Finite Group Databasebeta

← #46813571

Subgroups — #46813571

Order 34432.

668 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575425C211yesmaximal, characteristic
2C2×#22575413C211yesmaximal, characteristic, complemented
3C2×#22575409C211yesmaximal, characteristic, complemented
4#31770505C211yesmaximal, characteristic, complemented
5#31770505C211yesmaximal, characteristic, complemented
6#31770500C211yesmaximal, characteristic, complemented
7#31770499C211yesmaximal, characteristic, complemented
8#31770496C211yesmaximal, characteristic, complemented
9#31770485C211yesmaximal, characteristic, complemented
10#31770481C211yesmaximal, characteristic, complemented
11#31770477C211yesmaximal, characteristic, complemented
12#31770476C211yesmaximal, characteristic, complemented
13#31770463C211yesmaximal, characteristic, complemented
14C4×#16864381C211yesmaximal, characteristic, complemented
15#390×C269C211yesmaximal, characteristic, complemented
16C22×C22⋊D269C2211yescharacteristic
17C22×C22⋊D269C2211yescharacteristic
18C23×Dic269C2211yescharacteristic
19C23×Dic269C2211yescharacteristic
20C22×C4×D269C2211yescharacteristic
21#22575425C418yescomplemented
22#22575413—24no—
23#22575409—24no—
24C2×#16864381C2211yescharacteristic
25C2×#16864381C2211yescharacteristic
26C2×#16864381C2211yescharacteristic
27C2×#16864381C2211yescharacteristic
28C2×#16864381C2211yescharacteristic
29C2×#16864381C2211yescharacteristic
30C2×C22⋊Dic269C2211yescharacteristic
31C2×C22⋊Dic269C2211yescharacteristic
32C2×C22⋊Dic269C2211yescharacteristic
33C2×C22⋊Dic269C2211yescharacteristic
34C2×#16864379C2211yescharacteristic
35C2×#16864379C2211yescharacteristic
36C2×#16864379C2211yescharacteristic
37C2×#16864379C2211yescharacteristic
38C2×C4×Dic269C2211yescharacteristic
39C2×C4×Dic269C2211yescharacteristic
40C2×C4×Dic269C2211yescharacteristic
41#22575360C2211yescharacteristic
42#22575360C2211yescharacteristic
43#22575360C2211yescharacteristic
44#22575360C2211yescharacteristic
45#22575360C2211yescharacteristic
46#22575360C2211yescharacteristic
47C22×D4×C269C2211yescharacteristic
48C2×C4⋊C4×C269C2211yescharacteristic
49C2×C22⋊C4×C269C2211yescharacteristic
50C2×C22⋊C4×C269C2211yescharacteristic
51C2×C22⋊C4×C269C2211yescharacteristic
52C2×C22⋊C4×C269C2211yescharacteristic
53C2×C42×C269C2211yescharacteristic
54C23×D269C2311yescharacteristic
55C2×C22⋊D269—22no—
56C2×C22⋊D269—22no—
57C2×C22⋊D269—41no—
58C2×C22⋊D269—41no—
59C2×C22⋊D269C2×C414yes—
60C2×C22⋊D269C2×C414yescomplemented
61C22×Dic269—22no—
62C22×Dic269—22no—
63C22×Dic269—22no—
64C22×Dic269—41no—
65C22×Dic269—41no—
66C22×Dic269C2×C414yescomplemented
67C22×Dic269C2311yescharacteristic
68C22×Dic269C2311yescharacteristic
69C22×Dic269C2311yescharacteristic
70C22×Dic269C2311yescharacteristic
71C22×Dic269C2311yescharacteristic
72C22×Dic269C2311yescharacteristic
73C22×Dic269C2311yescharacteristic
74C2×C4×D269—22no—
75C2×C4×D269—41no—
76C2×C4×D269—41no—
77#16864381—22no—
78#16864381—22no—
79#16864381—22no—
80#16864381—22no—
81#16864381—41no—
82#16864381C2×C414yescomplemented
83C22⋊Dic269—22no—
84C22⋊Dic269—41no—
85C22⋊Dic269—41no—
86C22⋊Dic269C2×C414yescomplemented
87#16864379—22no—
88#16864379—41no—
89#16864379—41no—
90#16864379C2×C414yescomplemented
91C4×Dic269—22no—
92C4×Dic269—41no—
93C4×Dic269D414yescomplemented
94C24×C269C2311yescharacteristic
95C24×C269C2311yescharacteristic
96C2×D4×C269—41no—
97C2×D4×C269—41no—
98C2×D4×C269C2×C414yescomplemented
99C22×C4×C269C2311yescharacteristic
100C22×C4×C269C2311yescharacteristic
101C22×C4×C269C2311yescharacteristic
102C22×C4×C269C2311yescharacteristic
103C22×C4×C269C2311yescharacteristic
104C4⋊C4×C269—41no—
105C22⋊C4×C269—22no—
106C22⋊C4×C269—22no—
107C22⋊C4×C269—41no—
108C22⋊C4×C269—41no—
109C42×C269—41no—
110C22×D269—21no—
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—21no—
115C22×D269—21no—
116C22×D269C22×C412yes—
117C22⋊D269—28no—
118C22⋊D269—44no—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—21no—
122C2×Dic269—21no—
123C2×Dic269—21no—
124C2×Dic269—21no—
125C2×Dic269—28no—
126C2×Dic269—21no—
127C2×Dic269—21no—
128C2×Dic269—21no—
129C2×Dic269—21no—
130C2×Dic269—21no—
131C2×Dic269—21no—
132C2×Dic269—21no—
133C2×Dic269—21no—
134C2×Dic269—21no—
135C2×Dic269—44no—
136C2×Dic269C22×C412yes—
137C2×Dic269C22×C412yes—
138C2×Dic269C22×C412yes—
139C2×Dic269C2×D412yes—
140C2×Dic269C2×D412yes—
141C2×Dic269#4112yes—
142C4×D269—44no—
143C23×C269—21no—
144C23×C269—21no—
145C23×C269—21no—
146C23×C269—21no—
147C23×C269—21no—
148C23×C269—21no—
149C23×C269—21no—
150C23×C269—21no—
151C23×C269—21no—
152C23×C269—21no—
153C23×C269—21no—
154C23×C269—21no—
155C23×C269C22×C412yes—
156C23×C269C22×C412yes—
157C23×C269C2411yescharacteristic
158D4×C269—44no—
159C2×C4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—21no—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—21no—
166C2×C4×C269—21no—
167C2×C4×C269—21no—
168C2×C4×C269—21no—
169C2×C4×C269—21no—
170C2×C4×C269—21no—
171C2×C4×C269—21no—
172C2×C4×C269C22×C412yes—
173C2×C4×C269C2×D412yes—
174C22×C269—22no—
175C22×C269—22no—
176C22×C269—22no—
177C22×C269—22no—
178C22×C269—22no—
179C22×C269—22no—
180C22×C269—41no—
181C22×C269—41no—
182C22×C269—41no—
183C22×C269—41no—
184C22×C269—41no—
185C22×C269—41no—
186C22×C269—41no—
187C22×C269—41no—
188C22×C269C23×C411yescharacteristic
189C22×C269C22×D411yescharacteristic
190C22×C269C2×#4111yescharacteristic
191C22×C269H(5,𝔽2)11yescharacteristic
192C22×C269H(5,𝔽2)11yescharacteristic
193C22×C269H(5,𝔽2)11yescharacteristic
194C22×C2692−1+411yescharacteristic
195C2×D269—22no—
196C2×D269—22no—
197C2×D269—22no—
198C2×D269—41no—
199C2×D269—41no—
200C2×D269—41no—
201C2×D269—41no—
202C4×C269—22no—
203C4×C269—22no—
204C4×C269—41no—
205C4×C269—41no—
206C4×C269—41no—
207Dic269—22no—
208Dic269—22no—
209Dic269—22no—
210Dic269—41no—
211Dic269—41no—
212Dic269—41no—
213Dic269C4×D414yes—
214C2×C269—42no—
215C2×C269—42no—
216C2×C269C2×C4×D411yescharacteristic
217C2×C269#51811yescharacteristic
218C2×C269#52011yescharacteristic
219C2×C269#53511yescharacteristic
220C2×C269#53711yescharacteristic
221C2×C269#54011yescharacteristic
222C2×C269#54211yescharacteristic
223D269—42no—
224C269#232411yescharacteristic, complemented, Sylow
225#2324—2691nomaximal, Sylow
226C2×#121—2691no—
227C2×C4×D4—2691no—
228C2×#117—2691no—
229#390—2691no—
230#390—2691no—
231#390D26911yescharacteristic, complemented
232#389—2691no—
233#388—2691no—
234#388—2691no—
235#386—2691no—
236#386—2691no—
237#385—2691no—
238#385—2691no—
239#380—2691no—
240C4×C4⋊C4—2691no—
241C22×D4—2691no—
242C22×D4—2691no—
243C22×D4C2×D26911yescharacteristic
244C23×C4—2691no—
245C23×C4—2691no—
246C23×C4—2691no—
247#121—2698no—
248C4×D4—5384no—
249#117—5384no—
250C2×C4⋊C4—2691no—
251C2×C4⋊C4—2691no—
252C2×C4⋊C4—2691no—
253C2×C4⋊C4—2691no—
254C2×C4⋊C4—2691no—
255C2×C4⋊C4—2691no—
256C2×C4⋊C4C2×D26911yescharacteristic
257C2×C22⋊C4—2691no—
258C2×C22⋊C4—2691no—
259C2×C22⋊C4—2691no—
260C2×C22⋊C4—2691no—
261C2×C22⋊C4—2691no—
262C2×C22⋊C4—2691no—
263C2×C22⋊C4—2691no—
264C2×C22⋊C4—2691no—
265C2×C22⋊C4C2×D26911yescharacteristic
266C2×C22⋊C4C2×D26911yescharacteristic
267C2×C22⋊C4C2×D26911yescharacteristic
268C2×C22⋊C4C2×D26911yescharacteristic
269C2×C42—2691no—
270C2×C42—2691no—
271C2×C42—2691no—
272C2×C42C2×D26911yescharacteristic
273#95—2691no—
274#95—2691no—
275#95—2691no—
276#95—2691no—
277#95—2691no—
278#95—2691no—
279C24—2691no—
280C24C22×D26911yescharacteristic
281C24C22×D26911yescharacteristic
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—2694no—
285C2×D4—2694no—
286C2×D4—5382no—
287C2×D4—5382no—
288C2×D4—10761no—
289C2×D4—10761no—
290C2×D4C4×D26914yescomplemented
291C22×C4—2691no—
292C22×C4—2694no—
293C22×C4—2691no—
294C22×C4—2691no—
295C22×C4—2691no—
296C22×C4—2691no—
297C22×C4—2691no—
298C22×C4—2691no—
299C22×C4—5382no—
300C22×C4—5382no—
301C22×C4—5382no—
302C22×C4—5382no—
303C22×C4—10761no—
304C22×C4—10761no—
305C22×C4—10761no—
306C22×C4—10761no—
307C22×C4C22×D26911yescharacteristic
308C22×C4C22×D26911yescharacteristic
309C22×C4C22×D26911yescharacteristic
310C22×C4C22×D26911yescharacteristic
311C22×C4C22×D26911yescharacteristic
312C4⋊C4—41no—
313C4⋊C4—2694no—
314C4⋊C4—5382no—
315C4⋊C4—5382no—
316C4⋊C4—5382no—
317C4⋊C4—5382no—
318C4⋊C4—10761no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—2694no—
324C22⋊C4—2694no—
325C22⋊C4—5382no—
326C22⋊C4—5382no—
327C22⋊C4—10761no—
328C22⋊C4—10761no—
329C22⋊C4—10761no—
330C22⋊C4—10761no—
331C42—41no—
332C42—2694no—
333C42—5382no—
334C42—10761no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—2692no—
348C23—5381no—
349C23—5381no—
350C23—5381no—
351C23—5381no—
352C23—5381no—
353C23—5381no—
354C23C2×C4×D26912yes—
355C23C2×C4×D26912yes—
356C23C23×D26911yescharacteristic
357D4—44no—
358D4—5388no—
359D4—10764no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—2692no—
374C2×C4—2692no—
375C2×C4—2692no—
376C2×C4—2692no—
377C2×C4—2692no—
378C2×C4—2692no—
379C2×C4—5388no—
380C2×C4—5381no—
381C2×C4—5381no—
382C2×C4—5381no—
383C2×C4—5381no—
384C2×C4—5381no—
385C2×C4—5381no—
386C2×C4—5381no—
387C2×C4—5381no—
388C2×C4—5381no—
389C2×C4—5381no—
390C2×C4—5381no—
391C2×C4—5381no—
392C2×C4—5381no—
393C2×C4—5381no—
394C2×C4—5381no—
395C2×C4—10764no—
396C2×C4—10764no—
397C2×C4C2×C4×D26912yes—
398C2×C4D4×D26912yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—5382no—
414C22—5382no—
415C22—5382no—
416C22—10761no—
417C22—10761no—
418C22—10761no—
419C22—10761no—
420C22C22×C4×D26911yescharacteristic
421C22C2×D4×D26911yescharacteristic
422C22#41×D26911yescharacteristic
423C22#2257549811yescharacteristic
424C22#2257549811yescharacteristic
425C22#2257549811yescharacteristic
426C22#2257550011yescharacteristic
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—2694no—
433C4—5382no—
434C4—5382no—
435C4—5382no—
436C4—10761no—
437C4—10761no—
438C4—10761no—
439C2—42no—
440C2—42no—
441C2—10762no—
442C2C4×D4×D26911yescharacteristic
443C2#3177107811yescharacteristic
444C2#3177108411yescharacteristic
445C2#3177116811yescharacteristic
446C2#3177118211yescharacteristic
447C2#3177120511yescharacteristic
448C2#3177121911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
Fitting#390×C269C2
socle, cosocleC23×C269C24
prime radicalC2×#22575425C2
hypercenter#390D269
hypocenterC269#2324
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#22575413C2semidirect
3. C2×#22575409C2semidirect
4. #31770505C2semidirect
5. #31770505C2semidirect
6. #31770500C2semidirect
7. #31770499C2semidirect
8. #31770496C2semidirect
9. #31770485C2semidirect
10. #31770481C2semidirect
11. #31770477C2semidirect
12. #31770476C2semidirect
13. #31770463C2semidirect
14. C4×#16864381C2semidirect
15. #390×C269C2semidirect
21. #22575425C4semidirect
60. C2×C22⋊D269C2×C4semidirect
66. C22×Dic269C2×C4semidirect
82. #16864381C2×C4semidirect
86. C22⋊Dic269C2×C4semidirect
90. #16864379C2×C4semidirect
93. C4×Dic269D4semidirect
98. C2×D4×C269C2×C4semidirect
224. C269#2324semidirect
231. #390D269semidirect
290. C2×D4C4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754252
C2×#225754132
C2×#225754092
#317705052
#317705002
#317704992
#317704962
#317704852
#317704812
#317704772
#317704762
#317704632
C4×#168643812
#390×C2692
#2324269