Finite Group Databasebeta

← #29598255

Subgroups — #29598255

Order 15616.

482 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456243C211yesmaximal, characteristic
2C2×#16839570C211yesmaximal, characteristic, complemented
3C2×#16839569C211yesmaximal, characteristic, complemented
4C2×#16839568C211yesmaximal, characteristic, complemented
5C2×#16839537C211yesmaximal, characteristic, complemented
6#21445989C211yesmaximal, characteristic, complemented
7C2×C61×#359C211yesmaximal, characteristic, complemented
8C2×#14519799C2211yescharacteristic
9C2×#14519797—21no—
10#16840308—21no—
11C22×#11758185C2211yescharacteristic
12#16840147C411yescharacteristic
13#16840145C411yescharacteristic
14#16840143—21no—
15#16840140C411yescharacteristic
16#16840137C411yescharacteristic
17#16840133—21no—
18C2×#14519713C2211yescharacteristic
19C2×#14519712—21no—
20#16839785—21no—
21C2×C8×Dic61C2211yescharacteristic
22#16839570—22no—
23#16839569—22no—
24#16839568—22no—
25#16839537—21no—
26#16839537—21no—
27C2×#135×C61C2211yescharacteristic
28C22×C16×C61C2211yescharacteristic
29C2×#108×C61C2211yescharacteristic
30C61×#359—22no—
31C2×#11758222—21no—
32#14519799—21no—
33#14519799—21no—
34#14519799—21no—
35#14519799—21no—
36#14519797—42no—
37#14519797—42no—
38C2×Dic244C2×C411yescharacteristic
39C2×#11758202—21no—
40C2×#11758190—21no—
41C2×#11758200—21no—
42C2×#11758201—21no—
43C2×D488C2×C411yescharacteristic
44C2×#11758199—21no—
45C2×#11758194—21no—
46C22×#92418D411yescharacteristic
47#14519775—21no—
48#14519773—21no—
49#14519766—21no—
50#14519764—21no—
51#14519749—42no—
52C4×C22⋊D61—42no—
53C2×C4×Dic61D411yescharacteristic
54C2×#11758185—21no—
55C2×#11758185—21no—
56C2×#11758185D412yescomplemented
57#14519713—21no—
58#14519713—21no—
59#14519712—42no—
60#14519706—42no—
61C8×Dic61C2×C411yescharacteristic
62C8×Dic61C2×C411yescharacteristic
63C8×Dic61D411yescharacteristic
64C8×Dic61D411yescharacteristic
65C2×#41×C61—21no—
66#135×C61—21no—
67#135×C61—21no—
68#135×C61—21no—
69#135×C61—21no—
70C2×Q16×C61C2×C411yescharacteristic
71C2×SD16×C61—21no—
72C2×D8×C61C2×C411yescharacteristic
73C2×M16×C61—21no—
74C22×C8×C61C2311yescharacteristic
75C2×C16×C61—21no—
76C2×C16×C61—21no—
77C2×C16×C61D412yescomplemented
78#108×C61—21no—
79#108×C61—21no—
80#41×C61—41no—
81#41×C61—41no—
82#41×C61—41no—
83#41×C61—41no—
84C2×Q8×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61C2×D411yescharacteristic
89C2×C22⋊D61—41no—
90C2×C22⋊D61—42no—
91C22×Dic61SD1612yes—
92#11758222—41no—
93#11758222—41no—
94#11758222—41no—
95#11758222—41no—
96C2×D244—21no—
97C2×C4×D61—41no—
98C2×Dic122—21no—
99Q16×C61—21no—
100Q16×C61—21no—
101SD16×C61—41no—
102SD16×C61—41no—
103D8×C61—21no—
104D8×C61—21no—
105M16×C61—41no—
106M16×C61—41no—
107C2×C8×C61—41no—
108C2×C8×C61C22×C411yescharacteristic
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61#4111yescharacteristic
113C2×C8×C61#4111yescharacteristic
114C22⋊Dic61—42no—
115#11758202—42no—
116#11758201—42no—
117#11758200—42no—
118#11758199—42no—
119#11758198—42no—
120#11758196—42no—
121C4×Dic61C22⋊C411yescharacteristic
122C4×Dic61C22⋊C411yescharacteristic
123C4×Dic61D812yes—
124#11758194—41no—
125#11758194—41no—
126C2×#92418—41no—
127C2×#92418C22⋊C411yescharacteristic
128C2×#92418C22⋊C411yescharacteristic
129C2×#92418D812yes—
130C2×#92418SD1612yes—
131Dic244—21no—
132Dic244—21no—
133D488—21no—
134D488—21no—
135#11758190—41no—
136#11758190—41no—
137C16×C61—22no—
138#11758185—22no—
139C23×C61—41no—
140C23×C61C2×SD1611yescharacteristic
141C22×D61—41no—
142Q8×C61—41no—
143Q8×C61—41no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—81no—
147D4×C61—81no—
148C2×C4×C61—41no—
149C2×C4×C61—81no—
150C2×C4×C61—81no—
151C2×C4×C61C2×C22⋊C411yescharacteristic
152C2×C4×C61#12411yescharacteristic
153C2×C4×C61#12711yescharacteristic
154C2×C4×C61C2×D811yescharacteristic
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C2×SD1611yescharacteristic
157C22⋊D61—84no—
158C22⋊D61—81no—
159C22⋊D61—81no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61#10212yes—
163C2×Dic61#10212yes—
164D244—41no—
165D244—41no—
166C4×D61—81no—
167C4×D61—81no—
168Dic122—41no—
169Dic122—41no—
170C8×C61—81no—
171C8×C61C4×D411yescharacteristic
172C8×C61C4×D411yescharacteristic
173C8×C61#12111yescharacteristic
174C8×C61#12111yescharacteristic
175#92418—81no—
176#92418#10212yes—
177#92418#10212yes—
178#13209—611nomaximal, Sylow
179C22×C61—22no—
180C22×C61—82no—
181C22×C61—81no—
182C22×C61C2×#10211yescharacteristic
183C22×C61#48611yescharacteristic
184C22×C61#49211yescharacteristic
185C2×D61—82no—
186C2×D61—81no—
187C4×C61—81no—
188C4×C61#39011yescharacteristic
189C4×C61C2×#10211yescharacteristic
190C4×C61#48611yescharacteristic
191C4×C61#49311yescharacteristic
192Dic61—22no—
193Dic61—22no—
194Dic61—81no—
195#3147—611no—
196C2×#359—611no—
197C2×#359—611no—
198C2×#359—611no—
199C2×#359D6111yescharacteristic, complemented
200C2×#346—611no—
201#1848—611no—
202C2×C61—21no—
203C2×C61—21no—
204C2×C61—161no—
205C2×C61#195711yescharacteristic
206C2×C61#214312yes—
207D61—161no—
208C2×#135—611no—
209C2×#135—1221no—
210C2×#135C2×D6111yescharacteristic
211#532—1221no—
212C22×C16—611no—
213C22×C16C2×D6111yescharacteristic
214#495—611no—
215#495—611no—
216#495—1221no—
217#494—611no—
218#493—611no—
219#492—1221no—
220C2×#108—611no—
221C2×#108—1221no—
222C2×#108C2×D6111yescharacteristic
223#423—1221no—
224C2×C4×C8—611no—
225#359—22no—
226#359—1222no—
227#359—1222no—
228#359—1222no—
229#346—1221no—
230#346—1221no—
231C61#1320911yescharacteristic, complemented, Sylow
232C2×#41—21no—
233C2×#41—1221no—
234#135—21no—
235#135—21no—
236#135—21no—
237#135—21no—
238#135—1221no—
239#135—1221no—
240#135—1221no—
241#135—1221no—
242#135—2442no—
243#135—2442no—
244C2×Q16—611no—
245C2×Q16—1221no—
246C2×Q16C4×D6111yescharacteristic
247C2×SD16—21no—
248C2×SD16—1221no—
249C2×SD16—1221no—
250C2×SD16—1221no—
251C2×D8—611no—
252C2×D8—1221no—
253C2×D8C4×D6111yescharacteristic
254C2×M16—21no—
255C2×M16—1221no—
256C22×C8—611no—
257C22×C8C22×D6111yescharacteristic
258C4⋊Q8—1221no—
259#127—1221no—
260#124—1221no—
261#124—1221no—
262#121—2442no—
263C4×D4—2442no—
264C2×C42—611no—
265C2×C16—21no—
266C2×C16—21no—
267C2×C16—612no—
268C2×C16—1221no—
269C2×C16—1221no—
270C2×C16C22⋊D6112yescomplemented
271#108—21no—
272#108—21no—
273#108—1221no—
274#108—1221no—
275#108—2442no—
276C4⋊C8—2442no—
277C4×C8—611no—
278C4×C8—611no—
279C4×C8—611no—
280C4×C8—611no—
281#41—41no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—2441no—
286#41—2441no—
287#41—2441no—
288#41—2441no—
289C2×Q8—21no—
290C2×Q8—1221no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—1221no—
294C2×D4—2442no—
295C2×D4—2441no—
296C22×C4—41no—
297C22×C4—612no—
298C22×C4—2441no—
299C22×C4D4×D6111yescharacteristic
300Q16—21no—
301Q16—21no—
302Q16—1221no—
303Q16—1221no—
304Q16—2442no—
305SD16—41no—
306SD16—41no—
307SD16—2442no—
308SD16—2442no—
309SD16—2441no—
310SD16—2441no—
311D8—21no—
312D8—21no—
313D8—1221no—
314D8—1221no—
315D8—2442no—
316M16—41no—
317M16—41no—
318M16—2441no—
319M16—2441no—
320C2×C8—41no—
321C2×C8—611no—
322C2×C8—611no—
323C2×C8—612no—
324C2×C8—612no—
325C2×C8—2441no—
326C2×C8C2×C4×D6111yescharacteristic
327C2×C8#1175822211yescharacteristic
328C2×C8D4×D6111yescharacteristic
329C2×C8D4×D6111yescharacteristic
330C2×C8#1175822411yescharacteristic
331C2×C8C2×C22⋊D6111yescharacteristic
332C4⋊C4—2442no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C42—611no—
336C42—611no—
337C42—612no—
338C16—22no—
339C16—1222no—
340C23—41no—
341C23—2441no—
342C23SD16×D6111yescharacteristic
343Q8—41no—
344Q8—41no—
345Q8—2441no—
346Q8—2441no—
347D4—41no—
348D4—41no—
349D4—81no—
350D4—81no—
351D4—4884no—
352D4—4881no—
353D4—4881no—
354D4—2441no—
355D4—2441no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—612no—
360C2×C4—612no—
361C2×C4—4881no—
362C2×C4—4881no—
363C2×C4—2442no—
364C2×C4—2441no—
365C2×C4C22⋊C4×D6111yescharacteristic
366C2×C4#1451976311yescharacteristic
367C2×C4#1451977311yescharacteristic
368C2×C4D8×D6111yescharacteristic
369C2×C4D8×D6111yescharacteristic
370C2×C4SD16×D6111yescharacteristic
371C8—81no—
372C8—612no—
373C8—612no—
374C8—4881no—
375C8#1451973711yescharacteristic
376C8C4×C22⋊D6111yescharacteristic
377C8#1451974911yescharacteristic
378C8#1451975011yescharacteristic
379C22—22no—
380C22—82no—
381C22—81no—
382C22—4882no—
383C22—4881no—
384C22#102×D6111yescharacteristic
385C22#1684010711yescharacteristic
386C22#1684013311yescharacteristic
387C4—81no—
388C4—1222no—
389C4—1222no—
390C4—4881no—
391C4#1683966211yescharacteristic
392C4#102×D6111yescharacteristic
393C4#1684010711yescharacteristic
394C4#1684013711yescharacteristic
395C2—21no—
396C2—21no—
397C2—161no—
398C2—9761no—
399C2#2144662311yescharacteristic
400C2#2144754512yes—
subgroupG/H
centerC2×C4D8×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#359C2
socleC22×C61C2×#102
prime radical#21456243C2
cosocleC22×C8×C61C23
hypercenterC2×#359D61
hypocenterC61#13209
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#359

Complements

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

NH ≅ G/Ntype
2. C2×#16839570C2semidirect
3. C2×#16839569C2semidirect
4. C2×#16839568C2semidirect
5. C2×#16839537C2semidirect
6. #21445989C2semidirect
7. C2×C61×#359C2semidirect
56. C2×#11758185D4semidirect
77. C2×C16×C61D4semidirect
199. C2×#359D61semidirect
231. C61#13209semidirect
270. C2×C16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214562432
C2×#168395702
C2×#168395692
C2×#168395682
C2×#168395372
#214459892
C2×C61×#3592
#1320961