacteristic
180C2×C4×C61C2×D811yescharacteristic
181C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C22⋊D61—42no—
184C22⋊D61—42no—
185C2×Dic61—41no—
186D244—22no—
187D244—41no—
188C4×D61—42no—
189C4×D61—42no—
190Dic122—22no—
191Dic122—41no—
192C8×C61#12111yescharacteristic
193C8×C61#12111yescharacteristic
194C8×C61C2×D814yes—
195C8×C61C22×D411yescharacteristic
196C8×C61C2×#4111yescharacteristic
197#92418—42no—
198#33596—611nomaximal, Sylow
199C22×C61—41no—
200C22×C61—82no—
201C22×C61—81no—
202C22×C61#45911yescharacteristic
203C22×C61#45911yescharacteristic
204C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
205C2×D61—42no—
206C2×D61—161no—
207C4×C61—42no—
208C4×C61—81no—
209C4×C61—82no—
210C4×C61#45911yescharacteristic
211C4×C61#45911yescharacteristic
212C4×C61C2×#12111yescharacteristic
213C4×C61C22×D811yescharacteristic
214Dic61—42no—
215#3414—611no—
216C2×#510—611no—
217#3156D6111yescharacteristic, complemented
218#2957—611no—
219#2214—614no—
220#2212—614no—
221#2153—611no—
222#2148—611no—
223C2×#362—611no—
224C2×C61—21no—
225C2×C61—21no—
226C2×C61—21no—
227C2×C61—161no—
228C2×C61C2×#45911yescharacteristic
229D61—82no—
230C2×#137—21no—
231C2×#135—611no—
232C2×#135C2×D6111yescharacteristic
233C22×Q16C2×D6111yescharacteristic
234#568—1221no—
235#536—21no—
236#510—614no—
237#510—612no—
238#510—1222no—
239#510—1221no—
240#509—612no—
241#509—1221no—
242#508—1222no—
243C2×Q32—1221no—
244C2×SD32—1221no—
245C2×M32—611no—
246C2×M32—611no—
247#497—22no—
248#495C2×D6112yescomplemented
249#494C2×D6112yescomplemented
250#444—1222no—
251#443—614no—
252C2×#108—611no—
253#421—1221no—
254#405C2×D6111yescharacteristic
255#364—614no—
256#362—614no—
257#362—612no—
258#361—612no—
259C61#3359611yescharacteristic, complemented, Sylow
260C2×#41—21no—
261C2×#41—1221no—
262C22×Q8—21no—
263#137—42no—
264#137—42no—
265#135—21no—
266#135—21no—
267#135—21no—
268#135—21no—
269#135—612no—
270#135—612no—
271#135—1221no—
272#135—1221no—
273C2×Q16—21no—
274C2×Q16—21no—
275C2×Q16—21no—
276C2×Q16—21no—
277C2×Q16—21no—
278C2×Q16—22no—
279C2×Q16—611no—
280C2×Q16C22×D6112yes—
281C2×Q16C22×D6111yescharacteristic
282C2×SD16—21no—
283C2×SD16—22no—
284C2×SD16—1221no—
285C2×D8—611no—
286C2×D8C22×D6111yescharacteristic
287#131—1224no—
288#131—2442no—
289C2×M16—1221no—
290C2×M16—2442no—
291C2×M16C22⋊D6111yescharacteristic
292C22×C8C22×D6111yescharacteristic
293C4⋊Q8—22no—
294#124—22no—
295#123—42no—
296C22⋊Q8—42no—
297#117C22⋊D6111yescharacteristic
298Q32—1224no—
299Q32—1222no—
300SD32—1222no—
301SD32—1224no—
302SD32—1222no—
303D16—1222no—
304M32—612no—
305M32—612no—
306M32—1221no—
307M32—1221no—
308C2×C16—1221no—
309C2×C16—1221no—
310#108—612no—
311#108—1221no—
312#104—1224no—
313#97C22⋊D6112yes—
314C4×C8C22×D6112yes—
315#41—41no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—1222no—
320#41—1222no—
321#41—2441no—
322#41—2441no—
323C2×Q8—22no—
324C2×Q8—21no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—1221no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4—1221no—
334C2×D4—2441no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—2441no—
338C22×C4C2×C22⋊D6111yescharacteristic
339Q16—22no—
340Q16—21no—
341Q16—22no—
342Q16—21no—
343Q16—41no—
344Q16—41no—
345Q16—44no—
346Q16—612no—
347Q16—1221no—
348SD16—41no—
349SD16—41no—
350SD16—44no—
351SD16—1222no—
352SD16—1222no—
353D8—21no—
354D8—21no—
355D8—612no—
356D8—1221no—
357M16—1222no—
358M16—2441no—
359M16—2444no—
360M16#1175819912yes—
361M16#1175819912yes—
362C2×C8—2441no—
363C2×C8—2444no—
364C2×C8D4×D6111yescharacteristic
365C2×C8D4×D6111yescharacteristic
366C2×C8#1175822411yescharacteristic
367C2×C8#1175822411yescharacteristic
368C2×C8C2×C22⋊D6111yescharacteristic
369C2×C8C2×C22⋊D6112yes—
370C2×C8C23×D6111yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—44no—
373C22⋊C4—21no—
374C22⋊C4—44no—
375C42C2×C22⋊D6112yes—
376C16—1222no—
377C16—1222no—
378C23—41no—
379C23—2441no—
380C23#1451980311yescharacteristic
381Q8—41no—
382Q8—42no—
383Q8—42no—
384Q8—41no—
385Q8—81no—
386Q8—81no—
387Q8—1222no—
388Q8—2441no—
389D4—41no—
390D4—41no—
391D4—81no—
392D4—81no—
393D4—1222no—
394D4—2442no—
395D4—2442no—
396D4—2441no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—2441no—
405C2×C4—2442no—
406C2×C4—2442no—
407C2×C4#1451974811yescharacteristic
408C2×C4#1451974811yescharacteristic
409C2×C4C2×#1175819911yescharacteristic
410C2×C4C2×#1175819911yescharacteristic
411C2×C4#1451980311yescharacteristic
412C2×C4C22×C22⋊D6111yescharacteristic
413C8—2442no—
414C8#1451974811yescharacteristic
415C8#1451974811yescharacteristic
416C8C2×#1175819914yes—
417C8C2×D4×D6111yescharacteristic
418C8C2×#1175822411yescharacteristic
419C22—41no—
420C22—82no—
421C22—81no—
422C22—9761no—
423C22—2442no—
424C22#1683995411yescharacteristic
425C22#1683995411yescharacteristic
426C22C2×#1451980311yescharacteristic
427C4—42no—
428C4—81no—
429C4—82no—
430C4—2442no—
431C4#1683995411yescharacteristic
432C4#1683995411yescharacteristic
433C4C2×#1451974811yescharacteristic
434C4C22×#1175819911yescharacteristic
435C2—21no—
436C2—21no—
437C2—21no—
438C2—161no—
439C2—4882no—
440C2C2×#1683995411yescharacteristic
subgroupG/H
centerC2C2×#16839954
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3156C2
socleC2×C61C2×#459
prime radical#21458953C2
hypercenter#3156D61
hypocenterC61#33596
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #3156

Complements

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

NH ≅ G/Ntype
2. C2×#16840203C2semidirect
3. #21454737C2semidirect
4. #21447897C2semidirect
5. #21447885C2semidirect
6. #21447585C2semidirect
7. #21447568C2semidirect
8. C2×#16839578C2semidirect
9. C61×#3156C2semidirect
27. #16839583C22semidirect
28. #16839578C22semidirect
29. #16839578C22semidirect
30. #16839575C22semidirect
36. C61×#495C22semidirect
37. C61×#494C22semidirect
217. #3156D61semidirect
248. #495C2×D61semidirect
249. #494C2×D61semidirect
259. C61#33596semidirect

Maximal subgroups

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

subgroupindex
#214589532
C2×#168402032
#214547372
#214478972
#214478852
#214475852
#214475682
C2×#168395782
C61×#31562
#3359661