Finite Group Databasebeta

← #46821005

Subgroups — #46821005

Order 34432.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771152C211yesmaximal, characteristic
2C4×Q8×D269C211yesmaximal, characteristic, complemented
3#31770920C211yesmaximal, characteristic, complemented
4#31770917C211yesmaximal, characteristic, complemented
5#31770899C211yesmaximal, characteristic, complemented
6#31770898C211yesmaximal, characteristic, complemented
7#31770804C211yesmaximal, characteristic, complemented
8#31770803C211yesmaximal, characteristic, complemented
9#31770745C211yesmaximal, characteristic, complemented
10#31770742C211yesmaximal, characteristic, complemented
11#31770707C211yesmaximal, characteristic, complemented
12#31770705C211yesmaximal, characteristic, complemented
13#31770703C211yesmaximal, characteristic, complemented
14#31770660C211yesmaximal, characteristic, complemented
15#440×C269C211yesmaximal, characteristic, complemented
16C2×#16864402—21no—
17C2×Q8×D269C2211yescharacteristic
18C2×#16864391C2211yescharacteristic
19C2×#16864390C2211yescharacteristic
20C2×#16864389C2211yescharacteristic
21#22575452C2211yescharacteristic
22#22575449C2211yescharacteristic
23#22575439—21no—
24#22575428C2211yescharacteristic
25#22575426—21no—
26#22575424C2211yescharacteristic
27Q8×Dic269C2211yescharacteristic
28C4×Dic538—21no—
29C4×Dic538C2211yescharacteristic
30#22575418—21no—
31#22575418C2211yescharacteristic
32C4×D1076C2211yescharacteristic
33C4×C22⋊D269—21no—
34C4⋊C4×D269—21no—
35C4⋊C4×D269C2211yescharacteristic
36C42×D269—21no—
37C42×D269C2211yescharacteristic
38#22575387C2211yescharacteristic
39#22575386C2211yescharacteristic
40#22575386C2211yescharacteristic
41#22575382C2211yescharacteristic
42#22575382C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575376C2211yescharacteristic
46#22575375C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575373C2211yescharacteristic
49#22575365C2211yescharacteristic
50#22575365C2211yescharacteristic
51#22575364C2211yescharacteristic
52C2×SD16×C269C2211yescharacteristic
53C4×Q8×C269C2211yescharacteristic
54C4×D4×C269C2211yescharacteristic
55#107×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58#97×C269C2211yescharacteristic
59#16864402—44no—
60C2×Dic538—21no—
61C2×Dic538—21no—
62C2×Dic538C2311yescharacteristic
63Q8×D269—22no—
64Q8×D269—22no—
65C2×D1076—21no—
66C2×D1076C2311yescharacteristic
67C2×C22⋊D269—21no—
68C2×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C4×D269—21no—
71C2×C4×D269—21no—
72C2×C4×D269—21no—
73C2×C4×D269D411yescharacteristic
74C2×C4×D269D411yescharacteristic
75C2×C4×D269C2311yescharacteristic
76#16864391—22no—
77#16864390—22no—
78#16864389—22no—
79C2×#14524130C2311yescharacteristic
80C2×#14524130C2311yescharacteristic
81C4⋊Dic269—21no—
82C4⋊Dic269C2311yescharacteristic
83C4⋊Dic269C2311yescharacteristic
84#16864381—21no—
85#16864381—21no—
86#16864381—21no—
87#16864381—21no—
88C22⋊Dic269—21no—
89#16864379—21no—
90#16864379—21no—
91C4×Dic269—21no—
92C4×Dic269—21no—
93C4×Dic269—21no—
94C4×Dic269D411yescharacteristic
95C4×Dic269D411yescharacteristic
96C4×Dic269C2311yescharacteristic
97C2×Q8×C269C2311yescharacteristic
98C2×D4×C269C2311yescharacteristic
99C22×C4×C269—21no—
100SD16×C269—22no—
101C2×C8×C269C2311yescharacteristic
102C2×C8×C269C2311yescharacteristic
103C4⋊C4×C269—21no—
104C4⋊C4×C269C2311yescharacteristic
105C4⋊C4×C269C2311yescharacteristic
106C22⋊C4×C269—21no—
107C42×C269—21no—
108C42×C269C2311yescharacteristic
109C22×D269—21no—
110C22×D269C2×D411yescharacteristic
111Dic538—24no—
112Dic538—21no—
113Dic538—42no—
114Dic538#4112yescomplemented
115D1076—21no—
116D1076—21no—
117D1076—42no—
118C22⋊D269—42no—
119C22⋊D269—42no—
120C22⋊D269—42no—
121C2×Dic269—21no—
122C2×Dic269—21no—
123C2×Dic269—21no—
124C2×Dic269C2×D411yescharacteristic
125C2×Dic269C2×D411yescharacteristic
126C2×Dic269C2×D411yescharacteristic
127C4×D269—24no—
128C4×D269—22no—
129C4×D269—21no—
130C4×D269—21no—
131C4×D269—22no—
132C4×D269—42no—
133C4×D269—42no—
134#14524130—21no—
135#14524130—21no—
136C23×C269—21no—
137Q8×C269—21no—
138Q8×C269#4112yescomplemented
139D4×C269—21no—
140D4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—42no—
145C2×C4×C269C2×D411yescharacteristic
146C2×C4×C269C2×D411yescharacteristic
147C2×C4×C269C2411yescharacteristic
148C8×C269—21no—
149C8×C269—21no—
150C22×C269—41no—
151C22×C269—41no—
152C22×C269—41no—
153C22×C269C22×D411yescharacteristic
154C2×D269—21no—
155C2×D269—21no—
156C2×D269—41no—
157C2×D269—41no—
158C2×D269—41no—
159C2×D269AGL(1,ℤ/8)12yes—
160C4×C269—22no—
161C4×C269—21no—
162C4×C269—21no—
163C4×C269—41no—
164C4×C269—41no—
165C4×C269C2×#4111yescharacteristic
166C4×C269C2×#4111yescharacteristic
167Dic269—22no—
168Dic269—21no—
169Dic269—21no—
170Dic269—41no—
171Dic269—41no—
172Dic269#13712yes—
173C2×C269—81no—
174C2×C269#52511yescharacteristic
175C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C269C2×#13711yescharacteristic
177D269—22no—
178D269—81no—
179C269#311211yescharacteristic, complemented, Sylow
180#3112—2691nomaximal, Sylow
181#532—2691no—
182C2×C4×Q8—2691no—
183#480—2691no—
184#480—2691no—
185#477—2691no—
186#477—2691no—
187#460—2691no—
188#460—2691no—
189#448—2691no—
190#448—2691no—
191#440—2691no—
192#440—2691no—
193#440—2691no—
194#440D26911yescharacteristic, complemented
195#432—2691no—
196C2×#41—5381no—
197C22×Q8—2691no—
198C2×SD16—2691no—
199C2×SD16—2691no—
200C2×SD16—2691no—
201C2×SD16C2×D26911yescharacteristic
202C4⋊Q8—2691no—
203#127—2691no—
204#124—5381no—
205#121—2691no—
206#121—2691no—
207#121—5381no—
208C4×Q8—2691no—
209C4×Q8—2691no—
210C4×Q8—2691no—
211C4×Q8—5381no—
212C4×Q8—5381no—
213C4×Q8C2×D26911yescharacteristic
214C4×D4—2691no—
215C4×D4—5381no—
216C4×D4C2×D26911yescharacteristic
217C2×C4⋊C4—2691no—
218C2×C4⋊C4—5381no—
219C2×C42—2691no—
220C2×C42—5381no—
221#107—2691no—
222#107—2691no—
223#107—2691no—
224#107C2×D26911yescharacteristic
225C4⋊C8—2691no—
226C4⋊C8—2691no—
227#103—2691no—
228#103—2691no—
229#103—2691no—
230#103C2×D26911yescharacteristic
231#102—2691no—
232#102—2691no—
233#102—2691no—
234#102C2×D26911yescharacteristic
235C22⋊C8—2691no—
236C22⋊C8—2691no—
237#97—2691no—
238#97C2×D26911yescharacteristic
239#41—10764no—
240C2×Q8—2691no—
241C2×Q8—5381no—
242C2×Q8—5381no—
243C2×Q8—5382no—
244C2×Q8—5382no—
245C2×Q8C22×D26911yescharacteristic
246C2×D4—2691no—
247C2×D4—5381no—
248C2×D4—5381no—
249C2×D4—5381no—
250C2×D4—5381no—
251C2×D4C22×D26911yescharacteristic
252C22×C4—21no—
253C22×C4—2691no—
254C22×C4—2691no—
255C22×C4—2691no—
256C22×C4—5381no—
257C22×C4—5381no—
258C22×C4—5381no—
259SD16—22no—
260SD16—5382no—
261SD16—5382no—
262SD16—5382no—
263C2×C8—2691no—
264C2×C8—2691no—
265C2×C8C22×D26911yescharacteristic
266C2×C8C22×D26911yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—2691no—
269C4⋊C4—2691no—
270C4⋊C4—5381no—
271C4⋊C4—5381no—
272C4⋊C4—5381no—
273C4⋊C4—5381no—
274C4⋊C4—5381no—
275C4⋊C4C22×D26911yescharacteristic
276C4⋊C4C22×D26911yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—5381no—
279C22⋊C4—5381no—
280C22⋊C4—5381no—
281C42—21no—
282C42—2691no—
283C42—2691no—
284C42—2691no—
285C42—5381no—
286C42—5381no—
287C42—5381no—
288C42C22×D26911yescharacteristic
289C23—21no—
290C23—2691no—
291C23—5381no—
292Q8—21no—
293Q8—2692no—
294Q8—5384no—
295Q8—5381no—
296Q8—10762no—
297Q8#1686440212yescomplemented
298D4—21no—
299D4—21no—
300D4—5381no—
301D4—5381no—
302D4—10762no—
303D4—10762no—
304D4—10762no—
305D4—10762no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—2691no—
311C2×C4—2691no—
312C2×C4—2691no—
313C2×C4—5382no—
314C2×C4—5382no—
315C2×C4—5384no—
316C2×C4—5381no—
317C2×C4—5381no—
318C2×C4—5381no—
319C2×C4—5381no—
320C2×C4—5381no—
321C2×C4—10762no—
322C2×C4—10762no—
323C2×C4D4×D26911yescharacteristic
324C2×C4D4×D26911yescharacteristic
325C2×C4C23×D26911yescharacteristic
326C8—21no—
327C8—21no—
328C8—5381no—
329C8—5381no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—2692no—
334C22—5381no—
335C22—5381no—
336C22—10761no—
337C22—10761no—
338C22—10761no—
339C22C2×D4×D26911yescharacteristic
340C4—21no—
341C4—22no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—2692no—
346C4—5381no—
347C4—5382no—
348C4—5381no—
349C4—10761no—
350C4—10761no—
351C4#41×D26911yescharacteristic
352C4C2×#1686440211yescharacteristic
353C2—81no—
354C2—5382no—
355C2—21521no—
356C2#3177111411yescharacteristic
357C2AGL(1,ℤ/8)×D26911yescharacteristic
358C2#137×D26911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#440×C269C2
socleC22×C269C22×D4
prime radical#31771152C2
hypercenter#440D269
hypocenterC269#3112
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D269C2semidirect
3. #31770920C2semidirect
4. #31770917C2semidirect
5. #31770899C2semidirect
6. #31770898C2semidirect
7. #31770804C2semidirect
8. #31770803C2semidirect
9. #31770745C2semidirect
10. #31770742C2semidirect
11. #31770707C2semidirect
12. #31770705C2semidirect
13. #31770703C2semidirect
14. #31770660C2semidirect
15. #440×C269C2semidirect
114. Dic538#41semidirect
138. Q8×C269#41semidirect
179. C269#3112semidirect
194. #440D269semidirect
297. Q8#16864402semidirect

Maximal subgroups

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

subgroupindex
#317711522
C4×Q8×D2692
#317709202
#317709172
#317708992
#317708982
#317708042
#317708032
#317707452
#317707422
#317707072
#317707052
#317707032
#317706602
#440×C2692
#3112269