Finite Group Databasebeta

← #46822545

Subgroups — #46822545

Order 34432.

482 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575485C211yesmaximal, characteristic, complemented
2#31771274C211yesmaximal, characteristic
3Q8×C22⋊D269C211yesmaximal, characteristic, complemented
4#31770976C211yesmaximal, characteristic, complemented
5#31770851C211yesmaximal, characteristic, complemented
6#31770848C211yesmaximal, characteristic, complemented
7#31770813C211yesmaximal, characteristic, complemented
8#31770803C211yesmaximal, characteristic, complemented
9#31770767C211yesmaximal, characteristic, complemented
10#31770766C211yesmaximal, characteristic, complemented
11#31770744C211yesmaximal, characteristic, complemented
12#31770742C211yesmaximal, characteristic, complemented
13Q16×Dic269C211yesmaximal, characteristic, complemented
14#31770681C211yesmaximal, characteristic, complemented
15C2×#137×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic
18#22575485—24no—
19C2×#16864392C2211yescharacteristic
20C2×#16864391C2211yescharacteristic
21C2×#16864390C2211yescharacteristic
22C2×#16864389C2211yescharacteristic
23C2×#16864389C2211yescharacteristic
24C2×#16864385C2211yescharacteristic
25#22575451—21no—
26#22575451C2211yescharacteristic
27#22575449—21no—
28#22575449C2211yescharacteristic
29#22575432—21no—
30#22575432C2211yescharacteristic
31#22575429—21no—
32#22575429C2211yescharacteristic
33#22575427—21no—
34#22575427C2211yescharacteristic
35#22575424—21no—
36#22575424C2211yescharacteristic
37Q8×Dic269C2211yescharacteristic
38Q8×Dic269C2211yescharacteristic
39C4×C22⋊D269—21no—
40C4×C22⋊D269—21no—
41C4×C22⋊D269C2211yescharacteristic
42#22575387C2211yescharacteristic
43#22575382C2211yescharacteristic
44#22575378C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575377C2211yescharacteristic
47#22575377C2211yescharacteristic
48#22575375C2211yescharacteristic
49#22575374C2211yescharacteristic
50C22⋊#14524130C2211yescharacteristic
51#22575365C2211yescharacteristic
52C8×Dic269C2211yescharacteristic
53C2×#41×C269C2211yescharacteristic
54C22×Q8×C269C2211yescharacteristic
55#137×C269—24no—
56C2×Q16×C269C2211yescharacteristic
57C2×Q16×C269C2211yescharacteristic
58C2×SD16×C269C2211yescharacteristic
59C2×SD16×C269C2211yescharacteristic
60C2×M16×C269C2211yescharacteristic
61#16864403—22no—
62#16864403—41no—
63C2×Dic538—21no—
64C2×Dic538C2311yescharacteristic
65Q8×D269—22no—
66Q8×D269—41no—
67C2×D1076—21no—
68C2×D1076C2311yescharacteristic
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C2×C22⋊D269—21no—
72C2×C22⋊D269—21no—
73C2×C22⋊D269D411yescharacteristic
74C2×C4×D269—21no—
75C2×C4×D269—21no—
76C2×C4×D269C2311yescharacteristic
77#16864392—22no—
78#16864391—22no—
79#16864390—41no—
80#16864389—22no—
81#16864389—41no—
82#16864385—22no—
83C2×#14524130C2311yescharacteristic
84C2×#14524130C2311yescharacteristic
85C4⋊Dic269—21no—
86C4⋊Dic269—21no—
87C4⋊Dic269C2311yescharacteristic
88C4⋊Dic269C2311yescharacteristic
89#16864381—21no—
90#16864381—21no—
91#16864381—21no—
92#16864381—21no—
93#16864381D411yescharacteristic
94C22⋊Dic269—21no—
95C22⋊Dic269—21no—
96C22⋊Dic269D411yescharacteristic
97#16864379—21no—
98#16864379—21no—
99#16864379D411yescharacteristic
100C4×Dic269—21no—
101C4×Dic269—21no—
102C4×Dic269C2311yescharacteristic
103#41×C269—22no—
104#41×C269—41no—
105C2×Q8×C269—21no—
106C2×Q8×C269—22no—
107C2×Q8×C269—41no—
108C2×Q8×C269C2311yescharacteristic
109C2×Q8×C269C2311yescharacteristic
110C2×Q8×C269C2311yescharacteristic
111C2×D4×C269—21no—
112C2×D4×C269C2311yescharacteristic
113C22×C4×C269—21no—
114C22×C4×C269—21no—
115C22×C4×C269C2311yescharacteristic
116Q16×C269—22no—
117Q16×C269D414yescomplemented
118SD16×C269—22no—
119SD16×C269—22no—
120M16×C269—22no—
121C2×C8×C269C2311yescharacteristic
122C2×C8×C269C2311yescharacteristic
123C22×D269—21no—
124C22×D269C2×D411yescharacteristic
125Dic538—21no—
126Dic538—21no—
127Dic538—42no—
128D1076—21no—
129D1076—21no—
130D1076—42no—
131C22⋊D269—22no—
132C22⋊D269—42no—
133C22⋊D269—81no—
134C22⋊D269—81no—
135C22⋊D269—81no—
136C2×Dic269—21no—
137C2×Dic269—21no—
138C2×Dic269—21no—
139C2×Dic269C2×D411yescharacteristic
140C2×Dic269C2×D411yescharacteristic
141C2×Dic269C2×D411yescharacteristic
142C4×D269—21no—
143C4×D269—21no—
144C4×D269—42no—
145C4×D269—42no—
146#14524130—21no—
147#14524130—21no—
148C23×C269—21no—
149C23×C269C2×D411yescharacteristic
150Q8×C269—21no—
151Q8×C269—21no—
152Q8×C269—21no—
153Q8×C269—21no—
154Q8×C269—42no—
155Q8×C269C2×D412yes—
156Q8×C269C2×D412yes—
157D4×C269—21no—
158D4×C269—21no—
159D4×C269—42no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—21no—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—42no—
166C2×C4×C269—42no—
167C2×C4×C269C2×D411yescharacteristic
168C2×C4×C269C2411yescharacteristic
169C8×C269—21no—
170C8×C269C2×D412yes—
171C22×C269—21no—
172C22×C269—21no—
173C22×C269—21no—
174C22×C269—41no—
175C22×C269—41no—
176C22×C269—41no—
177C22×C269C22×D411yescharacteristic
178C2×D269—21no—
179C2×D269—21no—
180C2×D269—21no—
181C2×D269—41no—
182C2×D269—41no—
183C2×D269—41no—
184C4×C269—21no—
185C4×C269—22no—
186C4×C269—22no—
187C4×C269—41no—
188C4×C269C22×D411yescharacteristic
189C4×C269H(5,𝔽2)11yescharacteristic
190Dic269—21no—
191Dic269—21no—
192Dic269—41no—
193Dic269—41no—
194Dic269—41no—
195Dic269#13712yescomplemented
196C2×C269—41no—
197C2×C269—81no—
198C2×C269D4211yescharacteristic
199C2×C269C2×#13711yescharacteristic
200C2×C269#57711yescharacteristic
201D269—41no—
202D269—81no—
203C269#327811yescharacteristic, complemented, Sylow
204#3278—2691nomaximal, Sylow
205C2×#137—2691no—
206C2×#137D26911yescharacteristic, complemented
207#550—2691no—
208D4×Q8—2691no—
209#492—2691no—
210#468—2691no—
211#468—2691no—
212#462—2691no—
213#460—2691no—
214#452—2691no—
215#452—2691no—
216#448—2691no—
217#448—2691no—
218C4×Q16—2691no—
219#436—2691no—
220C2×#41—2691no—
221C2×#41C2×D26911yescharacteristic
222C22×Q8—2691no—
223C22×Q8C2×D26911yescharacteristic
224#137—24no—
225#137—5384no—
226C2×Q16—2691no—
227C2×Q16C2×D26911yescharacteristic
228C2×Q16C2×D26911yescharacteristic
229C2×SD16—2691no—
230C2×SD16—2691no—
231C2×SD16—2691no—
232C2×SD16—2691no—
233C2×SD16C2×D26911yescharacteristic
234C2×SD16C2×D26911yescharacteristic
235C2×M16—2691no—
236C2×M16C2×D26911yescharacteristic
237C4⋊Q8—2691no—
238C4⋊Q8—5381no—
239#127—2691no—
240#127—5381no—
241C22⋊Q8—2691no—
242C22⋊Q8—2691no—
243C22⋊Q8—5381no—
244C22⋊Q8—5381no—
245#121—2691no—
246#121—2691no—
247#121—5381no—
248#121—5381no—
249C4×Q8—2691no—
250C4×Q8—2691no—
251C4×D4—2691no—
252C4×D4—5381no—
253C4×D4—5381no—
254#107—2691no—
255C4⋊C8—2691no—
256#103—2691no—
257#103—2691no—
258#103—2691no—
259#103—2691no—
260#102—2691no—
261#102—2691no—
262C22⋊C8—2691no—
263C22⋊C8—2691no—
264C4×C8—2691no—
265#41—22no—
266#41—41no—
267#41—5382no—
268#41—10761no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2691no—
273C2×Q8—5381no—
274C2×Q8—5382no—
275C2×Q8—10761no—
276C2×Q8C22×D26911yescharacteristic
277C2×Q8C22×D26911yescharacteristic
278C2×Q8C22×D26911yescharacteristic
279C2×D4—21no—
280C2×D4—2691no—
281C2×D4—2691no—
282C2×D4—5381no—
283C2×D4—5381no—
284C2×D4—5381no—
285C2×D4—5381no—
286C2×D4—5381no—
287C2×D4C22×D26911yescharacteristic
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—2691no—
291C22×C4—5381no—
292C22×C4—5381no—
293C22×C4C22×D26911yescharacteristic
294Q16—22no—
295Q16—5382no—
296Q16C22⋊D26914yescomplemented
297SD16—22no—
298SD16—22no—
299SD16—5382no—
300SD16—5382no—
301SD16—10761no—
302SD16—10761no—
303M16—22no—
304M16—5382no—
305C2×C8—2691no—
306C2×C8—2691no—
307C2×C8C22×D26911yescharacteristic
308C2×C8C22×D26911yescharacteristic
309C4⋊C4—2691no—
310C4⋊C4—2691no—
311C4⋊C4—2691no—
312C4⋊C4—5381no—
313C4⋊C4—5381no—
314C4⋊C4—5381no—
315C4⋊C4—5381no—
316C4⋊C4—5381no—
317C4⋊C4—5381no—
318C22⋊C4—2691no—
319C22⋊C4—2691no—
320C22⋊C4—5381no—
321C22⋊C4—5381no—
322C22⋊C4—5381no—
323C22⋊C4—5381no—
324C42—2691no—
325C42—5381no—
326C42—5381no—
327C23—21no—
328C23—2691no—
329C23—5381no—
330C23D4×D26911yescharacteristic
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—42no—
336Q8—5381no—
337Q8—5381no—
338Q8—10762no—
339Q8C2×C22⋊D26912yes—
340Q8C2×C22⋊D26912yes—
341D4—21no—
342D4—21no—
343D4—42no—
344D4—5382no—
345D4—5381no—
346D4—5381no—
347D4—10762no—
348D4—10762no—
349D4—21521no—
350D4—21521no—
351D4—21521no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—2691no—
360C2×C4—2691no—
361C2×C4—2691no—
362C2×C4—5381no—
363C2×C4—5381no—
364C2×C4—5381no—
365C2×C4—5381no—
366C2×C4—5381no—
367C2×C4—10762no—
368C2×C4—10762no—
369C2×C4D4×D26911yescharacteristic
370C2×C4C23×D26911yescharacteristic
371C8—21no—
372C8—5381no—
373C8—5381no—
374C8C2×C22⋊D26912yes—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—5381no—
382C22—5381no—
383C22—5381no—
384C22—10761no—
385C22—10761no—
386C22—10761no—
387C22C2×D4×D26911yescharacteristic
388C4—21no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—2692no—
393C4—5381no—
394C4—5381no—
395C4—10761no—
396C4—10761no—
397C4—10761no—
398C4C22×C22⋊D26911yescharacteristic
399C4#2257549811yescharacteristic
400C2—41no—
401C2—81no—
402C2—10761no—
403C2—21521no—
404C2D4×C22⋊D26911yescharacteristic
405C2#137×D26911yescharacteristic
406C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#137×C269C2
socleC22×C269C22×D4
prime radical#31771274C2
hypercenterC2×#137D269
hypocenterC269#3278
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22575485C2semidirect
3. Q8×C22⋊D269C2semidirect
4. #31770976C2semidirect
5. #31770851C2semidirect
6. #31770848C2semidirect
7. #31770813C2semidirect
8. #31770803C2semidirect
9. #31770767C2semidirect
10. #31770766C2semidirect
11. #31770744C2semidirect
12. #31770742C2semidirect
13. Q16×Dic269C2semidirect
14. #31770681C2semidirect
15. C2×#137×C269C2semidirect
117. Q16×C269D4semidirect
195. Dic269#137semidirect
203. C269#3278semidirect
206. C2×#137D269semidirect
296. Q16C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754852
#317712742
Q8×C22⋊D2692
#317709762
#317708512
#317708482
#317708132
#317708032
#317707672
#317707662
#317707442
#317707422
Q16×Dic2692
#317706812
C2×#137×C2692
#3278269