Finite Group Databasebeta

← #29834596

Subgroups — #29834596

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
1C2×#16840198C211yesmaximal, characteristic, complemented
2#21458922C211yesmaximal, characteristic, complemented
3#21455539C211yesmaximal, characteristic, complemented
4#21447947C214yesmaximal, complemented
5#21447937C214yesmaximal, complemented
6#21447613C211yesmaximal, characteristic, complemented
7#21447566C211yesmaximal, characteristic, complemented
8C2×#16839575C211yesmaximal, characteristic, complemented
9C2×C61×#472C211yesmaximal, characteristic, complemented
10#16840560—21no—
11C2×#14519799C2211yescharacteristic
12#16840198—22no—
13#16840198—22no—
14C2×#14519721C2212yescomplemented
15C2×#14519718C2212yescomplemented
16#16840179—22no—
17#16840179C2214yescomplemented
18C2×#14519717—22no—
19C2×#14519717C2211yescharacteristic
20C2×#14519717C2211yescharacteristic
21#16840037C2212yescomplemented
22#16840032C2212yescomplemented
23#16840025—22no—
24C2×#14519713C2211yescharacteristic
25#16839744—21no—
26#16839591C2214yescomplemented
27#16839577C2212yescomplemented
28#16839575C2214yescomplemented
29#16839574C2212yescomplemented
30C2×AGL(1,ℤ/8)×C61—21no—
31C22×D8×C61C2211yescharacteristic
32C2×#131×C61C2211yescharacteristic
33C61×#472—22no—
34C61×#472C2214yes—
35C2×#108×C61C2211yescharacteristic
36C2×#100×C61—21no—
37C2×#11758222—21no—
38#14519808—41no—
39#14519807—22no—
40#14519803—41no—
41#14519802—22no—
42#14519799—21no—
43#14519799—21no—
44#14519799—21no—
45#14519799—21no—
46#14519797—42no—
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×D488C2311yescharacteristic
50C2×#11758194—21no—
51#14519721—24no—
52#14519718—24no—
53#14519717—21no—
54#14519717—24no—
55#14519717D412yes—
56#14519717D412yes—
57#14519717C2312yes—
58C2×#11758185—21no—
59C2×#11758185—24no—
60C2×#11758185C2312yes—
61#14519713—21no—
62#14519713C2312yes—
63#14519695—22no—
64#14519693—22no—
65C2×#41×C61D411yescharacteristic
66C22×D4×C61—21no—
67AGL(1,ℤ/8)×C61—24no—
68AGL(1,ℤ/8)×C61—42no—
69C2×SD16×C61—41no—
70C2×D8×C61—21no—
71C2×D8×C61—21no—
72C2×D8×C61—21no—
73C2×D8×C61—21no—
74C2×D8×C61—21no—
75C2×D8×C61—41no—
76C2×D8×C61C2312yes—
77#131×C61—21no—
78#131×C61—21no—
79#131×C61D412yes—
80#131×C61C2312yes—
81C2×M16×C61—21no—
82C2×M16×C61—21no—
83C2×M16×C61D411yescharacteristic
84C22×C8×C61—21no—
85C22×C8×C61C2311yescharacteristic
86#108×C61—21no—
87#108×C61C2312yes—
88#100×C61—24no—
89C24×C61—41no—
90#41×C61—21no—
91#41×C61—21no—
92#41×C61D812yes—
93#41×C61C2×D412yes—
94C2×Q8×C61D811yescharacteristic
95C2×D4×C61—21no—
96C2×D4×C61—22no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C2×D4×C61—41no—
102C2×D4×C61D811yescharacteristic
103C22×C4×C61—21no—
104C22×C4×C61C2×D411yescharacteristic
105C2×C22⋊D61—41no—
106#11758222—41no—
107#11758222—41no—
108#11758222—41no—
109#11758222—41no—
110C2×D244—21no—
111C2×C4×D61—41no—
112C2×Dic122—21no—
113SD16×C61—44no—
114D8×C61—22no—
115D8×C61—22no—
116D8×C61—44no—
117D8×C61—41no—
118D8×C61—41no—
119M16×C61—21no—
120M16×C61—22no—
121M16×C61—22no—
122M16×C61—22no—
123M16×C61—41no—
124M16×C61C2×D412yes—
125C2×C8×C61—21no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—41no—
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61#4111yescharacteristic
133C2×C8×C61#4111yescharacteristic
134C2×C8×C61C2411yescharacteristic
135#11758202—42no—
136#11758201—42no—
137#11758200—42no—
138#11758199—42no—
139#11758194—22no—
140#11758194—41no—
141C2×#92418—41no—
142Dic244—21no—
143Dic244—21no—
144D488—21no—
145D488—21no—
146#11758190—41no—
147#11758190—41no—
148#11758185—22no—
149#11758185C2×D414yes—
150C23×C61—21no—
151C23×C61—42no—
152C23×C61—81no—
153C23×C61—81no—
154C23×C61—162no—
155C23×C61#13511yescharacteristic
156C22×D61—41no—
157Q8×C61—21no—
158Q8×C61C2×D812yes—
159D4×C61—22no—
160D4×C61—22no—
161D4×C61—21no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—81no—
165D4×C61—81no—
166D4×C61C2×D812yes—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
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—81no—
177C22⋊D61—81no—
178C2×Dic61—41no—
179D244—41no—
180D244—41no—
181C4×D61—81no—
182C4×D61—81no—
183Dic122—41no—
184Dic122—41no—
185C8×C61—22no—
186C8×C61—42no—
187C8×C61#12111yescharacteristic
188C8×C61#12111yescharacteristic
189C8×C61C22×D411yescharacteristic
190C8×C61C2×#4111yescharacteristic
191#92418—42no—
192#33618—611nomaximal, Sylow
193C22×C61—22no—
194C22×C61—41no—
195C22×C61—81no—
196C22×C61—82no—
197C22×C61—162no—
198C22×C61—162no—
199C22×C61—162no—
200C22×C61#46611yescharacteristic
201C22×C61#46611yescharacteristic
202C22×C61C2×#13511yescharacteristic
203C2×D61—81no—
204C2×D61—161no—
205C4×C61—22no—
206C4×C61#46611yescharacteristic
207C4×C61#46611yescharacteristic
208C4×C61C2×#12111yescharacteristic
209C4×C61C22×D811yescharacteristic
210Dic61—81no—
211C2×#509—611no—
212#3407—611no—
213#3067—611no—
214C2×#472D6111yescharacteristic, complemented
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—42no—
224C2×C61—162no—
225C2×C61C2×#46611yescharacteristic
226D61—161no—
227#575—1221no—
228C2×AGL(1,ℤ/8)—21no—
229C2×#135—611no—
230C22×D8C2×D6111yescharacteristic
231C2×#131C2×D6111yescharacteristic
232#509—1222no—
233#509—1222no—
234C2×SD32—612no—
235C2×D16—612no—
236#504—614no—
237#504—1222no—
238C2×M32—611no—
239C2×M32—611no—
240C2×M32—1222no—
241#473—612no—
242#472—22no—
243#472—612no—
244#472C2×D6114yes—
245#471—1222no—
246C2×#108—611no—
247C2×#108C2×D6111yescharacteristic
248#413—1221no—
249C2×#100—21no—
250#368—614no—
251#362—612no—
252#361—614no—
253#361—612no—
254C61#3361811yescharacteristic, complemented, Sylow
255C2×#41—1221no—
256C2×#41C22⋊D6111yescharacteristic
257C22×D4—21no—
258#137—1222no—
259#137—2441no—
260AGL(1,ℤ/8)—24no—
261AGL(1,ℤ/8)—42no—
262AGL(1,ℤ/8)—1222no—
263AGL(1,ℤ/8)—2441no—
264#135—1221no—
265#135—1221no—
266#135—1221no—
267#135—1221no—
268#135—2442no—
269C2×Q16—611no—
270C2×SD16—41no—
271C2×SD16—1221no—
272C2×D8—21no—
273C2×D8—21no—
274C2×D8—21no—
275C2×D8—21no—
276C2×D8—21no—
277C2×D8—41no—
278C2×D8—611no—
279C2×D8C22×D6112yes—
280#131—21no—
281#131—21no—
282#131C22⋊D6112yes—
283#131C22×D6112yes—
284C2×M16—21no—
285C2×M16—21no—
286C2×M16—1221no—
287C2×M16C22⋊D6111yescharacteristic
288C22×C8—21no—
289C22×C8C22×D6111yescharacteristic
290SD32—1224no—
291D16—1224no—
292M32—612no—
293M32—612no—
294M32—612no—
295M32—1224no—
296M32—1221no—
297C2×C16—612no—
298C2×C16—1224no—
299C2×C16—1221no—
300#108—21no—
301#108—612no—
302#108—1221no—
303#108C22×D6112yes—
304#101—1222no—
305#100—24no—
306#100—1222no—
307C24—41no—
308#41—21no—
309#41—21no—
310#41—2441no—
311#41—2441no—
312#41—2441no—
313#41—2441no—
314#41#1175819912yes—
315#41C2×C22⋊D6112yes—
316C2×Q8—1221no—
317C2×Q8#1175819911yescharacteristic
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—1221no—
326C2×D4—2441no—
327C2×D4#1175819911yescharacteristic
328C22×C4—21no—
329C22×C4—2441no—
330C22×C4C2×C22⋊D6111yescharacteristic
331Q16—1221no—
332Q16—1221no—
333Q16—2442no—
334SD16—44no—
335SD16—2442no—
336SD16—2442no—
337SD16—2441no—
338SD16—2441no—
339D8—22no—
340D8—22no—
341D8—44no—
342D8—41no—
343D8—41no—
344D8—1221no—
345D8—1221no—
346D8—2442no—
347M16—21no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—41no—
352M16—1222no—
353M16—2441no—
354M16C2×C22⋊D6112yes—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—2441no—
360C2×C8D4×D6111yescharacteristic
361C2×C8D4×D6111yescharacteristic
362C2×C8#1175822411yescharacteristic
363C2×C8#1175822411yescharacteristic
364C2×C8C2×C22⋊D6111yescharacteristic
365C2×C8C23×D6111yescharacteristic
366C16—614no—
367C16—1222no—
368C23—21no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23—162no—
373C23—2441no—
374C23#1451979711yescharacteristic
375Q8—21no—
376Q8—2441no—
377Q8—2441no—
378Q8C2×#1175819912yes—
379D4—22no—
380D4—22no—
381D4—21no—
382D4—42no—
383D4—42no—
384D4—81no—
385D4—81no—
386D4—4881no—
387D4—4881no—
388D4—2441no—
389D4—2441no—
390D4C2×#1175819912yes—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—4881no—
395C2×C4—4881no—
396C2×C4—2441no—
397C2×C4#1451974911yescharacteristic
398C2×C4#1451974911yescharacteristic
399C2×C4C2×#1175819911yescharacteristic
400C2×C4C2×#1175819911yescharacteristic
401C2×C4#1451979711yescharacteristic
402C2×C4C22×C22⋊D6111yescharacteristic
403C8—22no—
404C8—42no—
405C8—2442no—
406C8#1451974911yescharacteristic
407C8#1451974911yescharacteristic
408C8C2×D4×D6111yescharacteristic
409C8C2×#1175822411yescharacteristic
410C22—22no—
411C22—41no—
412C22—81no—
413C22—82no—
414C22—162no—
415C22—162no—
416C22—162no—
417C22—9761no—
418C22—4881no—
419C22#1683999611yescharacteristic
420C22#1683999611yescharacteristic
421C22C2×#1451979711yescharacteristic
422C4—22no—
423C4—4881no—
424C4#1683999611yescharacteristic
425C4#1683999611yescharacteristic
426C4C2×#1451974911yescharacteristic
427C4C22×#1175819911yescharacteristic
428C2—21no—
429C2—21no—
430C2—21no—
431C2—42no—
432C2—162no—
433C2—9761no—
434C2C2×#1683999611yescharacteristic
subgroupG/H
centerC2C2×#16839996
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#472C2
socleC2×C61C2×#466
hypercenterC2×#472D61
hypocenterC61#33618
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. C2×#16840198C2semidirect
2. #21458922C2semidirect
3. #21455539C2semidirect
4. #21447947C2semidirect
5. #21447937C2semidirect
6. #21447613C2semidirect
7. #21447566C2semidirect
8. C2×#16839575C2semidirect
9. C2×C61×#472C2semidirect
14. C2×#14519721C22semidirect
15. C2×#14519718C22semidirect
17. #16840179C22semidirect
21. #16840037C22semidirect
22. #16840032C22semidirect
26. #16839591C22semidirect
27. #16839577C22semidirect
28. #16839575C22semidirect
29. #16839574C22semidirect
214. C2×#472D61semidirect
254. C61#33618semidirect

Maximal subgroups

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

subgroupindex
C2×#168401982
#214589222
#214555392
#214479472
#214479372
#214476132
#214475662
C2×#168395752
C2×C61×#4722
#3361861