Finite Group Databasebeta

← #46822786

Subgroups — #46822786

Order 34432.

398 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#31771308C211yesmaximal, characteristic, complemented
2#31771263C211yesmaximal, characteristic, complemented
3#31770944C211yesmaximal, characteristic, complemented
4#31770922C211yesmaximal, characteristic, complemented
5#31770909C211yesmaximal, characteristic, complemented
6#31770840C211yesmaximal, characteristic, complemented
7#31770837C211yesmaximal, characteristic, complemented
8#31770817C211yesmaximal, characteristic, complemented
9#31770751C211yesmaximal, characteristic, complemented
10#31770745C211yesmaximal, characteristic, complemented
11C4×#16864391C211yesmaximal, characteristic, complemented
12C8×D1076C211yesmaximal, characteristic, complemented
13#31770598C211yesmaximal, characteristic, complemented
14#103×D269C211yesmaximal, characteristic, complemented
15#488×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic
18C2×#16864391C2211yescharacteristic
19C2×#16864389C2211yescharacteristic
20C2×#16864387C2211yescharacteristic
21C2×C8×D269C2211yescharacteristic
22C2×C8×D269C2211yescharacteristic
23#22575448—21no—
24#22575439—21no—
25#22575438—21no—
26#22575430—21no—
27#22575429C2211yescharacteristic
28#22575428—21no—
29#22575428C2211yescharacteristic
30#22575428C2211yescharacteristic
31#22575426—21no—
32#22575424C2211yescharacteristic
33C4×Dic538C2211yescharacteristic
34C4×D1076C2211yescharacteristic
35C4×D1076C2211yescharacteristic
36#22575415—21no—
37#22575413—21no—
38#22575411C2211yescharacteristic
39C4⋊C4×D269C2211yescharacteristic
40#22575386C2211yescharacteristic
41#22575385C2211yescharacteristic
42#22575384C2211yescharacteristic
43C4⋊#14524130C2211yescharacteristic
44#22575378C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575376C2211yescharacteristic
47#22575376C2211yescharacteristic
48#22575375C2211yescharacteristic
49#22575374C2211yescharacteristic
50#22575365C2211yescharacteristic
51#22575365C2211yescharacteristic
52#125×C269C2211yescharacteristic
53#124×C269C2211yescharacteristic
54#103×C269C2211yescharacteristic
55#103×C269C2211yescharacteristic
56#102×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58C4×C8×C269C2211yescharacteristic
59#16864401—22no—
60#16864401—41no—
61C2×Dic538—21no—
62C2×Dic538C2311yescharacteristic
63Q8×D269—22no—
64Q8×D269—41no—
65C2×D1076—21no—
66C2×D1076—21no—
67C2×D1076D411yescharacteristic
68C2×D1076C2311yescharacteristic
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C22×Dic269—21no—
72C2×C4×D269—21no—
73C2×C4×D269—21no—
74C2×C4×D269—21no—
75C2×C4×D269C2311yescharacteristic
76C2×C4×D269C2311yescharacteristic
77#16864391—22no—
78#16864389—41no—
79#16864387—41no—
80C2×#14524130C2311yescharacteristic
81C2×#14524130C2311yescharacteristic
82C8×D269—41no—
83C8×D269—41no—
84C4⋊Dic269D411yescharacteristic
85C4⋊Dic269C2311yescharacteristic
86C4⋊Dic269C2311yescharacteristic
87#16864381—21no—
88#16864381—21no—
89#16864381—21no—
90C22⋊Dic269—21no—
91#16864379—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96#16864379D411yescharacteristic
97#16864379D411yescharacteristic
98C4×Dic269—21no—
99C4×Dic269—21no—
100C2×Q8×C269C2311yescharacteristic
101C2×D4×C269C2311yescharacteristic
102C2×C8×C269C2311yescharacteristic
103C2×C8×C269C2311yescharacteristic
104C4⋊C4×C269—21no—
105C4⋊C4×C269—21no—
106C4⋊C4×C269C2311yescharacteristic
107C4⋊C4×C269C2311yescharacteristic
108C22⋊C4×C269—21no—
109C22⋊C4×C269—21no—
110C42×C269C2311yescharacteristic
111C22×D269—21no—
112C22×D269C2×D411yescharacteristic
113C22×D269C2×D411yescharacteristic
114Dic538—21no—
115Dic538—42no—
116Dic538#4112yes—
117D1076—22no—
118D1076—21no—
119D1076—21no—
120D1076—42no—
121D1076—81no—
122C22⋊D269—42no—
123C22⋊D269—42no—
124C2×Dic269—21no—
125C2×Dic269—21no—
126C2×Dic269—21no—
127C2×Dic269—42no—
128C2×Dic269C2×D411yescharacteristic
129C2×Dic269C2×D411yescharacteristic
130C4×D269—21no—
131C4×D269—21no—
132C4×D269—21no—
133C4×D269—21no—
134C4×D269—42no—
135C4×D269—81no—
136C4×D269—81no—
137#14524130—21no—
138#14524130—21no—
139C23×C269—21no—
140Q8×C269—21no—
141Q8×C269—21no—
142D4×C269—21no—
143D4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269C2×D411yescharacteristic
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2411yescharacteristic
150C8×C269—21no—
151C8×C269—21no—
152C22×C269—41no—
153C22×C269—41no—
154C22×C269—41no—
155C22×C269C22×D411yescharacteristic
156C2×D269—21no—
157C2×D269—21no—
158C2×D269—21no—
159C2×D269—21no—
160C2×D269—21no—
161C2×D269—41no—
162C2×D269—41no—
163C2×D269—41no—
164C2×D269#13512yes—
165C4×C269—21no—
166C4×C269—21no—
167C4×C269—41no—
168C4×C269—41no—
169C4×C269—41no—
170C4×C269C2×#4111yescharacteristic
171C4×C269H(5,𝔽2)11yescharacteristic
172Dic269—21no—
173Dic269—21no—
174Dic269—22no—
175Dic269—41no—
176Dic269—41no—
177C2×C269—81no—
178C2×C269#54611yescharacteristic
179C2×C269C2×#13511yescharacteristic
180C2×C269#57711yescharacteristic
181D269—22no—
182D269—41no—
183D269—81no—
184C269#330111yescharacteristic, complemented, Sylow
185#3301—2691nomaximal, Sylow
186#553—2691no—
187#548—2691no—
188#488D26911yescharacteristic, complemented
189#485—2691no—
190#481—2691no—
191#479—2691no—
192#466—2691no—
193#465—2691no—
194#463—2691no—
195#449—2691no—
196#448—2691no—
197C4×SD16—2691no—
198C8×D4—2691no—
199#416—2691no—
200C2×#103—2691no—
201C2×#41—2691no—
202C22×Q8—2691no—
203C2×SD16—2691no—
204C2×SD16—2691no—
205C2×D8—2691no—
206C22×C8—2691no—
207C22×C8—2691no—
208#126—5381no—
209#125C2×D26911yescharacteristic
210#124—5381no—
211#124C2×D26911yescharacteristic
212#123—5381no—
213C22⋊Q8—2691no—
214C22⋊Q8—5381no—
215#121—2691no—
216#121—2691no—
217#121—2691no—
218#121—5381no—
219#121—5381no—
220C4×Q8—2691no—
221C4×D4—2691no—
222C4×D4—2691no—
223C4×D4—5381no—
224C4×D4—5381no—
225#117—2691no—
226C2×C4⋊C4—2691no—
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—
243C4×C8C2×D26911yescharacteristic
244#41—5382no—
245#41—10761no—
246C2×Q8—2691no—
247C2×Q8—5381no—
248C2×Q8—5382no—
249C2×Q8—10761no—
250C2×Q8C22×D26911yescharacteristic
251C2×D4—2691no—
252C2×D4—2691no—
253C2×D4—5381no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4—5381no—
257C2×D4C22×D26911yescharacteristic
258C22×C4—2691no—
259C22×C4—2691no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4—5381no—
263C22×C4—5381no—
264SD16—5382no—
265SD16—10761no—
266D8—10761no—
267C2×C8—2691no—
268C2×C8—2691no—
269C2×C8—10761no—
270C2×C8—10761no—
271C2×C8C22×D26911yescharacteristic
272C2×C8C22×D26911yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—2691no—
276C4⋊C4—2691no—
277C4⋊C4—2691no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4C22×D26911yescharacteristic
282C4⋊C4C22×D26911yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—2691no—
286C22⋊C4—2691no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4—5381no—
293C42—5381no—
294C42—5381no—
295C42C22×D26911yescharacteristic
296C23—21no—
297C23—2691no—
298C23—2691no—
299C23—5381no—
300Q8—21no—
301Q8—21no—
302Q8—2692no—
303Q8—5381no—
304Q8—10762no—
305D4—21no—
306D4—21no—
307D4—5382no—
308D4—5381no—
309D4—5381no—
310D4—10762no—
311D4—10762no—
312D4—10762no—
313D4—21521no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—2691no—
318C2×C4—2691no—
319C2×C4—5381no—
320C2×C4—5381no—
321C2×C4—5381no—
322C2×C4—5381no—
323C2×C4—5381no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—10762no—
327C2×C4—10762no—
328C2×C4—21521no—
329C2×C4—21521no—
330C2×C4D4×D26911yescharacteristic
331C2×C4D4×D26911yescharacteristic
332C2×C4C23×D26911yescharacteristic
333C8—21no—
334C8—21no—
335C8—5381no—
336C8—5381no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—2692no—
341C22—5381no—
342C22—5381no—
343C22—5381no—
344C22—5381no—
345C22—5381no—
346C22—10761no—
347C22—10761no—
348C22—10761no—
349C22C2×D4×D26911yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—5381no—
356C4—5381no—
357C4—5382no—
358C4—10761no—
359C4—10761no—
360C4#41×D26911yescharacteristic
361C4#2257549911yescharacteristic
362C2—81no—
363C2—5382no—
364C2—10761no—
365C2—21521no—
366C2#3177124811yescharacteristic
367C2#135×D26911yescharacteristic
368C2#3177141211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#488×C269C2
socleC22×C269C22×D4
hypercenter#488D269
hypocenterC269#3301
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #31771308C2semidirect
2. #31771263C2semidirect
3. #31770944C2semidirect
4. #31770922C2semidirect
5. #31770909C2semidirect
6. #31770840C2semidirect
7. #31770837C2semidirect
8. #31770817C2semidirect
9. #31770751C2semidirect
10. #31770745C2semidirect
11. C4×#16864391C2semidirect
12. C8×D1076C2semidirect
13. #31770598C2semidirect
14. #103×D269C2semidirect
15. #488×C269C2semidirect
184. C269#3301semidirect
188. #488D269semidirect

Maximal subgroups

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

subgroupindex
#317713082
#317712632
#317709442
#317709222
#317709092
#317708402
#317708372
#317708172
#317707512
#317707452
C4×#168643912
C8×D10762
#317705982
#103×D2692
#488×C2692
#3301269