Finite Group Databasebeta

← #29834595

Subgroups — #29834595

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458921C211yesmaximal, characteristic, complemented
2#21455535C211yesmaximal, characteristic, complemented
3C2×#16840029C211yesmaximal, characteristic, complemented
4#21447946C214yesmaximal, complemented
5#21447936C214yesmaximal, complemented
6#21447614C211yesmaximal, characteristic, complemented
7#21447568C211yesmaximal, characteristic, complemented
8C2×#16839575C211yesmaximal, characteristic, complemented
9C2×C61×#509C211yesmaximal, characteristic, complemented
10#16840560—21no—
11C2×#14519803—21no—
12C2×#14519783C2211yescharacteristic
13#16840180—22no—
14#16840178C2214yescomplemented
15C2×#14519717C2211yescharacteristic
16M32×D61—22no—
17#16840034C2212yescomplemented
18#16840029—22no—
19#16840029C2214yes—
20#16840029C2212yescomplemented
21#16840022—22no—
22C2×#14519713C2211yescharacteristic
23C2×#14519713C2211yescharacteristic
24#16839746—21no—
25C2×#14519694—21no—
26#16839592C2214yescomplemented
27#16839578C2212yescomplemented
28#16839575C2212yescomplemented
29#16839575C2214yescomplemented
30C2×#135×C61C2211yescharacteristic
31C22×D8×C61C2211yescharacteristic
32C61×#509—22no—
33C61×#509—22no—
34C2×SD32×C61C2212yescomplemented
35C2×D16×C61C2212yescomplemented
36C2×M32×C61C2211yescharacteristic
37C2×#11758222D411yescharacteristic
38#14519808—22no—
39#14519807—41no—
40#14519803—22no—
41#14519803—24no—
42#14519803—42no—
43#14519802—41no—
44#14519797—42no—
45C2×#11758200—41no—
46C2×#11758199—41no—
47#14519783—21no—
48#14519783—21no—
49#14519783D412yes—
50#14519783C2312yes—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194D411yescharacteristic
54C2×#11758189—21no—
55C2×C8×D61—21no—
56#14519717—21no—
57#14519717C2312yes—
58#14519716—24no—
59C2×#11758185—21no—
60C16×D61—24no—
61#14519713—21no—
62#14519713—21no—
63#14519713C2312yes—
64#14519713C2312yes—
65#14519696—22no—
66#14519694—24no—
67#14519694—22no—
68C2×#41×C61—21no—
69C22×D4×C61—21no—
70#135×C61—21no—
71#135×C61—21no—
72#135×C61—21no—
73#135×C61—21no—
74C2×Q16×C61C2311yescharacteristic
75C2×SD16×C61—21no—
76C2×D8×C61—21no—
77C2×D8×C61—21no—
78C2×D8×C61—21no—
79C2×D8×C61—21no—
80C2×D8×C61—21no—
81C2×D8×C61C2312yes—
82C2×D8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84SD32×C61—24no—
85D16×C61—24no—
86M32×C61D412yes—
87M32×C61D412yes—
88C2×C16×C61C2312yes—
89C24×C61—41no—
90#41×C61—41no—
91#41×C61—41no—
92#41×C61—41no—
93#41×C61—41no—
94C2×Q8×C61—21no—
95C2×D4×C61—22no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C2×D4×C61—41no—
102C2×D4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61C2×D411yescharacteristic
105C2×C22⋊D61—21no—
106#11758222—21no—
107#11758222—21no—
108#11758222D812yes—
109#11758222C2×D412yes—
110C2×D244D811yescharacteristic
111C2×C4×D61—21no—
112C2×Dic122D811yescharacteristic
113Q16×C61—21no—
114Q16×C61—21no—
115SD16×C61—41no—
116SD16×C61—41no—
117D8×C61—21no—
118D8×C61—22no—
119D8×C61—22no—
120D8×C61—21no—
121D8×C61—41no—
122D8×C61—41no—
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61#4111yescharacteristic
127C2×C8×C61#4111yescharacteristic
128C2×C8×C61C2411yescharacteristic
129#11758202—42no—
130#11758201—42no—
131#11758200—44no—
132#11758200—42no—
133#11758199—44no—
134#11758199—42no—
135#11758194—21no—
136#11758194—22no—
137#11758194—22no—
138#11758194—41no—
139#11758194—41no—
140#11758194C2×D412yes—
141C2×#92418—21no—
142C2×#92418—41no—
143C2×#92418—41no—
144#11758189—22no—
145#11758189—22no—
146C8×D61—22no—
147C8×D61—22no—
148C16×C61C2×D414yes—
149#11758185—22no—
150C23×C61—41no—
151C23×C61—42no—
152C23×C61—81no—
153C23×C61—81no—
154C23×C61—162no—
155C23×C61#13511yescharacteristic
156C22×D61—21no—
157Q8×C61—41no—
158Q8×C61—41no—
159D4×C61—42no—
160D4×C61—41no—
161D4×C61—42no—
162D4×C61—41no—
163D4×C61—81no—
164D4×C61—81no—
165D4×C61—81no—
166D4×C61—81no—
167C2×C4×C61—41no—
168C2×C4×C61—81no—
169C2×C4×C61—81no—
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61C2×D811yescharacteristic
173C2×C4×C61C2×D811yescharacteristic
174C2×C4×C61#13511yescharacteristic
175C2×C4×C61C22×D411yescharacteristic
176C22⋊D61—22no—
177C22⋊D61—22no—
178C2×Dic61—21no—
179D244—21no—
180D244C2×D812yes—
181C4×D61—22no—
182C4×D61—22no—
183Dic122—21no—
184Dic122C2×D812yes—
185C8×C61#12111yescharacteristic
186C8×C61#12111yescharacteristic
187C8×C61C22×D411yescharacteristic
188C8×C61C2×#4111yescharacteristic
189#92418—22no—
190#92418—42no—
191#92418—42no—
192#33618—611nomaximal, Sylow
193C22×C61—41no—
194C22×C61—81no—
195C22×C61—82no—
196C22×C61—161no—
197C22×C61—162no—
198C22×C61—162no—
199C22×C61—162no—
200C22×C61#46611yescharacteristic
201C22×C61#46611yescharacteristic
202C22×C61C2×#13511yescharacteristic
203C2×D61—22no—
204C2×D61—81no—
205C4×C61—81no—
206C4×C61#46611yescharacteristic
207C4×C61#46611yescharacteristic
208C4×C61C2×#12111yescharacteristic
209C4×C61C22×D811yescharacteristic
210Dic61—22no—
211C2×#509D6111yescharacteristic, complemented
212#3407—611no—
213#3067—611no—
214C2×#472—611no—
215#2223—614no—
216#2221—614no—
217#2161—611no—
218#2148—611no—
219C2×#361—611no—
220C2×C61—21no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61—161no—
224C2×C61—162no—
225C2×C61C2×#46611yescharacteristic
226D61—42no—
227#575—1221no—
228C2×AGL(1,ℤ/8)—1221no—
229C2×#135C2×D6111yescharacteristic
230C22×D8C2×D6111yescharacteristic
231C2×#131—611no—
232#509—22no—
233#509—22no—
234C2×SD32C2×D6112yescomplemented
235C2×D16C2×D6112yescomplemented
236#504—614no—
237#504—1222no—
238C2×M32—611no—
239C2×M32—1222no—
240C2×M32C2×D6111yescharacteristic
241#473—612no—
242#472—614no—
243#472—612no—
244#472—1222no—
245#471—1222no—
246C2×#108—611no—
247C2×#108—611no—
248#413—1221no—
249C2×#100—1221no—
250#368—614no—
251#362—612no—
252#361—614no—
253#361—612no—
254C61#3361811yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C2×#41—611no—
257C22×D4—21no—
258#137—1222no—
259#137—2441no—
260AGL(1,ℤ/8)—1224no—
261AGL(1,ℤ/8)—1222no—
262AGL(1,ℤ/8)—2442no—
263AGL(1,ℤ/8)—2441no—
264#135—21no—
265#135—21no—
266#135—21no—
267#135—21no—
268#135—2442no—
269C2×Q16C22×D6111yescharacteristic
270C2×SD16—21no—
271C2×SD16—2441no—
272C2×D8—21no—
273C2×D8—21no—
274C2×D8—21no—
275C2×D8—21no—
276C2×D8—21no—
277C2×D8—2441no—
278C2×D8C22×D6112yes—
279C2×D8C22×D6111yescharacteristic
280#131—612no—
281#131—612no—
282#131—1221no—
283#131—1221no—
284C2×M16—611no—
285C2×M16—1221no—
286C2×M16—1221no—
287C2×M16—1221no—
288C22×C8—1221no—
289C22×C8C22×D6111yescharacteristic
290SD32—24no—
291D16—24no—
292M32—612no—
293M32—1221no—
294M32—1224no—
295M32C22⋊D6112yes—
296M32C22⋊D6112yes—
297C2×C16—1221no—
298C2×C16—1224no—
299C2×C16C22×D6112yes—
300#108—612no—
301#108—612no—
302#108—1221no—
303#108—1221no—
304#101—1222no—
305#100—1224no—
306#100—1222no—
307C24—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312#41—612no—
313#41—612no—
314#41—1221no—
315#41—1221no—
316C2×Q8—21no—
317C2×Q8—611no—
318C2×D4—22no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—611no—
327C2×D4—1221no—
328C22×C4—41no—
329C22×C4—1221no—
330C22×C4D4×D6111yescharacteristic
331Q16—21no—
332Q16—21no—
333Q16—2442no—
334SD16—41no—
335SD16—41no—
336SD16—2444no—
337SD16—2442no—
338SD16—2442no—
339D8—21no—
340D8—22no—
341D8—22no—
342D8—21no—
343D8—41no—
344D8—41no—
345D8—2444no—
346D8—2442no—
347M16—612no—
348M16—1221no—
349M16—1222no—
350M16—1222no—
351M16—1222no—
352M16—1222no—
353M16—2441no—
354M16—2441no—
355C2×C8—1221no—
356C2×C8—1222no—
357C2×C8—1222no—
358C2×C8—2441no—
359C2×C8—2441no—
360C2×C8D4×D6111yescharacteristic
361C2×C8#1175822411yescharacteristic
362C2×C8#1175822411yescharacteristic
363C2×C8C2×C22⋊D6111yescharacteristic
364C2×C8C2×C22⋊D6111yescharacteristic
365C2×C8C23×D6111yescharacteristic
366C16—1222no—
367C16C2×C22⋊D6114yes—
368C23—41no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23—162no—
373C23—1221no—
374C23#1451979611yescharacteristic
375Q8—41no—
376Q8—41no—
377Q8—612no—
378Q8—1221no—
379D4—42no—
380D4—41no—
381D4—42no—
382D4—41no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—612no—
388D4—1222no—
389D4—1222no—
390D4—1221no—
391C2×C4—41no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—1221no—
395C2×C4—1222no—
396C2×C4—1222no—
397C2×C4#1451974811yescharacteristic
398C2×C4#1451974811yescharacteristic
399C2×C4D8×D6111yescharacteristic
400C2×C4D8×D6111yescharacteristic
401C2×C4#1451979611yescharacteristic
402C2×C4C2×D4×D6111yescharacteristic
403C8—1222no—
404C8—2442no—
405C8—2442no—
406C8#1451974811yescharacteristic
407C8#1451974811yescharacteristic
408C8C22×C22⋊D6111yescharacteristic
409C8C2×#1175822411yescharacteristic
410C22—41no—
411C22—81no—
412C22—82no—
413C22—161no—
414C22—162no—
415C22—162no—
416C22—162no—
417C22—1222no—
418C22—4881no—
419C22#1683999511yescharacteristic
420C22#1683999511yescharacteristic
421C22C2×#1451979611yescharacteristic
422C4—81no—
423C4—1222no—
424C4#1683999511yescharacteristic
425C4#1683999511yescharacteristic
426C4C2×#1451974811yescharacteristic
427C4C2×D8×D6111yescharacteristic
428C2—21no—
429C2—21no—
430C2—21no—
431C2—162no—
432C2—161no—
433C2—2442no—
434C2C2×#1683999511yescharacteristic
subgroupG/H
centerC2C2×#16839995
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC2×C61C2×#466
hypercenterC2×#509D61
hypocenterC61#33618
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. #21458921C2semidirect
2. #21455535C2semidirect
3. C2×#16840029C2semidirect
4. #21447946C2semidirect
5. #21447936C2semidirect
6. #21447614C2semidirect
7. #21447568C2semidirect
8. C2×#16839575C2semidirect
9. C2×C61×#509C2semidirect
14. #16840178C22semidirect
17. #16840034C22semidirect
20. #16840029C22semidirect
26. #16839592C22semidirect
27. #16839578C22semidirect
28. #16839575C22semidirect
29. #16839575C22semidirect
34. C2×SD32×C61C22semidirect
35. C2×D16×C61C22semidirect
211. C2×#509D61semidirect
234. C2×SD32C2×D61semidirect
235. C2×D16C2×D61semidirect
254. C61#33618semidirect

Maximal subgroups

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

subgroupindex
#214589212
#214555352
C2×#168400292
#214479462
#214479362
#214476142
#214475682
C2×#168395752
C2×C61×#5092
#3361861