Finite Group Databasebeta

← #46822862

Subgroups — #46822862

Order 34432.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771303C211yesmaximal, characteristic, complemented
2#31771254C211yesmaximal, characteristic, complemented
3#31770972C211yesmaximal, characteristic, complemented
4#31770945C211yesmaximal, characteristic, complemented
5#31770935C211yesmaximal, characteristic, complemented
6#31770903C211yesmaximal, characteristic, complemented
7#31770851C211yesmaximal, characteristic, complemented
8#31770837C211yesmaximal, characteristic, complemented
9#31770809C211yesmaximal, characteristic, complemented
10#31770751C211yesmaximal, characteristic, complemented
11#31770750C211yesmaximal, characteristic, complemented
12#31770714C211yesmaximal, characteristic, complemented
13#31770674C211yesmaximal, characteristic, complemented
14#31770598C211yesmaximal, characteristic, complemented
15#417×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17C2×#16864391C2211yescharacteristic
18C2×#16864389C2211yescharacteristic
19C2×#16864387C2211yescharacteristic
20C2×C8×D269C2211yescharacteristic
21#22575451C2211yescharacteristic
22#22575448—21no—
23#22575444C2211yescharacteristic
24#22575438—21no—
25#22575436—21no—
26#22575434—21no—
27#22575432C2211yescharacteristic
28#22575429C2211yescharacteristic
29#22575428C2211yescharacteristic
30#22575427—21no—
31#22575427C2211yescharacteristic
32#22575425—21no—
33D4×Dic269C2211yescharacteristic
34#22575415C2211yescharacteristic
35C4×C22⋊D269—21no—
36C4×C22⋊D269C2211yescharacteristic
37#22575411C2211yescharacteristic
38C2×#16864381—21no—
39#22575387C2211yescharacteristic
40#22575384C2211yescharacteristic
41#22575384C2211yescharacteristic
42#22575382C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575376C2211yescharacteristic
46#22575376C2211yescharacteristic
47#22575375C2211yescharacteristic
48#22575374C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic
50#22575365C2211yescharacteristic
51#22575363C2211yescharacteristic
52C2×#41×C269C2211yescharacteristic
53C2×M16×C269C2211yescharacteristic
54C2×C4⋊C4×C269C2211yescharacteristic
55#103×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59#16864401—22no—
60#16864401—41no—
61C2×Dic538C2311yescharacteristic
62C2×D1076—21no—
63C2×D1076C2311yescharacteristic
64C2×C22⋊D269—21no—
65C2×C22⋊D269—21no—
66C2×C22⋊D269D411yescharacteristic
67C22×Dic269—21no—
68C22×Dic269—21no—
69C2×C4×D269—21no—
70C2×C4×D269C2311yescharacteristic
71#16864391—41no—
72#16864389—41no—
73#16864387—22no—
74C2×#14524130C2311yescharacteristic
75C2×#14524130C2311yescharacteristic
76C8×D269—41no—
77C4⋊Dic269—21no—
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85#16864381D411yescharacteristic
86C22⋊Dic269—21no—
87C22⋊Dic269—21no—
88C22⋊Dic269D411yescharacteristic
89#16864379—21no—
90#16864379—21no—
91#16864379—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379D411yescharacteristic
95C4×Dic269—21no—
96C4×Dic269C2311yescharacteristic
97#41×C269—22no—
98#41×C269—41no—
99C2×Q8×C269C2311yescharacteristic
100C2×D4×C269—21no—
101C2×D4×C269C2311yescharacteristic
102C22×C4×C269—21no—
103C22×C4×C269—21no—
104C22×C4×C269C2311yescharacteristic
105M16×C269—41no—
106C2×C8×C269C2311yescharacteristic
107C2×C8×C269C2311yescharacteristic
108C4⋊C4×C269—21no—
109C4⋊C4×C269C2311yescharacteristic
110C4⋊C4×C269C2311yescharacteristic
111C22×D269—21no—
112C22×D269C2×D411yescharacteristic
113Dic538—21no—
114Dic538—21no—
115D1076—21no—
116D1076—21no—
117D1076—81no—
118C22⋊D269—41no—
119C22⋊D269—42no—
120C22⋊D269—42no—
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—81no—
132#14524130—21no—
133#14524130—21no—
134C23×C269—21no—
135C23×C269C2×D411yescharacteristic
136Q8×C269—21no—
137Q8×C269—21no—
138D4×C269—21no—
139D4×C269—42no—
140D4×C269#4112yes—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—42no—
147C2×C4×C269—81no—
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2411yescharacteristic
150C8×C269—21no—
151C8×C269—21no—
152C22×C269—21no—
153C22×C269—22no—
154C22×C269—21no—
155C22×C269—21no—
156C22×C269—41no—
157C22×C269—41no—
158C22×C269C22×D411yescharacteristic
159C2×D269—21no—
160C2×D269—21no—
161C2×D269—21no—
162C2×D269—41no—
163C2×D269—41no—
164C2×D269—41no—
165C4×C269—21no—
166C4×C269—41no—
167C4×C269—41no—
168C4×C269—41no—
169C4×C269C2×#4111yescharacteristic
170C4×C269H(5,𝔽2)11yescharacteristic
171Dic269—21no—
172Dic269—21no—
173Dic269—21no—
174Dic269—41no—
175Dic269—41no—
176Dic269—41no—
177C2×C269—41no—
178C2×C269—42no—
179C2×C269#54611yescharacteristic
180C2×C269#57511yescharacteristic
181C2×C269#57711yescharacteristic
182D269—41no—
183D269—81no—
184C269#330711yescharacteristic, complemented, Sylow
185#3307—2691nomaximal, Sylow
186#553—2691no—
187#547—2691no—
188#491—2691no—
189#485—2691no—
190#483—2691no—
191#478—2691no—
192#468—2691no—
193#465—2691no—
194#461—2691no—
195#449—2691no—
196#449—2691no—
197#442—2691no—
198#435—2691no—
199#417D26911yescharacteristic, complemented
200#416—2691no—
201C2×#41—2691no—
202C2×#41C2×D26911yescharacteristic
203C2×SD16—2691no—
204C2×SD16—2691no—
205C2×D8—2691no—
206C2×M16C2×D26911yescharacteristic
207C22×C8—2691no—
208C4⋊Q8—2691no—
209#126—5381no—
210#125—2691no—
211#123—5381no—
212#123—5381no—
213#123—5381no—
214C22⋊Q8—2691no—
215C22⋊Q8—2691no—
216#121—2691no—
217#121—2691no—
218#121—5381no—
219#121—5381no—
220C4×D4—2691no—
221C4×D4—2691no—
222C4×D4—2691no—
223C4×D4—5381no—
224#117—2691no—
225C2×C4⋊C4—5381no—
226C2×C4⋊C4C2×D26911yescharacteristic
227#107—2691no—
228#106—2691no—
229#106—2691no—
230C4⋊C8—2691no—
231#103—2691no—
232#103—2691no—
233#103—2691no—
234#103—2691no—
235#103C2×D26911yescharacteristic
236#103C2×D26911yescharacteristic
237#102—2691no—
238#102—2691no—
239#102C2×D26911yescharacteristic
240#102C2×D26911yescharacteristic
241C22⋊C8—2691no—
242C22⋊C8—2691no—
243#97—2691no—
244#41—22no—
245#41—41no—
246#41—5382no—
247#41—10761no—
248C2×Q8—2691no—
249C2×Q8C22×D26911yescharacteristic
250C2×D4—21no—
251C2×D4—2691no—
252C2×D4—2691no—
253C2×D4—5381no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4C22×D26911yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—2691no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4—5381no—
263C22×C4C22×D26911yescharacteristic
264SD16—10761no—
265SD16—10761no—
266D8—5382no—
267M16—41no—
268C2×C8—2691no—
269C2×C8—2691no—
270C2×C8—10761no—
271C2×C8C22×D26911yescharacteristic
272C2×C8C22×D26911yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—2691no—
275C4⋊C4—2691no—
276C4⋊C4—2691no—
277C4⋊C4—5381no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4—5381no—
282C4⋊C4—5381no—
283C4⋊C4C22×D26911yescharacteristic
284C4⋊C4C22×D26911yescharacteristic
285C22⋊C4—2691no—
286C22⋊C4—2691no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4—5381no—
293C22⋊C4—5381no—
294C42—2691no—
295C42—5381no—
296C23—21no—
297C23—2691no—
298C23—5381no—
299C23D4×D26911yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—5381no—
303Q8—5381no—
304D4—21no—
305D4—42no—
306D4—5381no—
307D4—5381no—
308D4—10761no—
309D4—10762no—
310D4—10762no—
311D4—21521no—
312D4#1686440212yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—2691no—
321C2×C4—2691no—
322C2×C4—2691no—
323C2×C4—5381no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—5381no—
328C2×C4—10762no—
329C2×C4—10762no—
330C2×C4—21521no—
331C2×C4D4×D26911yescharacteristic
332C2×C4C23×D26911yescharacteristic
333C8—21no—
334C8—21no—
335C8—5381no—
336C8—5381no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—5381no—
344C22—5381no—
345C22—5381no—
346C22—10761no—
347C22—10761no—
348C22—10761no—
349C22C2×D4×D26911yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—5381no—
355C4—5381no—
356C4—5381no—
357C4—10761no—
358C4—10761no—
359C4—10761no—
360C4C2×#1686440211yescharacteristic
361C4#2257549811yescharacteristic
362C2—41no—
363C2—42no—
364C2—10761no—
365C2—21521no—
366C2#3177124511yescharacteristic
367C2#3177140411yescharacteristic
368C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#417×C269C2
socleC22×C269C22×D4
hypercenter#417D269
hypocenterC269#3307
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #31771303C2semidirect
2. #31771254C2semidirect
3. #31770972C2semidirect
4. #31770945C2semidirect
5. #31770935C2semidirect
6. #31770903C2semidirect
7. #31770851C2semidirect
8. #31770837C2semidirect
9. #31770809C2semidirect
10. #31770751C2semidirect
11. #31770750C2semidirect
12. #31770714C2semidirect
13. #31770674C2semidirect
14. #31770598C2semidirect
15. #417×C269C2semidirect
184. C269#3307semidirect
199. #417D269semidirect

Maximal subgroups

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

subgroupindex
#317713032
#317712542
#317709722
#317709452
#317709352
#317709032
#317708512
#317708372
#317708092
#317707512
#317707502
#317707142
#317706742
#317705982
#417×C2692
#3307269