Finite Group Databasebeta

← #29836376

Subgroups — #29836376

Order 15616.

528 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456402C211yesmaximal, characteristic, complemented
2#21456396C211yesmaximal, characteristic
3#21447972C211yesmaximal, characteristic, complemented
4#21447971C211yesmaximal, characteristic, complemented
5#21447954C212yesmaximal, complemented
6#21447931C212yesmaximal, complemented
7#21447929C212yesmaximal, complemented
8#21447862C212yesmaximal, complemented
9#21447856C211yesmaximal, characteristic, complemented
10#21447573C211yesmaximal, characteristic, complemented
11C61×#2207C211yesmaximal, characteristic, complemented
12#16840316—21no—
13#16840310—21no—
14C2×#14519721C2212yes—
15C2×#14519716C2212yes—
16#16840167C2211yescharacteristic
17#16840166C2211yescharacteristic
18#16840140C2211yescharacteristic
19#16840111—21no—
20#16840108—21no—
21#16840004C2211yescharacteristic
22#16840002—21no—
23#16840000C2211yescharacteristic
24#16839995—21no—
25#16839995C2212yes—
26C4×Dic244C2211yescharacteristic
27C4×#11758202—21no—
28D8×Dic61—21no—
29D8×Dic61C2212yes—
30C4×C8×D61C2211yescharacteristic
31#16839588C2212yes—
32#16839587C2212yes—
33C4⋊#11758185C2211yescharacteristic
34#16839580C2212yes—
35#16839567C2212yes—
36#16839566C2212yes—
37#16839563C2212yes—
38C2×SD32×C61C2212yes—
39C61×#493C2211yescharacteristic
40C4×Q16×C61C2211yescharacteristic
41C61×C4⋊C16C2211yescharacteristic
42C61×#357C2212yes—
43C2×#11758224—41no—
44C2×Dic244C2311yescharacteristic
45C2×#11758202—21no—
46C2×C8×D61D411yescharacteristic
47C2×C8×D61C2311yescharacteristic
48#14519777—21no—
49#14519776—21no—
50#14519775—21no—
51#14519767—41no—
52#14519764—41no—
53#14519757—21no—
54#14519755—41no—
55#14519753—21no—
56#14519748—22no—
57#14519748—41no—
58C4×Dic122—21no—
59C4×Dic122—41no—
60D4×Dic61—22no—
61D4×Dic61—41no—
62#14519734—41no—
63C42×D61D411yescharacteristic
64#14519721—42no—
65#14519716—42no—
66C2×#11758185C2312yes—
67#14519711—21no—
68#14519711C2312yes—
69#14519711C2311yescharacteristic
70#14519710—21no—
71#14519702—21no—
72#14519701—21no—
73#14519700—21no—
74#14519698—22no—
75#14519698—22no—
76C4×#92418D411yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C2×Q16×C61C2311yescharacteristic
80C2×D8×C61—21no—
81C2×D8×C61C2312yes—
82#127×C61—21no—
83C4×Q8×C61—21no—
84SD32×C61—42no—
85C2×C16×C61C2312yes—
86#107×C61C2311yescharacteristic
87#103×C61—21no—
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—21no—
90C2×D4×C61—22no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×C22⋊D61—42no—
94C22×Dic61—42no—
95#11758224—84no—
96C2×C4×D61D811yescharacteristic
97C2×C4×D61D811yescharacteristic
98C2×C4×D61C2×D411yescharacteristic
99C2×Dic122—21no—
100C2×Dic122—41no—
101C2×Dic122—41no—
102Q16×C61—21no—
103Q16×C61—21no—
104D8×C61—22no—
105D8×C61—22no—
106D8×C61—42no—
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—41no—
111C42×C61—41no—
112C42×C61C2×D411yescharacteristic
113C22⋊Dic61—42no—
114C22⋊Dic61—42no—
115#11758202—42no—
116#11758198—41no—
117#11758198—41no—
118C4⋊Dic61—22no—
119C4⋊Dic61—21no—
120C4⋊Dic61—41no—
121C4⋊Dic61—41no—
122#11758196—41no—
123#11758196—41no—
124#11758196—41no—
125#11758196—41no—
126#11758196—41no—
127C4×Dic61—41no—
128C4×Dic61D811yescharacteristic
129C4×Dic61D811yescharacteristic
130C4×Dic61C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133Dic244—21no—
134Dic244—21no—
135C8×D61—21no—
136C8×D61—21no—
137C8×D61—41no—
138C16×C61—22no—
139#11758185—22no—
140C23×C61—42no—
141C22×D61C2×D811yescharacteristic
142Q8×C61—41no—
143Q8×C61—41no—
144D4×C61—42no—
145D4×C61—42no—
146D4×C61—82no—
147D4×C61—82no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C22⋊D61—84no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—42no—
157C2×Dic61—84no—
158C2×Dic61C2×D811yescharacteristic
159C2×Dic61C2×D811yescharacteristic
160C2×Dic61C2×D811yescharacteristic
161C4×D61—21no—
162C4×D61—21no—
163C4×D61—22no—
164C4×D61—22no—
165Dic122—41no—
166Dic122—41no—
167Dic122—82no—
168Dic122—82no—
169C8×C61—21no—
170C8×C61H(5,𝔽2)11yescharacteristic
171C8×C612−1+411yescharacteristic
172#92418—21no—
173#92418—21no—
174#33820—611nomaximal, Sylow
175C22×C61—82no—
176C22×C61—82no—
177C22×C61—82no—
178C22×C61C22×D811yescharacteristic
179C2×D61—21no—
180C2×D61—21no—
181C2×D61—21no—
182C4×C61—21no—
183C4×C61—81no—
184C4×C61—81no—
185C4×C61#51012yes—
186C4×C61#53711yescharacteristic
187C4×C61#57511yescharacteristic
188Dic61—21no—
189Dic61—21no—
190Dic61—81no—
191Dic61—82no—
192Dic61—81no—
193Dic61#50912yes—
194#3170—611no—
195#3169—611no—
196#2227—611no—
197#2227—611no—
198#2224—612no—
199#2220—612no—
200#2220—612no—
201#2208—612no—
202#2207—611no—
203#2207D6111yescharacteristic, complemented
204#2150—611no—
205C2×C61—162no—
206C2×C61#324511yescharacteristic
207C2×C61C2×#50911yescharacteristic
208C2×C61C2×#51011yescharacteristic
209D61—41no—
210#533—1221no—
211#532—1221no—
212C2×SD32—612no—
213C2×SD32C2×D6112yes—
214C2×M32—612no—
215#500—611no—
216#500—611no—
217#494—611no—
218#493C2×D6111yescharacteristic
219#487—1221no—
220#486—1221no—
221#467—611no—
222#467—611no—
223#467—1221no—
224#466—612no—
225#466—1221no—
226C4×Q16—611no—
227C4×Q16—1221no—
228C4×Q16C2×D6111yescharacteristic
229C4×D8—612no—
230C4×D8—1221no—
231C2×C4×C8—611no—
232#366—612no—
233#366—612no—
234C4⋊C16—612no—
235C4⋊C16—611no—
236C4⋊C16C2×D6111yescharacteristic
237#358—612no—
238#358—612no—
239#357—612no—
240#357C2×D6112yes—
241C61#3382011yescharacteristic, complemented, Sylow
242C2×#41—2441no—
243C2×Q16—611no—
244C2×Q16—1221no—
245C2×Q16C22×D6111yescharacteristic
246C2×D8—21no—
247C2×D8C22×D6112yes—
248C22×C8—611no—
249C22×C8—611no—
250C4⋊Q8—1221no—
251C4⋊Q8—1221no—
252C4⋊Q8—1221no—
253#127—21no—
254#125—2441no—
255#124—2441no—
256C22⋊Q8—1221no—
257C22⋊Q8—1221no—
258C22⋊Q8—2441no—
259#121—1222no—
260#121—2441no—
261C4×Q8—21no—
262C4×Q8—1221no—
263C4×Q8—2441no—
264C4×D4—1222no—
265C4×D4—2441no—
266#117—2441no—
267C2×C42—611no—
268SD32—42no—
269SD32—2442no—
270M32—2442no—
271C2×C16—612no—
272C2×C16C22×D6112yes—
273#107—611no—
274#107—612no—
275#107—1221no—
276#107—1221no—
277#107C22×D6111yescharacteristic
278#103—21no—
279#103—1221no—
280#103—1221no—
281#103—1221no—
282#102—1222no—
283#102—1222no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8C22×D6111yescharacteristic
288#41—4884no—
289C2×Q8—21no—
290C2×Q8—1221no—
291C2×Q8—2441no—
292C2×Q8—2441no—
293C2×D4—22no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—2442no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—2442no—
301Q16—21no—
302Q16—21no—
303Q16—1221no—
304Q16—1221no—
305Q16—2442no—
306D8—22no—
307D8—22no—
308D8—42no—
309C2×C8—611no—
310C2×C8—611no—
311C2×C8—1221no—
312C2×C8—1221no—
313C2×C8—2441no—
314C2×C8D4×D6111yescharacteristic
315C2×C8C23×D6111yescharacteristic
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—1222no—
319C4⋊C4—1221no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C22⋊C4—2441no—
328C22⋊C4—2441no—
329C22⋊C4—2442no—
330C22⋊C4—2442no—
331C42—41no—
332C42—611no—
333C42—611no—
334C42—611no—
335C42—2441no—
336C42D4×D6111yescharacteristic
337C16—22no—
338C16—1222no—
339C23—42no—
340C23—611no—
341Q8—41no—
342Q8—41no—
343Q8—4882no—
344Q8—4882no—
345Q8—2441no—
346Q8—2441no—
347D4—42no—
348D4—42no—
349D4—82no—
350D4—82no—
351D4—4884no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—611no—
355C2×C4—611no—
356C2×C4—611no—
357C2×C4—1222no—
358C2×C4—1222no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—4884no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4—2442no—
365C2×C4D8×D6111yescharacteristic
366C2×C4D8×D6111yescharacteristic
367C2×C4C2×D4×D6111yescharacteristic
368C8—21no—
369C8—1221no—
370C8—1221no—
371C8#1451982211yescharacteristic
372C8#1451982411yescharacteristic
373C22—82no—
374C22—82no—
375C22—82no—
376C22—1221no—
377C22—1221no—
378C22—1221no—
379C22C2×D8×D6111yescharacteristic
380C4—21no—
381C4—81no—
382C4—81no—
383C4—612no—
384C4—1221no—
385C4—1221no—
386C4—4881no—
387C4—4882no—
388C4—4881no—
389C4#1684020112yes—
390C4#1684034111yescharacteristic
391C4#1684055811yescharacteristic
392C2—162no—
393C2—2441no—
394C2#2145712711yescharacteristic
395C2#509×D6111yescharacteristic
396C2C2×#1684020111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2207C2
socleC22×C61C22×D8
prime radical#21456396C2
hypercenter#2207D61
hypocenterC61#33820
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #21456402C2semidirect
3. #21447972C2semidirect
4. #21447971C2semidirect
5. #21447954C2semidirect
6. #21447931C2semidirect
7. #21447929C2semidirect
8. #21447862C2semidirect
9. #21447856C2semidirect
10. #21447573C2semidirect
11. C61×#2207C2semidirect
203. #2207D61semidirect
241. C61#33820semidirect

Maximal subgroups

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

subgroupindex
#214564022
#214563962
#214479722
#214479712
#214479542
#214479312
#214479292
#214478622
#214478562
#214475732
C61×#22072
#3382061