Finite Group Databasebeta

← #29836798

Subgroups — #29836798

Order 15616.

646 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458961C212yesmaximal, complemented
2#21457604C212yesmaximal, complemented
3#21447946C212yesmaximal, complemented
4#21447885C212yesmaximal, complemented
5#21447796C212yesmaximal, complemented
6#21447775C212yesmaximal, complemented
7#21447701C211yesmaximal, characteristic, complemented
8#21447657C211yesmaximal, characteristic, complemented
9C61×#2249C211yesmaximal, characteristic, complemented
10#16840575C2212yes—
11#16840568—22no—
12#16840563C2212yescomplemented
13#16840203C2212yescomplemented
14#16840201—22no—
15#16840199—22no—
16#16840198C2212yes—
17#16840194C2211yescharacteristic
18#16840193—22no—
19SD32×D61—22no—
20#16840178C2212yescomplemented
21#16840034C2212yescomplemented
22#16840029C2212yes—
23#16840022—22no—
24#16839939—22no—
25#16839934—22no—
26#16839880C2211yescharacteristic
27#16839880C2212yescomplemented
28#16839592C2211yescharacteristic, complemented
29#16839583C2211yescharacteristic, complemented
30#16839578C2212yescomplemented
31#16839575C2212yescomplemented
32#16839570C2212yescomplemented
33#16839545C2212yescomplemented
34C2×Q32×C61C2211yescharacteristic, complemented
35C2×SD32×C61C2212yescomplemented
36C2×D16×C61C2211yescharacteristic
37C61×#495C2212yescomplemented
38C4×C16×C61C2211yescharacteristic, complemented
39#14519825—22no—
40#14519822—22no—
41#14519809—42no—
42#14519808—22no—
43#14519808—22no—
44#14519806—42no—
45#14519804—42no—
46#14519803—22no—
47#14519803—22no—
48#14519801—42no—
49#14519800—22no—
50#14519799D411yescharacteristic
51#14519799C2312yes—
52#14519798—42no—
53#14519796—22no—
54Q16×D61—22no—
55SD16×D61—42no—
56D8×D61—22no—
57#14519783—22no—
58#14519783C2312yes—
59Dic488D411yescharacteristic
60#14519723—22no—
61#14519722D412yes—
62#14519721—22no—
63#14519720—22no—
64D976D411yescharacteristic
65#14519718—22no—
66#14519717C2312yes—
67#14519716—22no—
68C16×D61—22no—
69#14519713D411yescharacteristic
70#14519713C2312yes—
71#14519705—22no—
72#14519705D412yes—
73#14519696—22no—
74#14519694—22no—
75C2×Q16×C61C2312yes—
76C2×SD16×C61—22no—
77C2×D8×C61C2312yes—
78#124×C61—22no—
79Q32×C61—41no—
80SD32×C61—22no—
81SD32×C61—22no—
82D16×C61—41no—
83C2×C16×C61C2311yescharacteristic
84C2×C16×C61C2311yescharacteristic
85C4×C8×C61C2311yescharacteristic
86C2×Q8×C61—22no—
87C2×D4×C61—22no—
88C2×C22⋊D61—82no—
89#11758226—42no—
90#11758226—42no—
91Q8×D61—42no—
92Q8×D61—42no—
93#11758224—42no—
94#11758224—42no—
95D4×D61—42no—
96D4×D61—42no—
97#11758222—22no—
98#11758222—82no—
99#11758222C2×D412yes—
100Q16×C61—22no—
101Q16×C61—22no—
102SD16×C61—42no—
103SD16×C61—42no—
104D8×C61—22no—
105D8×C61—22no—
106C2×C8×C61C2×D411yescharacteristic
107C2×C8×C61C2411yescharacteristic
108C22⋊C4×C61—82no—
109C42×C61C2×D411yescharacteristic
110#11758202—42no—
111#11758202—42no—
112#11758201—42no—
113#11758201—42no—
114#11758200—42no—
115#11758200—42no—
116#11758199—42no—
117#11758199—42no—
118#11758194—22no—
119#11758194C2×D412yes—
120Dic244D811yescharacteristic
121Dic244C2×D412yes—
122D488D811yescharacteristic
123D488C2×D412yes—
124#11758190—22no—
125#11758190D812yes—
126#11758189—22no—
127#11758189—42no—
128C8×D61—22no—
129C8×D61—42no—
130C16×C61—21no—
131C16×C61C2×D411yescharacteristic
132C16×C61C2×D411yescharacteristic
133#11758185—22no—
134C23×C61—42no—
135C22×D61—82no—
136Q8×C61—42no—
137Q8×C61—42no—
138D4×C61—42no—
139D4×C61—42no—
140C2×C4×C61—21no—
141C2×C4×C61—42no—
142C2×C4×C61C22×D411yescharacteristic
143C22⋊D61—22no—
144C22⋊D61—42no—
145C22⋊D61—82no—
146C22⋊D61—82no—
147C2×Dic61—82no—
148D244—22no—
149D244—82no—
150D244C2×D812yes—
151C4×D61—22no—
152C4×D61—42no—
153C4×D61—82no—
154C4×D61—82no—
155Dic122—22no—
156Dic122—82no—
157Dic122C2×D812yes—
158C8×C61C2×D811yescharacteristic
159C8×C61C2×D811yescharacteristic
160C8×C61C22×D411yescharacteristic
161C8×C61H(5,𝔽2)11yescharacteristic
162#92418—22no—
163#92418—42no—
164#33857—611nomaximal, Sylow
165C22×C61—82no—
166C22×C61—162no—
167C22×C61#57611yescharacteristic
168C2×D61—22no—
169C2×D61—42no—
170C2×D61—322no—
171C4×C61—41no—
172C4×C61—82no—
173C4×C61D4211yescharacteristic
174C4×C61C22×D811yescharacteristic
175Dic61—22no—
176Dic61—42no—
177#3416—612no—
178#3290—612no—
179#2249D6111yescharacteristic, complemented
180#2223—612no—
181#2212—612no—
182#2195—612no—
183#2192—612no—
184#2178—611no—
185#2170—611no—
186C2×C61—21no—
187C2×C61—162no—
188C2×C61D4×D811yescharacteristic
189D61—42no—
190D61—82no—
191#578—612no—
192#577—1222no—
193#576—612no—
194#510—612no—
195#510—1222no—
196#509—612no—
197#509—1222no—
198#508—611no—
199#508—1222no—
200C2×Q32C2×D6111yescharacteristic, complemented
201C2×SD32—1222no—
202C2×SD32C2×D6112yescomplemented
203C2×D16C2×D6111yescharacteristic
204#504—612no—
205#495C2×D6112yescomplemented
206#473—612no—
207#472—612no—
208#471—1222no—
209#455—1222no—
210#454—1222no—
211#443—611no—
212#443—612no—
213#368—611no—
214#364—611no—
215#362—612no—
216#361—612no—
217#359—612no—
218#350—612no—
219C4×C16C2×D6111yescharacteristic, complemented
220C61#3385711yescharacteristic, complemented, Sylow
2212−1+4—1222no—
222H(5,𝔽2)—1222no—
223#137—1222no—
224#137—1222no—
225#137—2442no—
226#137—2442no—
227AGL(1,ℤ/8)—1222no—
228AGL(1,ℤ/8)—1222no—
229AGL(1,ℤ/8)—2442no—
230AGL(1,ℤ/8)—2442no—
231#135—611no—
232#135—612no—
233#135—1222no—
234#135—1222no—
235#135—2442no—
236C2×Q16—1222no—
237C2×Q16C22×D6112yes—
238C2×SD16—22no—
239C2×SD16—2442no—
240C2×D8—1222no—
241C2×D8C22×D6112yes—
242#131—612no—
243#131—1222no—
244#124—22no—
245Q32—41no—
246Q32—611no—
247Q32—1222no—
248SD32—22no—
249SD32—22no—
250SD32—612no—
251SD32—1222no—
252SD32—1222no—
253D16—41no—
254D16—611no—
255D16—1222no—
256M32—612no—
257M32—1222no—
258C2×C16—1222no—
259C2×C16C22×D6111yescharacteristic
260C2×C16C22×D6111yescharacteristic
261#108—611no—
262#108—612no—
263#104—612no—
264#104—1222no—
265#101—1222no—
266#100—1222no—
267C4×C8C22×D6111yescharacteristic
268#41—612no—
269#41—1222no—
270#41—4882no—
271#41—2442no—
272#41—2442no—
273#41—2442no—
274#41—2442no—
275C2×Q8—22no—
276C2×Q8—2442no—
277C2×Q8—2442no—
278C2×D4—22no—
279C2×D4—4882no—
280C2×D4—2442no—
281C2×D4—2442no—
282Q16—22no—
283Q16—22no—
284Q16—612no—
285Q16—611no—
286Q16—2442no—
287Q16—2442no—
288SD16—42no—
289SD16—42no—
290SD16—612no—
291SD16—1222no—
292SD16—2442no—
293SD16—2442no—
294SD16—2442no—
295SD16—2442no—
296D8—22no—
297D8—22no—
298D8—612no—
299D8—611no—
300D8—2442no—
301D8—2442no—
302M16—612no—
303M16—1222no—
304M16—1222no—
305M16—2442no—
306C2×C8—1222no—
307C2×C8—2442no—
308C2×C8D4×D6111yescharacteristic
309C2×C8C23×D6111yescharacteristic
310C22⋊C4—82no—
311C42D4×D6111yescharacteristic
312C16—21no—
313C16—1222no—
314C16D4×D6111yescharacteristic
315C16D4×D6111yescharacteristic
316C23—42no—
317C23—4882no—
318Q8—42no—
319Q8—42no—
320Q8—612no—
321Q8—1222no—
322Q8—4882no—
323D4—42no—
324D4—42no—
325D4—612no—
326D4—1222no—
327D4—1222no—
328D4—4882no—
329D4—4882no—
330D4—4882no—
331D4—2442no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—1222no—
335C2×C4—4882no—
336C2×C4—4882no—
337C2×C4—4882no—
338C2×C4—2442no—
339C2×C4C2×D4×D6111yescharacteristic
340C8—1222no—
341C8—2442no—
342C8D8×D6111yescharacteristic
343C8D8×D6111yescharacteristic
344C8C2×D4×D6111yescharacteristic
345C8#1451982211yescharacteristic
346C22—82no—
347C22—162no—
348C22—1222no—
349C22—19522no—
350C22—2442no—
351C22#1684056311yescharacteristic
352C4—41no—
353C4—82no—
354C4—1222no—
355C4—2442no—
356C4#1684039711yescharacteristic
357C4C2×D8×D6111yescharacteristic
358C2—21no—
359C2—162no—
360C2—4882no—
361C2—2442no—
362C2#2145749411yescharacteristic
subgroupG/H
centerC2#21457494
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2249C2
socleC2×C61D4×D8
hypercenter#2249D61
hypocenterC61#33857
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2249

Complements

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

NH ≅ G/Ntype
1. #21458961C2semidirect
2. #21457604C2semidirect
3. #21447946C2semidirect
4. #21447885C2semidirect
5. #21447796C2semidirect
6. #21447775C2semidirect
7. #21447701C2semidirect
8. #21447657C2semidirect
9. C61×#2249C2semidirect
12. #16840563C22semidirect
13. #16840203C22semidirect
20. #16840178C22semidirect
21. #16840034C22semidirect
27. #16839880C22semidirect
28. #16839592C22semidirect
29. #16839583C22semidirect
30. #16839578C22semidirect
31. #16839575C22semidirect
32. #16839570C22semidirect
33. #16839545C22semidirect
34. C2×Q32×C61C22semidirect
35. C2×SD32×C61C22semidirect
37. C61×#495C22semidirect
38. C4×C16×C61C22semidirect
179. #2249D61semidirect
200. C2×Q32C2×D61semidirect
202. C2×SD32C2×D61semidirect
205. #495C2×D61semidirect
219. C4×C16C2×D61semidirect
220. C61#33857semidirect

Maximal subgroups

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

subgroupindex
#214589612
#214576042
#214479462
#214478852
#214477962
#214477752
#214477012
#214476572
C61×#22492
#3385761