Finite Group Databasebeta

← #29834908

Subgroups — #29834908

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455424C211yesmaximal, characteristic
2C2×#16839809C211yesmaximal, characteristic, complemented
3#21448037C214yesmaximal, complemented
4#21448032C214yesmaximal, complemented
5#21447593C211yesmaximal, characteristic, complemented
6#21447592C211yesmaximal, characteristic, complemented
7#21447568C211yesmaximal, characteristic, complemented
8C2×#16839543C211yesmaximal, characteristic, complemented
9C61×#2148C211yesmaximal, characteristic, complemented
10C22×#11758189C2211yescharacteristic
11C2×M16×D61—21no—
12#16840269—21no—
13C2×#14519717C2211yescharacteristic
14#16840017C2212yes—
15#16840012C2212yes—
16#16840005—22no—
17#16839809—22no—
18#16839809C2214yes—
19C2×#14519713C2211yescharacteristic
20C2×#14519712—21no—
21#16839800C2211yescharacteristic
22#16839758—21no—
23#16839586C2214yescomplemented
24#16839585C2214yescomplemented
25#16839578C2212yescomplemented
26#16839575C2212yescomplemented
27#16839543C2214yes—
28C2×#135×C61C2211yescharacteristic
29C2×M32×C61C2211yescharacteristic
30C2×#108×C61C2211yescharacteristic
31C61×#362C2212yescomplemented
32C61×#361C2212yescomplemented
33C22×C4×D61D411yescharacteristic
34C2×#11758194—21no—
35C2×#11758189—21no—
36C2×#11758189—21no—
37C2×#11758189—21no—
38C2×#11758189—21no—
39C2×#11758189—41no—
40C2×#11758189D412yes—
41C2×#11758189C2312yes—
42M16×D61—24no—
43M16×D61—42no—
44C22×#92418D411yescharacteristic
45C2×C8×D61—41no—
46#14519758—41no—
47#14519753—22no—
48#14519748—22no—
49#14519745—41no—
50C4×C22⋊D61—42no—
51#14519736—21no—
52#14519717—21no—
53#14519717C2312yes—
54C2×#11758185—21no—
55#14519713—21no—
56#14519713C2312yes—
57#14519712—24no—
58#14519711—21no—
59#14519709C2312yes—
60#14519701—22no—
61#14519698—22no—
62C2×#41×C61—21no—
63#135×C61—21no—
64#135×C61—21no—
65#135×C61—21no—
66#135×C61—21no—
67C2×Q16×C61C2311yescharacteristic
68C2×SD16×C61—21no—
69C2×D8×C61C2311yescharacteristic
70C2×M16×C61—21no—
71C22×C8×C61C2311yescharacteristic
72M32×C61—21no—
73M32×C61C2312yes—
74C2×C16×C61—21no—
75#108×C61—21no—
76#108×C61C2312yes—
77C23×D61SD1611yescharacteristic
78#41×C61—41no—
79#41×C61—41no—
80#41×C61—41no—
81#41×C61—41no—
82C2×Q8×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61C2×D411yescharacteristic
87C2×C22⋊D61—42no—
88C22×Dic61SD1611yescharacteristic
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—21no—
93C2×C4×D61SD1612yes—
94C2×C4×D61C2×D412yes—
95Q16×C61—21no—
96Q16×C61—21no—
97SD16×C61—41no—
98SD16×C61—41no—
99D8×C61—21no—
100D8×C61—21no—
101M16×C61—22no—
102M16×C61—41no—
103C2×C8×C61—41no—
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61#4111yescharacteristic
106C2×C8×C61#4111yescharacteristic
107C2×C8×C61#4111yescharacteristic
108C2×C8×C61#4111yescharacteristic
109C2×C8×C61C2411yescharacteristic
110C22⋊Dic61—41no—
111C22⋊Dic61—42no—
112#11758198—42no—
113C4⋊Dic61—22no—
114#11758196—42no—
115C4×Dic61—41no—
116#11758194—22no—
117#11758194—41no—
118C2×#92418—21no—
119C2×#92418—21no—
120C2×#92418—41no—
121C2×#92418C2×D412yes—
122#11758189—22no—
123#11758189—22no—
124#11758189—22no—
125#11758189—22no—
126#11758189—44no—
127C8×D61—44no—
128C16×C61—22no—
129#11758185—22no—
130C23×C61—41no—
131C23×C61C2×SD1611yescharacteristic
132C22×D61—21no—
133C22×D61—21no—
134C22×D61—81no—
135C22×D61C2×SD1612yes—
136Q8×C61—41no—
137Q8×C61—41no—
138D4×C61—41no—
139D4×C61—41no—
140D4×C61—81no—
141D4×C61—81no—
142C2×C4×C61—41no—
143C2×C4×C61—81no—
144C2×C4×C61—81no—
145C2×C4×C61#12311yescharacteristic
146C2×C4×C61#12311yescharacteristic
147C2×C4×C61C2×SD1611yescharacteristic
148C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C61C22×D411yescharacteristic
151C22⋊D61—84no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—42no—
155C2×Dic61C2×SD1612yes—
156C4×D61—22no—
157C4×D61—22no—
158C4×D61—22no—
159C4×D61—22no—
160C8×C61—42no—
161C8×C61#12311yescharacteristic
162C8×C61#12311yescharacteristic
163C8×C61C2×#4111yescharacteristic
164C8×C61C2×#4111yescharacteristic
165#92418—22no—
166#92418—42no—
167#33656—611nomaximal, Sylow
168C22×C61—41no—
169C22×C61—82no—
170C22×C61—81no—
171C22×C61#48111yescharacteristic
172C22×C61#48111yescharacteristic
173C22×C61C22×SD1611yescharacteristic
174C2×D61—22no—
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—81no—
178C4×C61—81no—
179C4×C61#48111yescharacteristic
180C4×C61#48111yescharacteristic
181C4×C61C2×#12311yescharacteristic
182C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
183Dic61—22no—
184Dic61—82no—
185#3054—611no—
186C2×#430—611no—
187#2239—614no—
188#2238—614no—
189#2155—611no—
190#2155—611no—
191#2148—611no—
192#2148D6111yescharacteristic, complemented
193C2×#349—611no—
194C2×C61—21no—
195C2×C61—21no—
196C2×C61—21no—
197C2×C61—161no—
198C2×C61C2×#48111yescharacteristic
199D61—42no—
200C2×#135C2×D6111yescharacteristic
201C22×M16—611no—
202C22×M16—1221no—
203#525—1221no—
204C2×M32—611no—
205C2×M32C2×D6111yescharacteristic
206#470—612no—
207#469—612no—
208#468—1222no—
209#430—614no—
210#430—1222no—
211C2×#108—611no—
212C2×#108—1221no—
213C2×#108C2×D6111yescharacteristic
214#427—611no—
215#416—1221no—
216#365—614no—
217#365—614no—
218#362—612no—
219#362C2×D6112yescomplemented
220#361—612no—
221#361C2×D6112yescomplemented
222#349—614no—
223C61#3365611yescharacteristic, complemented, Sylow
224C2×#41—21no—
225C23×C4—611no—
226#135—21no—
227#135—21no—
228#135—21no—
229#135—21no—
230C2×Q16C22×D6111yescharacteristic
231C2×SD16—21no—
232C2×D8C22×D6111yescharacteristic
233C2×M16—21no—
234C2×M16—612no—
235C2×M16—612no—
236C2×M16—1221no—
237C2×M16—1221no—
238C2×M16—1221no—
239C2×M16—1221no—
240C2×M16—1224no—
241C2×M16—1221no—
242C2×M16—2441no—
243C2×M16—2442no—
244C22×C8—611no—
245C22×C8—2441no—
246C22×C8C22×D6111yescharacteristic
247#123—2441no—
248C22⋊Q8—1222no—
249#121—1222no—
250#120—2441no—
251C4×D4—2442no—
252#117—1221no—
253M32—21no—
254M32—612no—
255M32—1221no—
256M32C22×D6112yes—
257C2×C16—21no—
258C2×C16—1221no—
259#108—21no—
260#108—612no—
261#108—1224no—
262#108—1221no—
263#108C22×D6112yes—
264#107—1221no—
265#106—612no—
266#103—1222no—
267#102—1222no—
268C24—611no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—2442no—
277C22×C4—41no—
278C22×C4—611no—
279C22×C4—612no—
280C22×C4—612no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4D4×D6111yescharacteristic
286Q16—21no—
287Q16—21no—
288SD16—41no—
289SD16—41no—
290D8—21no—
291D8—21no—
292M16—22no—
293M16—41no—
294M16—1222no—
295M16—1222no—
296M16—1222no—
297M16—1222no—
298M16—1222no—
299M16—2444no—
300M16—2441no—
301C2×C8—41no—
302C2×C8—612no—
303C2×C8—1221no—
304C2×C8—1221no—
305C2×C8—2444no—
306C2×C8—2441no—
307C2×C8#1175822211yescharacteristic
308C2×C8#1175822211yescharacteristic
309C2×C8D4×D6111yescharacteristic
310C2×C8#1175822411yescharacteristic
311C2×C8#1175822411yescharacteristic
312C2×C8C23×D6111yescharacteristic
313C4⋊C4—1222no—
314C4⋊C4—2442no—
315C22⋊C4—2442no—
316C22⋊C4—2442no—
317C22⋊C4—2441no—
318C42—2441no—
319C16—22no—
320C16—1222no—
321C23—41no—
322C23—612no—
323C23—1221no—
324C23—1221no—
325C23—4881no—
326C23SD16×D6111yescharacteristic
327Q8—41no—
328Q8—41no—
329D4—41no—
330D4—41no—
331D4—81no—
332D4—81no—
333D4—4884no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—612no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1222no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—2442no—
345C2×C4#1451975911yescharacteristic
346C2×C4#1451975911yescharacteristic
347C2×C4SD16×D6111yescharacteristic
348C2×C4#1451980111yescharacteristic
349C2×C4#1451980111yescharacteristic
350C2×C4C2×D4×D6111yescharacteristic
351C8—42no—
352C8—1222no—
353C8—2442no—
354C8#1451975911yescharacteristic
355C8#1451975911yescharacteristic
356C8C2×#1175822411yescharacteristic
357C8C2×#1175822211yescharacteristic
358C22—41no—
359C22—82no—
360C22—81no—
361C22—1222no—
362C22—4881no—
363C22—2442no—
364C22—2442no—
365C22#1684007911yescharacteristic
366C22#1684007911yescharacteristic
367C22C2×SD16×D6111yescharacteristic
368C4—81no—
369C4—1222no—
370C4—4882no—
371C4#1684007911yescharacteristic
372C4#1684007911yescharacteristic
373C4C2×#1451975911yescharacteristic
374C4C2×#1451980111yescharacteristic
375C2—21no—
376C2—21no—
377C2—21no—
378C2—161no—
379C2—2442no—
380C2C2×#1684007911yescharacteristic
subgroupG/H
centerC4#16840079
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2148C2
socleC2×C61C2×#481
prime radical#21455424C2
hypercenter#2148D61
hypocenterC61#33656
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. C2×#16839809C2semidirect
3. #21448037C2semidirect
4. #21448032C2semidirect
5. #21447593C2semidirect
6. #21447592C2semidirect
7. #21447568C2semidirect
8. C2×#16839543C2semidirect
9. C61×#2148C2semidirect
23. #16839586C22semidirect
24. #16839585C22semidirect
25. #16839578C22semidirect
26. #16839575C22semidirect
31. C61×#362C22semidirect
32. C61×#361C22semidirect
192. #2148D61semidirect
219. #362C2×D61semidirect
221. #361C2×D61semidirect
223. C61#33656semidirect

Maximal subgroups

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

subgroupindex
#214554242
C2×#168398092
#214480372
#214480322
#214475932
#214475922
#214475682
C2×#168395432
C61×#21482
#3365661