Finite Group Databasebeta

← #46821134

Subgroups — #46821134

Order 34432.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771213C211yesmaximal, characteristic, complemented
2#31771077C211yesmaximal, characteristic
3#31770949C211yesmaximal, characteristic, complemented
4#31770944C211yesmaximal, characteristic, complemented
5#31770907C211yesmaximal, characteristic, complemented
6#31770904C211yesmaximal, characteristic, complemented
7#31770809C211yesmaximal, characteristic, complemented
8#31770808C211yesmaximal, characteristic, complemented
9#31770757C211yesmaximal, characteristic
10#31770754C211yesmaximal, characteristic
11#31770705C211yesmaximal, characteristic, complemented
12#31770703C211yesmaximal, characteristic, complemented
13C4×#16864391C211yesmaximal, characteristic, complemented
14#31770664C211yesmaximal, characteristic, complemented
15C4×SD16×C269C211yesmaximal, characteristic, complemented
16C2×D4×D269C2211yescharacteristic
17C2×#16864391C2211yescharacteristic
18C2×#16864390C2211yescharacteristic
19C2×#16864389C2211yescharacteristic
20#22575451C2211yescharacteristic, complemented
21#22575448—21no—
22#22575447—21no—
23#22575445—21no—
24#22575444C2211yescharacteristic, complemented
25#22575443—21no—
26#22575433C2211yescharacteristic
27#22575429C2211yescharacteristic
28C4×Dic538C2211yescharacteristic, complemented
29C4×D1076C2211yescharacteristic
30D4×Dic269C2211yescharacteristic
31#22575415C2211yescharacteristic
32C4×C22⋊D269—21no—
33#22575413—21no—
34#22575411C2211yescharacteristic
35#22575410C2211yescharacteristic
36C2×#16864379—21no—
37C22⋊C4×D269—21no—
38#22575386C2211yescharacteristic, complemented
39#22575386C2211yescharacteristic, complemented
40#22575385C2211yescharacteristic, complemented
41#22575382C2211yescharacteristic, complemented
42#22575382C2211yescharacteristic, complemented
43#22575378C2211yescharacteristic, complemented
44#22575377C2211yescharacteristic, complemented
45#22575376C2211yescharacteristic, complemented
46#22575375C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575373C2211yescharacteristic
49#22575365C2211yescharacteristic
50#22575365C2211yescharacteristic
51#22575364C2211yescharacteristic, complemented
52C2×SD16×C269C2211yescharacteristic
53C4×Q8×C269C2211yescharacteristic, complemented
54C4×D4×C269C2211yescharacteristic
55#106×C269C2211yescharacteristic, complemented
56#103×C269C2211yescharacteristic, complemented
57#102×C269C2211yescharacteristic
58C4×C8×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60C2×Dic538C2311yescharacteristic
61C2×D1076C2311yescharacteristic
62C2×C22⋊D269—21no—
63D4×D269—22no—
64D4×D269—22no—
65C22×Dic269—21no—
66C2×C4×D269—21no—
67C2×C4×D269—21no—
68C2×C4×D269C2311yescharacteristic
69#16864391—22no—
70#16864390—22no—
71#16864389—22no—
72C2×#14524130C2311yescharacteristic
73C2×#14524130C2311yescharacteristic
74C4⋊Dic269—21no—
75C4⋊Dic269—21no—
76C4⋊Dic269C2311yescharacteristic
77C4⋊Dic269C2311yescharacteristic
78#16864381—21no—
79#16864381—21no—
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381D411yescharacteristic
84#16864381D411yescharacteristic
85C22⋊Dic269—21no—
86#16864379—21no—
87#16864379—21no—
88#16864379—21no—
89#16864379—21no—
90#16864379—21no—
91#16864379D411yescharacteristic
92#16864379D411yescharacteristic
93C4×Dic269—21no—
94C4×Dic269—21no—
95C4×Dic269C2311yescharacteristic
96C2×Q8×C269C2311yescharacteristic
97C2×D4×C269C2311yescharacteristic
98C22×C4×C269—21no—
99SD16×C269—22no—
100C2×C8×C269C2311yescharacteristic
101C2×C8×C269C2311yescharacteristic
102C4⋊C4×C269—21no—
103C4⋊C4×C269C2311yescharacteristic
104C4⋊C4×C269C2311yescharacteristic
105C22⋊C4×C269—21no—
106C42×C269—21no—
107C42×C269C2311yescharacteristic
108C22×D269—21no—
109C22×D269—24no—
110C22×D269—81no—
111C22×D269—81no—
112C22×D269C2×D411yescharacteristic
113Dic538—21no—
114Dic538—21no—
115D1076—21no—
116D1076#4112yes—
117C22⋊D269—24no—
118C2×Dic269—21no—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—81no—
122C2×Dic269C2×D411yescharacteristic
123C2×Dic269C2×D411yescharacteristic
124C2×Dic269C2×D411yescharacteristic
125C4×D269—21no—
126C4×D269—21no—
127C4×D269—81no—
128C4×D269—81no—
129#14524130—21no—
130#14524130—21no—
131C23×C269—21no—
132Q8×C269—21no—
133Q8×C269—21no—
134D4×C269—21no—
135D4×C269#4112yes—
136C2×C4×C269—21no—
137C2×C4×C269—21no—
138C2×C4×C269—21no—
139C2×C4×C269—81no—
140C2×C4×C269C2×D411yescharacteristic
141C2×C4×C269C2×D411yescharacteristic
142C2×C4×C269C2411yescharacteristic
143C8×C269—21no—
144C8×C269—21no—
145C22×C269—22no—
146C22×C269—41no—
147C22×C269—41no—
148C22×C269C22×D411yescharacteristic
149C2×D269—21no—
150C2×D269—22no—
151C2×D269—21no—
152C2×D269—21no—
153C2×D269—41no—
154C2×D269—41no—
155C2×D269—84no—
156C4×C269—21no—
157C4×C269—21no—
158C4×C269—41no—
159C4×C269—41no—
160C4×C269—41no—
161C4×C269C2×#4111yescharacteristic
162C4×C269C2×#4111yescharacteristic
163Dic269—21no—
164Dic269—21no—
165Dic269—21no—
166Dic269—41no—
167Dic269—41no—
168Dic269—41no—
169C2×C269—42no—
170C2×C269#52511yescharacteristic
171C2×C269#57711yescharacteristic
172C2×C269#57711yescharacteristic
173D269—41no—
174D269—42no—
175C269#312311yescharacteristic, complemented, Sylow
176#3123—2691nomaximal, Sylow
177#541—2691no—
178#518—2691no—
179#485—2691no—
180#485—2691no—
181#478—2691no—
182#478—2691no—
183#461—2691no—
184#461—2691no—
185#450—2691no—
186#450—2691no—
187#440—2691no—
188#440—2691no—
189C4×SD16—2691no—
190C4×SD16D26911yescharacteristic, complemented
191#433—2691no—
192C22×D4—2691no—
193C2×SD16—2691no—
194C2×SD16—2691no—
195C2×SD16—2691no—
196C2×SD16C2×D26911yescharacteristic
197C4⋊Q8—2691no—
198#126—5381no—
199#126—5381no—
200#125—2691no—
201#125—5381no—
202#125—5381no—
203C22⋊Q8—2691no—
204C22⋊Q8—2691no—
205C4×Q8—2691no—
206C4×Q8C2×D26911yescharacteristic, complemented
207C4×D4—2691no—
208C4×D4—2691no—
209C4×D4—2691no—
210C4×D4—5381no—
211C4×D4—5381no—
212C4×D4C2×D26911yescharacteristic
213#117—2691no—
214#117—2691no—
215C2×C22⋊C4—5381no—
216C2×C22⋊C4—5381no—
217#107—2691no—
218#107—2691no—
219#106—2691no—
220#106C2×D26911yescharacteristic, complemented
221C4⋊C8—2691no—
222C4⋊C8—2691no—
223#103—2691no—
224#103—2691no—
225#103—2691no—
226#103C2×D26911yescharacteristic, complemented
227#102—2691no—
228#102—2691no—
229#102—2691no—
230#102C2×D26911yescharacteristic
231C22⋊C8—2691no—
232C22⋊C8—2691no—
233#97—2691no—
234C4×C8C2×D26911yescharacteristic, complemented
235C24—5381no—
236C2×Q8—2691no—
237C2×Q8C22×D26911yescharacteristic
238C2×D4—2691no—
239C2×D4—5381no—
240C2×D4—5382no—
241C2×D4—5382no—
242C2×D4C22×D26911yescharacteristic
243C22×C4—21no—
244C22×C4—2691no—
245C22×C4—5381no—
246C22×C4—5381no—
247C22×C4—5381no—
248SD16—22no—
249SD16—5382no—
250SD16—5382no—
251SD16—5382no—
252C2×C8—2691no—
253C2×C8—2691no—
254C2×C8C22×D26911yescharacteristic
255C2×C8C22×D26911yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—2691no—
258C4⋊C4—2691no—
259C4⋊C4—2691no—
260C4⋊C4—2691no—
261C4⋊C4—5381no—
262C4⋊C4—5381no—
263C4⋊C4—5381no—
264C4⋊C4—5381no—
265C4⋊C4—5381no—
266C4⋊C4—5381no—
267C4⋊C4—5381no—
268C4⋊C4C22×D26911yescharacteristic
269C4⋊C4C22×D26911yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—2691no—
272C22⋊C4—2691no—
273C22⋊C4—5381no—
274C22⋊C4—5381no—
275C22⋊C4—5381no—
276C22⋊C4—5381no—
277C22⋊C4—5381no—
278C22⋊C4—5381no—
279C42—21no—
280C42—2691no—
281C42—5381no—
282C42—5381no—
283C42C22×D26911yescharacteristic
284C23—21no—
285C23—2691no—
286C23—5384no—
287C23—5381no—
288C23—21521no—
289C23—21521no—
290Q8—21no—
291Q8—21no—
292Q8—5381no—
293Q8—5381no—
294D4—21no—
295D4—2692no—
296D4—5384no—
297D4—5381no—
298D4#1686440212yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—2691no—
304C2×C4—2691no—
305C2×C4—2691no—
306C2×C4—5381no—
307C2×C4—5381no—
308C2×C4—5381no—
309C2×C4—5381no—
310C2×C4—5381no—
311C2×C4—21521no—
312C2×C4—21521no—
313C2×C4—21521no—
314C2×C4D4×D26911yescharacteristic
315C2×C4D4×D26911yescharacteristic
316C2×C4C23×D26911yescharacteristic
317C8—21no—
318C8—21no—
319C8—5381no—
320C8—5381no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—5381no—
325C22—5381no—
326C22—5381no—
327C22—5382no—
328C22—10761no—
329C22—10761no—
330C22—21524no—
331C22C2×D4×D26911yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—5381no—
338C4—5381no—
339C4—5381no—
340C4—10761no—
341C4—10761no—
342C4—10761no—
343C4#41×D26911yescharacteristic
344C4C2×#1686440211yescharacteristic
345C2—42no—
346C2—10761no—
347C2—10762no—
348C2#3177111411yescharacteristic
349C2#3177141111yescharacteristic
350C2#3177141211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC4×SD16×C269C2
socleC22×C269C22×D4
prime radicalC2×D4×D269C22
hypercenterC4×SD16D269
hypocenterC269#3123
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #31771213C2semidirect
3. #31770949C2semidirect
4. #31770944C2semidirect
5. #31770907C2semidirect
6. #31770904C2semidirect
7. #31770809C2semidirect
8. #31770808C2semidirect
11. #31770705C2semidirect
12. #31770703C2semidirect
13. C4×#16864391C2semidirect
14. #31770664C2semidirect
15. C4×SD16×C269C2semidirect
20. #22575451C22semidirect
24. #22575444C22semidirect
28. C4×Dic538C22semidirect
38. #22575386C22semidirect
39. #22575386C22semidirect
40. #22575385C22semidirect
41. #22575382C22semidirect
42. #22575382C22semidirect
43. #22575378C22semidirect
44. #22575377C22semidirect
45. #22575376C22semidirect
51. #22575364C22semidirect
53. C4×Q8×C269C22semidirect
55. #106×C269C22semidirect
56. #103×C269C22semidirect
58. C4×C8×C269C22semidirect
175. C269#3123semidirect
190. C4×SD16D269semidirect
206. C4×Q8C2×D269semidirect
220. #106C2×D269semidirect
226. #103C2×D269semidirect
234. C4×C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712132
#317710772
#317709492
#317709442
#317709072
#317709042
#317708092
#317708082
#317707572
#317707542
#317707052
#317707032
C4×#168643912
#317706642
C4×SD16×C2692
#3123269