Finite Group Databasebeta

← #46822869

Subgroups — #46822869

Order 34432.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771262C211yesmaximal, characteristic, complemented
2#31771255C211yesmaximal, characteristic, complemented
3#31770963C211yesmaximal, characteristic, complemented
4#31770941C211yesmaximal, characteristic, complemented
5#31770917C211yesmaximal, characteristic, complemented
6#31770898C211yesmaximal, characteristic, complemented
7#31770849C211yesmaximal, characteristic, complemented
8#31770838C211yesmaximal, characteristic, complemented
9#31770818C211yesmaximal, characteristic, complemented
10#31770742C211yesmaximal, characteristic, complemented
11#31770706C211yesmaximal, characteristic, complemented
12#31770679C211yesmaximal, characteristic, complemented
13#31770606C211yesmaximal, characteristic, complemented
14#103×D269C211yesmaximal, characteristic, complemented
15#449×C269C211yesmaximal, characteristic, complemented
16C2×#16864401—21no—
17C2×Q8×D269C2211yescharacteristic
18C2×#16864390C2211yescharacteristic
19C2×#16864386C2211yescharacteristic
20C2×C8×D269C2211yescharacteristic
21#22575452C2211yescharacteristic
22#22575440—21no—
23#22575440C2211yescharacteristic
24#22575435—21no—
25#22575434—21no—
26#22575432—21no—
27#22575432C2211yescharacteristic
28#22575430—21no—
29#22575424—21no—
30#22575424C2211yescharacteristic
31#22575424C2211yescharacteristic
32Q8×Dic269C2211yescharacteristic
33D4×Dic269C2211yescharacteristic
34C4×C22⋊D269—21no—
35C4×C22⋊D269C2211yescharacteristic
36C2×C4⋊Dic269C2211yescharacteristic
37C4⋊C4×D269C2211yescharacteristic
38#22575387C2211yescharacteristic
39#22575387C2211yescharacteristic
40#22575386C2211yescharacteristic
41#22575382C2211yescharacteristic
42#22575378C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575376C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic
49#22575374C2211yescharacteristic
50#22575365C2211yescharacteristic
51#22575363C2211yescharacteristic
52C2×#41×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54C2×D8×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59#16864401—44no—
60C2×Dic538—21no—
61C2×Dic538—21no—
62C2×Dic538C2311yescharacteristic
63Q8×D269—22no—
64Q8×D269—41no—
65C2×C22⋊D269—21no—
66C2×C22⋊D269—21no—
67C2×C22⋊D269D411yescharacteristic
68C22×Dic269—21no—
69C22×Dic269—21no—
70C2×C4×D269—21no—
71C2×C4×D269—21no—
72C2×C4×D269C2311yescharacteristic
73#16864390—41no—
74#16864386—41no—
75C2×#14524130C2311yescharacteristic
76C2×#14524130C2311yescharacteristic
77C8×D269—41no—
78C4⋊Dic269—21no—
79C4⋊Dic269—21no—
80C4⋊Dic269C2311yescharacteristic
81C4⋊Dic269C2311yescharacteristic
82C4⋊Dic269C2311yescharacteristic
83#16864381—21no—
84#16864381—21no—
85#16864381—21no—
86#16864381D411yescharacteristic
87C22⋊Dic269—21no—
88C22⋊Dic269—21no—
89C22⋊Dic269—21no—
90C22⋊Dic269—21no—
91C22⋊Dic269—21no—
92C22⋊Dic269—21no—
93C22⋊Dic269D411yescharacteristic
94#16864379D411yescharacteristic
95C4×Dic269—21no—
96C4×Dic269C2311yescharacteristic
97#41×C269—22no—
98#41×C269—41no—
99C2×Q8×C269C2311yescharacteristic
100C2×D4×C269—21no—
101C2×D4×C269—21no—
102C2×D4×C269C2311yescharacteristic
103C2×D4×C269C2311yescharacteristic
104C22×C4×C269—21no—
105C22×C4×C269C2311yescharacteristic
106SD16×C269—22no—
107D8×C269—41no—
108C2×C8×C269C2311yescharacteristic
109C2×C8×C269C2311yescharacteristic
110C4⋊C4×C269C2311yescharacteristic
111C22⋊C4×C269—21no—
112C22×D269C2×D411yescharacteristic
113Dic538—21no—
114Dic538—21no—
115Dic538—42no—
116Dic538—42no—
117C22⋊D269—22no—
118C22⋊D269—42no—
119C22⋊D269—42no—
120C2×Dic269—21no—
121C2×Dic269—21no—
122C2×Dic269—21no—
123C2×Dic269—21no—
124C2×Dic269—42no—
125C2×Dic269—42no—
126C2×Dic269C2×D411yescharacteristic
127C2×Dic269C2×D411yescharacteristic
128C2×Dic269C2×D411yescharacteristic
129C4×D269—21no—
130C4×D269—21no—
131C4×D269—42no—
132C4×D269—42no—
133#14524130—21no—
134#14524130—21no—
135C23×C269—21no—
136C23×C269—21no—
137C23×C269C2×D411yescharacteristic
138Q8×C269—21no—
139Q8×C269#4112yescomplemented
140D4×C269—21no—
141D4×C269—21no—
142D4×C269—21no—
143D4×C269—21no—
144D4×C269—42no—
145D4×C269—42no—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—21no—
149C2×C4×C269—21no—
150C2×C4×C269—42no—
151C2×C4×C269C2×D411yescharacteristic
152C2×C4×C269C2411yescharacteristic
153C8×C269—21no—
154C8×C269—21no—
155C22×C269—21no—
156C22×C269—21no—
157C22×C269—21no—
158C22×C269—41no—
159C22×C269—41no—
160C22×C269—41no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269—41no—
164C22×C269C22×D411yescharacteristic
165C2×D269—21no—
166C2×D269—21no—
167C2×D269AGL(1,ℤ/8)12yes—
168C4×C269—21no—
169C4×C269—22no—
170C4×C269—41no—
171C4×C269C2×#4111yescharacteristic
172C4×C269H(5,𝔽2)11yescharacteristic
173Dic269—21no—
174Dic269—21no—
175Dic269—21no—
176Dic269—41no—
177Dic269—41no—
178Dic269—41no—
179Dic269—41no—
180C2×C269—41no—
181C2×C269—81no—
182C2×C269—81no—
183C2×C269#54611yescharacteristic
184C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C269#57811yescharacteristic
186D269—22no—
187C269#330811yescharacteristic, complemented, Sylow
188#3308—2691nomaximal, Sylow
189#548—2691no—
190#547—2691no—
191#489—2691no—
192#484—2691no—
193#480—2691no—
194#477—2691no—
195#468—2691no—
196#466—2691no—
197#463—2691no—
198#449D26911yescharacteristic, complemented
199#448—2691no—
200#440—2691no—
201#435—2691no—
202#417—2691no—
203C2×#103—2691no—
204C2×#41—5381no—
205C2×#41C2×D26911yescharacteristic
206C22×Q8—2691no—
207C2×SD16—2691no—
208C2×SD16C2×D26911yescharacteristic
209C2×D8C2×D26911yescharacteristic
210C2×M16—2691no—
211C22×C8—2691no—
212C4⋊Q8—2691no—
213#124—2691no—
214#124—5381no—
215#123—5381no—
216#123—5381no—
217C22⋊Q8—2691no—
218C22⋊Q8—5381no—
219C22⋊Q8—5381no—
220#121—2691no—
221#121—2691no—
222#121—5381no—
223#121C2×D26911yescharacteristic
224C4×Q8—2691no—
225C4×D4—2691no—
226C4×D4—2691no—
227C4×D4—5381no—
228C2×C4⋊C4—2691no—
229C2×C4⋊C4—2691no—
230#107—2691no—
231#107—2691no—
232#107—2691no—
233C4⋊C8—2691no—
234#103—2691no—
235#103—2691no—
236#103—2691no—
237#103—2691no—
238#103—2691no—
239#103C2×D26911yescharacteristic
240#102—2691no—
241#102—2691no—
242#102—2691no—
243#102C2×D26911yescharacteristic
244C22⋊C8—2691no—
245C22⋊C8C2×D26911yescharacteristic
246#97—2691no—
247#41—22no—
248#41—41no—
249#41—10764no—
250C2×Q8—2691no—
251C2×Q8—5381no—
252C2×Q8—5381no—
253C2×Q8—5382no—
254C2×Q8—10761no—
255C2×Q8C22×D26911yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—2691no—
259C2×D4—5381no—
260C2×D4—5381no—
261C2×D4C22×D26911yescharacteristic
262C2×D4C22×D26911yescharacteristic
263C22×C4—21no—
264C22×C4—2691no—
265C22×C4—5381no—
266C22×C4—5381no—
267C22×C4—5381no—
268C22×C4—5381no—
269C22×C4C22×D26911yescharacteristic
270SD16—22no—
271SD16—10761no—
272D8—41no—
273M16—10761no—
274C2×C8—2691no—
275C2×C8—2691no—
276C2×C8—10761no—
277C2×C8C22×D26911yescharacteristic
278C2×C8C22×D26911yescharacteristic
279C4⋊C4—2691no—
280C4⋊C4—2691no—
281C4⋊C4—2691no—
282C4⋊C4—2691no—
283C4⋊C4—5381no—
284C4⋊C4—5381no—
285C4⋊C4—5381no—
286C4⋊C4—5381no—
287C4⋊C4—5381no—
288C4⋊C4C22×D26911yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—2691no—
291C22⋊C4—2691no—
292C22⋊C4—5381no—
293C22⋊C4—5381no—
294C22⋊C4—5381no—
295C22⋊C4—5381no—
296C22⋊C4—5381no—
297C22⋊C4—5381no—
298C42—2691no—
299C42—5381no—
300C23—21no—
301C23—21no—
302C23—2691no—
303C23D4×D26911yescharacteristic
304Q8—21no—
305Q8—5381no—
306Q8—5381no—
307Q8—10762no—
308Q8—10762no—
309Q8#1686440112yescomplemented
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—42no—
316D4—5382no—
317D4—10762no—
318D4—10762no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—2691no—
325C2×C4—2691no—
326C2×C4—2691no—
327C2×C4—5381no—
328C2×C4—5381no—
329C2×C4—5381no—
330C2×C4—5381no—
331C2×C4—5381no—
332C2×C4—5381no—
333C2×C4—10762no—
334C2×C4—10762no—
335C2×C4—10762no—
336C2×C4—10762no—
337C2×C4D4×D26911yescharacteristic
338C2×C4C23×D26911yescharacteristic
339C8—21no—
340C8—21no—
341C8—5381no—
342C8—5381no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—2692no—
353C22—5381no—
354C22—5381no—
355C22C2×D4×D26911yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—5381no—
360C4—5381no—
361C4—5381no—
362C4—10761no—
363C4—10761no—
364C4—10761no—
365C4—10761no—
366C4C2×#1686440111yescharacteristic
367C4#2257549811yescharacteristic
368C2—41no—
369C2—81no—
370C2—81no—
371C2—5382no—
372C2#3177124211yescharacteristic
373C2AGL(1,ℤ/8)×D26911yescharacteristic
374C2#3177141911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#449×C269C2
socleC22×C269C22×D4
hypercenter#449D269
hypocenterC269#3308
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31771262C2semidirect
2. #31771255C2semidirect
3. #31770963C2semidirect
4. #31770941C2semidirect
5. #31770917C2semidirect
6. #31770898C2semidirect
7. #31770849C2semidirect
8. #31770838C2semidirect
9. #31770818C2semidirect
10. #31770742C2semidirect
11. #31770706C2semidirect
12. #31770679C2semidirect
13. #31770606C2semidirect
14. #103×D269C2semidirect
15. #449×C269C2semidirect
139. Q8×C269#41semidirect
187. C269#3308semidirect
198. #449D269semidirect
309. Q8#16864401semidirect

Maximal subgroups

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

subgroupindex
#317712622
#317712552
#317709632
#317709412
#317709172
#317708982
#317708492
#317708382
#317708182
#317707422
#317707062
#317706792
#317706062
#103×D2692
#449×C2692
#3308269