Finite Group Databasebeta

← #46822622

Subgroups — #46822622

Order 34432.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575473C211yesmaximal, characteristic, complemented
2C22×#16864392C211yesmaximal, characteristic, complemented
3#31771264C211yesmaximal, characteristic, complemented
4#31770987C211yesmaximal, characteristic, complemented
5#31770844C211yesmaximal, characteristic, complemented
6#31770839C211yesmaximal, characteristic, complemented
7#31770827C211yesmaximal, characteristic, complemented
8#31770824C211yesmaximal, characteristic, complemented
9#31770767C211yesmaximal, characteristic, complemented
10#31770765C211yesmaximal, characteristic, complemented
11#31770744C211yesmaximal, characteristic, complemented
12#31770743C211yesmaximal, characteristic, complemented
13C4×#16864392C211yesmaximal, characteristic, complemented
14D4×#14524130C211yesmaximal, characteristic, complemented
15#548×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×Dic538C2211yescharacteristic
18#22575473—24no—
19C2×#16864392—24no—
20C2×#16864392C2211yescharacteristic
21C2×#16864392C2211yescharacteristic
22C2×#16864392C2211yescharacteristic
23C2×#16864392C2211yescharacteristic
24C2×#16864390C2211yescharacteristic
25C2×#16864390C2211yescharacteristic
26C2×#16864389C2211yescharacteristic
27C2×#16864389C2211yescharacteristic
28C2×#16864387C2211yescharacteristic
29C22×#14524130C2211yescharacteristic
30C22×#14524130C2211yescharacteristic
31#22575441C2211yescharacteristic
32#22575439—21no—
33#22575432C2211yescharacteristic
34#22575431—21no—
35#22575427C2211yescharacteristic
36#22575425—21no—
37C4×Dic538C2211yescharacteristic
38#22575413—21no—
39#22575386C2211yescharacteristic
40C4⋊#14524130C2211yescharacteristic
41#22575377C2211yescharacteristic
42#22575377C2211yescharacteristic
43#22575376C2211yescharacteristic
44#22575376C2211yescharacteristic
45#22575374C2211yescharacteristic
46#22575373C2211yescharacteristic
47C22⋊#14524130C2211yescharacteristic
48C22⋊#14524130C2211yescharacteristic
49C4×#14524130C2211yescharacteristic
50C2×#41×C269C2211yescharacteristic
51C22×Q8×C269C2211yescharacteristic
52#124×C269—21no—
53#124×C269C2211yescharacteristic
54C22⋊Q8×C269—21no—
55C22⋊Q8×C269C2211yescharacteristic
56#121×C269—21no—
57#121×C269C2211yescharacteristic
58C4×Q8×C269C2211yescharacteristic
59C4×D4×C269—21no—
60C4×D4×C269C2211yescharacteristic
61#16864402—22no—
62#16864402—41no—
63C2×Dic538—22no—
64C2×Dic538—21no—
65C2×Dic538—41no—
66C2×Dic538C2311yescharacteristic
67C2×Dic538C2311yescharacteristic
68C2×D1076C2311yescharacteristic
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C22×Dic269—21no—
72C2×C4×D269—21no—
73#16864392—22no—
74#16864392—22no—
75#16864392—22no—
76#16864392D414yescomplemented
77#16864390—22no—
78#16864390—41no—
79#16864389—22no—
80#16864389—41no—
81#16864387—22no—
82C2×#14524130—22no—
83C2×#14524130—22no—
84C2×#14524130C2311yescharacteristic
85C2×#14524130C2311yescharacteristic
86C2×#14524130C2311yescharacteristic
87C2×#14524130C2311yescharacteristic
88C4⋊Dic269C2311yescharacteristic
89#16864381—21no—
90#16864381—21no—
91C22⋊Dic269—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379—21no—
95C4×Dic269—21no—
96#41×C269—22no—
97#41×C269—41no—
98C2×Q8×C269—21no—
99C2×Q8×C269—22no—
100C2×Q8×C269—41no—
101C2×Q8×C269C2311yescharacteristic
102C2×Q8×C269C2311yescharacteristic
103C2×D4×C269—21no—
104C2×D4×C269—21no—
105C2×D4×C269D411yescharacteristic
106C2×D4×C269C2311yescharacteristic
107C22×C4×C269—21no—
108C22×C4×C269—21no—
109C22×C4×C269C2311yescharacteristic
110C22×C4×C269C2311yescharacteristic
111C4⋊C4×C269—21no—
112C4⋊C4×C269—21no—
113C4⋊C4×C269D411yescharacteristic
114C4⋊C4×C269C2311yescharacteristic
115C22⋊C4×C269—21no—
116C22⋊C4×C269—21no—
117C22⋊C4×C269—21no—
118C22⋊C4×C269—21no—
119C22⋊C4×C269D411yescharacteristic
120C22⋊C4×C269D411yescharacteristic
121C42×C269—21no—
122C42×C269C2311yescharacteristic
123C22×D269—21no—
124Dic538—21no—
125Dic538—21no—
126Dic538—21no—
127Dic538—42no—
128Dic538C2×D412yes—
129D1076—21no—
130D1076—21no—
131C22⋊D269—42no—
132C22⋊D269—42no—
133C2×Dic269—21no—
134C2×Dic269—21no—
135C2×Dic269—21no—
136C2×Dic269—42no—
137C4×D269—42no—
138#14524130—21no—
139#14524130—21no—
140#14524130—21no—
141#14524130C2×D412yes—
142C23×C269—21no—
143C23×C269C2×D411yescharacteristic
144C23×C269C2×D411yescharacteristic
145Q8×C269—21no—
146Q8×C269—21no—
147Q8×C269—21no—
148Q8×C269—42no—
149Q8×C269C2×D412yes—
150D4×C269—22no—
151D4×C269—21no—
152D4×C269—21no—
153D4×C269—42no—
154D4×C269—42no—
155C2×C4×C269—21no—
156C2×C4×C269—21no—
157C2×C4×C269—21no—
158C2×C4×C269—21no—
159C2×C4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—42no—
163C2×C4×C269—42no—
164C2×C4×C269C2×D411yescharacteristic
165C2×C4×C269C2×D411yescharacteristic
166C2×C4×C269C2×D411yescharacteristic
167C2×C4×C269C2×D411yescharacteristic
168C2×C4×C269C2411yescharacteristic
169C22×C269—21no—
170C22×C269—21no—
171C22×C269—21no—
172C22×C269—21no—
173C22×C269—21no—
174C22×C269—41no—
175C22×C269—41no—
176C22×C269—41no—
177C22×C269#13512yescomplemented
178C22×C269C22×D411yescharacteristic
179C2×D269—41no—
180C2×D269—41no—
181C2×D269—41no—
182C4×C269—21no—
183C4×C269—21no—
184C4×C269—22no—
185C4×C269—21no—
186C4×C269—21no—
187C4×C269—41no—
188C4×C269—41no—
189C4×C269C22×D411yescharacteristic
190C4×C269H(5,𝔽2)11yescharacteristic
191Dic269—22no—
192Dic269—41no—
193Dic269—41no—
194C2×C269—22no—
195C2×C269—41no—
196C2×C269—81no—
197C2×C269D4211yescharacteristic
198C2×C269C2×#13511yescharacteristic
199C2×C269#57811yescharacteristic
200D269—81no—
201C269#328511yescharacteristic, complemented, Sylow
202#3285—2691nomaximal, Sylow
203C2×#135—2691no—
204C22×Q16—2691no—
205#548—2691no—
206#548D26911yescharacteristic, complemented
207#495—2691no—
208#467—2691no—
209#466—2691no—
210#464—2691no—
211#464—2691no—
212#452—2691no—
213#452—2691no—
214#448—2691no—
215#448—2691no—
216C4×Q16—2691no—
217C8×D4—2691no—
218C2×#41—2691no—
219C2×#41C2×D26911yescharacteristic
220C22×Q8—2691no—
221C22×Q8C2×D26911yescharacteristic
222#135—5384no—
223C2×Q16—2691no—
224C2×Q16—2691no—
225C2×Q16—2691no—
226C2×Q16—2691no—
227C2×Q16—5384no—
228C2×SD16—2691no—
229C2×SD16—2691no—
230C2×SD16—2691no—
231C2×SD16—2691no—
232C2×D8—2691no—
233C22×C8—2691no—
234C22×C8—2691no—
235#124—21no—
236#124—2691no—
237#124—5381no—
238#124C2×D26911yescharacteristic
239C22⋊Q8—21no—
240C22⋊Q8—2691no—
241C22⋊Q8—5381no—
242C22⋊Q8C2×D26911yescharacteristic
243#121—21no—
244#121—2691no—
245#121—5381no—
246#121C2×D26911yescharacteristic
247C4×Q8—2691no—
248C4×Q8C2×D26911yescharacteristic
249C4×D4—21no—
250C4×D4—5381no—
251C4×D4C2×D26911yescharacteristic
252#107—2691no—
253C4⋊C8—2691no—
254#103—2691no—
255#103—2691no—
256#103—2691no—
257#103—2691no—
258#102—2691no—
259#102—2691no—
260C22⋊C8—2691no—
261C22⋊C8—2691no—
262C4×C8—2691no—
263#41—22no—
264#41—41no—
265#41—5382no—
266#41—10761no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—2691no—
271C2×Q8—2691no—
272C2×Q8—5382no—
273C2×Q8—5381no—
274C2×Q8—10761no—
275C2×Q8C22×D26911yescharacteristic
276C2×Q8C22×D26911yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—2691no—
280C2×D4—5381no—
281C2×D4—5381no—
282C2×D4C22⋊D26911yescharacteristic
283C2×D4C22×D26911yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—5381no—
287C22×C4—5381no—
288C22×C4C22×D26911yescharacteristic
289C22×C4C22×D26911yescharacteristic
290Q16—2694no—
291Q16—5382no—
292Q16—5382no—
293Q16—5382no—
294SD16—5382no—
295SD16—5382no—
296SD16—10761no—
297SD16—10761no—
298D8—5382no—
299C2×C8—2691no—
300C2×C8—2691no—
301C2×C8—2691no—
302C2×C8—2691no—
303C2×C8—5382no—
304C2×C8—5382no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—2691no—
308C4⋊C4—5381no—
309C4⋊C4—5381no—
310C4⋊C4C22⋊D26911yescharacteristic
311C4⋊C4C22×D26911yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—5381no—
317C22⋊C4—5381no—
318C22⋊C4—5381no—
319C22⋊C4—5381no—
320C22⋊C4C22⋊D26911yescharacteristic
321C22⋊C4C22⋊D26911yescharacteristic
322C42—21no—
323C42—5381no—
324C42C22×D26911yescharacteristic
325C23—21no—
326C23—5381no—
327C23C2×C22⋊D26911yescharacteristic
328C23C2×C22⋊D26911yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—2692no—
334Q8—5381no—
335Q8—5381no—
336Q8—5381no—
337Q8—10762no—
338Q8D4×D26912yes—
339D4—22no—
340D4—21no—
341D4—21no—
342D4—42no—
343D4—42no—
344D4—5381no—
345D4—5381no—
346D4—10762no—
347D4—10762no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—5381no—
358C2×C4—5381no—
359C2×C4—5381no—
360C2×C4—10762no—
361C2×C4—10762no—
362C2×C4C2×C22⋊D26911yescharacteristic
363C2×C4C2×C22⋊D26911yescharacteristic
364C2×C4C2×C22⋊D26911yescharacteristic
365C2×C4C2×C22⋊D26911yescharacteristic
366C2×C4C23×D26911yescharacteristic
367C8—2692no—
368C8—5381no—
369C8—5381no—
370C8—5381no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—10761no—
380C22—10761no—
381C22—10761no—
382C22#2257547312yescomplemented
383C22C22×C22⋊D26911yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—22no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—5382no—
392C4—10761no—
393C4—10761no—
394C4C2×D4×D26911yescharacteristic
395C4#2257549811yescharacteristic
396C2—22no—
397C2—41no—
398C2—81no—
399C2—21521no—
400C2D4×C22⋊D26911yescharacteristic
401C2C2×#2257547311yescharacteristic
402C2#3177142011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#548×C269C2
socleC22×C269C22×D4
hypercenter#548D269
hypocenterC269#3285
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#22575473C2semidirect
2. C22×#16864392C2semidirect
3. #31771264C2semidirect
4. #31770987C2semidirect
5. #31770844C2semidirect
6. #31770839C2semidirect
7. #31770827C2semidirect
8. #31770824C2semidirect
9. #31770767C2semidirect
10. #31770765C2semidirect
11. #31770744C2semidirect
12. #31770743C2semidirect
13. C4×#16864392C2semidirect
14. D4×#14524130C2semidirect
15. #548×C269C2semidirect
76. #16864392D4semidirect
177. C22×C269#135semidirect
201. C269#3285semidirect
206. #548D269semidirect
382. C22#22575473semidirect

Maximal subgroups

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

subgroupindex
C2×#225754732
C22×#168643922
#317712642
#317709872
#317708442
#317708392
#317708272
#317708242
#317707672
#317707652
#317707442
#317707432
C4×#168643922
D4×#145241302
#548×C2692
#3285269