Finite Group Databasebeta

← #35228148

Subgroups — #35228148

Order 20864.

1094 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782390C211yesmaximal, characteristic, complemented
2#23782386C212yesmaximal, complemented
3#23782385C211yesmaximal, characteristic, complemented
4#23782352C214yesmaximal, complemented
5#23782350C212yesmaximal, complemented
6D4×Dic326C211yesmaximal, characteristic, complemented
7#23782321C212yesmaximal, complemented
8#23782314C212yesmaximal, complemented
9#23782220C211yesmaximal, characteristic, complemented
10#23782217C212yesmaximal, complemented
11#23782196C212yesmaximal, complemented
12#23782192C214yesmaximal, complemented
13#41×Dic163C212yesmaximal, complemented
14C4×#14579698C211yesmaximal, characteristic, complemented
15C4×#14579698C211yesmaximal, characteristic, complemented
16C2×#19104700C212yesmaximal, complemented
17#547×C163C211yesmaximal, characteristic, complemented
18C2×#14579698C2211yescharacteristic
19C2×#14579698C2211yescharacteristic
20C22×Dic326C2212yes—
21#19104735C2211yescharacteristic
22#19104734C2212yes—
23#19104733C2212yes—
24#19104730C2214yes—
25#19104728C2214yes—
26#19104728C2214yes—
27#19104726C2212yes—
28#19104726C2212yes—
29#19104725C2214yes—
30#19104725C2214yes—
31#19104722C2212yes—
32#19104721C2214yes—
33#19104718C2214yes—
34#19104716C2214yes—
35#19104715C2212yes—
36#19104714C2212yes—
37#19104714C2212yes—
38#19104712C2214yes—
39#19104712C2214yes—
40#19104711C2212yes—
41#19104707C2214yes—
42Q8×Dic163C2212yes—
43Q8×Dic163C2212yes—
44C4×Dic326C2211yescharacteristic
45C4×Dic326C2211yescharacteristic
46#19104700C2214yes—
47#19104700C2212yes—
48#19104700C2211yescharacteristic
49#19104700C2212yes—
50#19104700C2212yes—
51#19104700C2214yes—
52#19104700C2211yescharacteristic
53D4×Dic163C2214yes—
54D4×Dic163C2211yescharacteristic
55D4×Dic163C2212yes—
56D4×Dic163C2214yes—
57D4×Dic163C2211yescharacteristic
58C4×C22⋊D163C2212yes—
59C4×C22⋊D163C2212yes—
60(C22⋊D163)⋊C4C2212yes—
61(C22⋊D163)⋊C4C2212yes—
62#19104694C2214yes—
63#19104694C2212yes—
64#19104693C2212yes—
65#19104693C2211yescharacteristic
66#19104693C2211yescharacteristic
67#19104691C2214yes—
68#19104691C2212yes—
69#19104691C2212yes—
70C2×Dic163⋊C4C2214yes—
71C2×C4×Dic163C2214yes—
72C2×C4×Dic163C2212yes—
73C2×C4×Dic163C2212yes—
74C42×D163C2211yescharacteristic
75C2×#41×C163C2212yes—
76C4⋊Q8×C163C2211yescharacteristic
77#123×C163C2214yes—
78C22⋊Q8×C163C2212yes—
79#121×C163C2212yes—
80C4×D4×C163C2211yescharacteristic
81C4×D4×C163C2211yescharacteristic
82C2×C4⋊C4×C163C2212yes—
83#14579698—24no—
84#14579698—24no—
85C2×Dic326—28no—
86C2×Dic326C2314yes—
87C2×Dic326C2312yes—
88C2×Dic326C2311yescharacteristic
89C2×Dic326C2311yescharacteristic
90C2×C22⋊D163C2312yes—
91C2×C22⋊D163C2312yes—
92C22×Dic163C2314yes—
93C22×Dic163C2312yes—
94C22×Dic163C2312yes—
95C22×Dic163C2312yes—
96C22×Dic163C2312yes—
97C2×C4×D163C2311yescharacteristic
98C2×C4×D163C2311yescharacteristic
99C2×C4×D163C2311yescharacteristic
100C4⋊Dic163C2311yescharacteristic
101C4⋊Dic163C2311yescharacteristic
102C4⋊Dic163C2312yes—
103C4⋊Dic163C2312yes—
104C4⋊Dic163C2314yes—
105C4⋊Dic163C2314yes—
106Dic163⋊C4C2312yes—
107Dic163⋊C4C2314yes—
108Dic163⋊C4C2314yes—
109Dic163⋊C4C2314yes—
110Dic163⋊C4C2312yes—
111Dic163⋊C4C2312yes—
112Dic163⋊C4C2312yes—
113Dic163⋊C4C2314yes—
114Dic163⋊C4C2314yes—
115C22⋊Dic163C2312yes—
116C22⋊Dic163C2312yes—
117C22⋊Dic163C2314yes—
118C22⋊Dic163C2314yes—
119C22⋊Dic163C2314yes—
120(C2×D163)⋊C4C2312yes—
121(C2×D163)⋊C4C2312yes—
122(C2×D163)⋊C4C2314yes—
123C4×Dic163C2311yescharacteristic
124C4×Dic163C2311yescharacteristic
125C4×Dic163C2311yescharacteristic
126C4×Dic163C2312yes—
127C4×Dic163C2312yes—
128C4×Dic163C2312yes—
129C4×Dic163C2314yes—
130C4×Dic163C2312yes—
131C4×Dic163C2314yes—
132C4×Dic163C2312yes—
133C4×Dic163C2312yes—
134C4×Dic163C2314yes—
135#41×C163—28no—
136C2×Q8×C163C2312yes—
137C2×D4×C163C2311yescharacteristic
138C2×D4×C163C2314yes—
139C2×D4×C163C2311yescharacteristic
140C22×C4×C163C2312yes—
141C22×C4×C163C2314yes—
142C22×C4×C163C2312yes—
143C4⋊C4×C163C2311yescharacteristic
144C4⋊C4×C163C2312yes—
145C4⋊C4×C163C2314yes—
146C4⋊C4×C163C2311yescharacteristic
147C4⋊C4×C163C2312yes—
148C22⋊C4×C163C2312yes—
149C22⋊C4×C163C2312yes—
150C22⋊C4×C163C2314yes—
151C42×C163C2311yescharacteristic
152C22×D163C2411yescharacteristic
153Dic326—24no—
154Dic326—22no—
155Dic326—28no—
156Dic326#4114yescomplemented
157C22⋊D163—24no—
158C22⋊D163—24no—
159C2×Dic163—24no—
160C2×Dic163—28no—
161C2×Dic163—24no—
162C2×Dic163—24no—
163C2×Dic163—24no—
164C2×Dic163C2412yes—
165C2×Dic163C2411yescharacteristic
166C2×Dic163C2411yescharacteristic
167C2×Dic163C2412yes—
168C2×Dic163C2414yes—
169C2×Dic163C2412yes—
170C2×Dic163C2412yes—
171C2×Dic163C2411yescharacteristic
172C4×D163—22no—
173C4×D163—22no—
174C4×D163—22no—
175C23×C163C2412yes—
176C23×C163C2412yes—
177Q8×C163—24no—
178D4×C163—28no—
179D4×C163—22no—
180D4×C163#4114yescomplemented
181C2×C4×C163—28no—
182C2×C4×C163—24no—
183C2×C4×C163—24no—
184C2×C4×C163C2411yescharacteristic
185C2×C4×C163C2412yes—
186C2×C4×C163C2412yes—
187C2×C4×C163C2414yes—
188C2×C4×C163C2411yescharacteristic
189C2×C4×C163C2411yescharacteristic
190C22×C163—22no—
191C22×C163—22no—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163—22no—
195C22×C163C2×#4114yes—
196C22×C163C2511yescharacteristic
197C2×D163—21no—
198C2×D163—21no—
199C2×D163—21no—
200C4×C163—22no—
201C4×C163—24no—
202C4×C163—22no—
203C4×C163—21no—
204C4×C163C2×#4112yes—
205C4×C163C2×#4112yes—
206Dic163—21no—
207Dic163—21no—
208Dic163—22no—
209Dic163—24no—
210Dic163—22no—
211Dic163—22no—
212Dic163C2×#4114yes—
213Dic1632−1+412yes—
214C2×C163—24no—
215C2×C163—42no—
216C2×C163C22×#4111yescharacteristic
217C2×C163C22×#4111yescharacteristic
218C2×C163C2×2−1+411yescharacteristic
219D163—41no—
220C163#348211yescharacteristic, complemented, Sylow
221#3482—1631nomaximal, Sylow
222#557—1631no—
223#556—1632no—
224#556—1631no—
225#552—1634no—
226#552—1632no—
227D4×Q8—1631no—
228#548—1632no—
229#547—1632no—
230#547D16311yescharacteristic, complemented
231#533—1631no—
232#533—1632no—
233#529—1634no—
234#529—1632no—
235C4×#41—1632no—
236C4×#41—1631no—
237C4×#41—1631no—
238C2×C4×Q8—1632no—
239C2×#41—1631no—
240C2×#41—1631no—
241C2×#41C2×D16312yes—
242C22×Q8—1632no—
243C4⋊Q8—1632no—
244C4⋊Q8—1632no—
245C4⋊Q8—1631no—
246C4⋊Q8C2×D16311yescharacteristic
247#126—1634no—
248#126—1634no—
249#126—1634no—
250#125—1634no—
251#125—1632no—
252#125—1634no—
253#125—1632no—
254#124—1632no—
255#124—1634no—
256#123—1634no—
257#123—1634no—
258#123C2×D16314yes—
259C22⋊Q8—1632no—
260C22⋊Q8—1632no—
261C22⋊Q8—1632no—
262C22⋊Q8—1632no—
263C22⋊Q8—1634no—
264C22⋊Q8—1634no—
265C22⋊Q8C2×D16312yes—
266#121—1634no—
267#121C2×D16312yes—
268C4×Q8—1632no—
269C4×Q8—1632no—
270C4×Q8—1631no—
271C4×Q8—1634no—
272C4×Q8—1632no—
273C4×Q8—1632no—
274C4×Q8—1634no—
275C4×Q8—1632no—
276C4×Q8—1631no—
277C4×Q8—1631no—
278C4×Q8—1631no—
279C4×D4—1631no—
280C4×D4—1631no—
281C4×D4—1634no—
282C4×D4—1632no—
283C4×D4—1634no—
284C4×D4—1632no—
285C4×D4—1632no—
286C4×D4—1632no—
287C4×D4—1632no—
288C4×D4C2×D16311yescharacteristic
289C4×D4C2×D16311yescharacteristic
290#117—1634no—
291#117—1632no—
292#117—1631no—
293#117—1631no—
294#117—1632no—
295#117—1632no—
296#117—1632no—
297#117—1634no—
298C2×C4⋊C4—1634no—
299C2×C4⋊C4C2×D16312yes—
300C2×C42—1632no—
301C2×C42—1634no—
302C2×C42—1632no—
303C2×C42—1631no—
304#41—28no—
305#41—3264no—
306#41—3264no—
307C2×Q8—3268no—
308C2×Q8—1632no—
309C2×Q8—1634no—
310C2×Q8—1631no—
311C2×Q8—1631no—
312C2×Q8C22×D16312yes—
313C2×D4—1632no—
314C2×D4—1632no—
315C2×D4C22×D16314yes—
316C2×D4C22×D16311yescharacteristic
317C2×D4C22×D16311yescharacteristic
318C22×C4—1632no—
319C22×C4—1632no—
320C22×C4—1634no—
321C22×C4—1632no—
322C22×C4—1632no—
323C22×C4—1631no—
324C22×C4—1631no—
325C22×C4—1631no—
326C22×C4C22×D16314yes—
327C22×C4C22×D16312yes—
328C22×C4C22×D16312yes—
329C4⋊C4—1632no—
330C4⋊C4—1634no—
331C4⋊C4—1634no—
332C4⋊C4—1634no—
333C4⋊C4—1631no—
334C4⋊C4—1632no—
335C4⋊C4—1632no—
336C4⋊C4—1634no—
337C4⋊C4—1631no—
338C4⋊C4—1632no—
339C4⋊C4—1632no—
340C4⋊C4—1634no—
341C4⋊C4—1634no—
342C4⋊C4—1632no—
343C4⋊C4—1634no—
344C4⋊C4C22×D16311yescharacteristic
345C4⋊C4C22×D16312yes—
346C4⋊C4C22×D16314yes—
347C4⋊C4C22×D16311yescharacteristic
348C4⋊C4C22×D16312yes—
349C22⋊C4—1632no—
350C22⋊C4—1634no—
351C22⋊C4—1632no—
352C22⋊C4—1634no—
353C22⋊C4—1634no—
354C22⋊C4—1632no—
355C22⋊C4—1632no—
356C22⋊C4—1634no—
357C22⋊C4C22×D16312yes—
358C22⋊C4C22×D16312yes—
359C22⋊C4C22×D16314yes—
360C42—1631no—
361C42—1631no—
362C42—1632no—
363C42—1631no—
364C42—1632no—
365C42—1634no—
366C42—1632no—
367C42—1632no—
368C42—1634no—
369C42—1632no—
370C42—1632no—
371C42—1634no—
372C42C22×D16311yescharacteristic
373C23—1631no—
374C23C23×D16312yes—
375C23C23×D16312yes—
376Q8—24no—
377Q8—3268no—
378Q8—3264no—
379Q8—3262no—
380Q8—1634no—
381D4—28no—
382D4—22no—
383D4—3264no—
384D4—3264no—
385D4#1457969814yescomplemented
386C2×C4—28no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—3262no—
390C2×C4—3264no—
391C2×C4—3268no—
392C2×C4—3264no—
393C2×C4—3264no—
394C2×C4—3264no—
395C2×C4—3262no—
396C2×C4—3262no—
397C2×C4—1632no—
398C2×C4—1631no—
399C2×C4—1631no—
400C2×C4—1632no—
401C2×C4—1634no—
402C2×C4—1632no—
403C2×C4—1631no—
404C2×C4—1632no—
405C2×C4C23×D16312yes—
406C2×C4C23×D16312yes—
407C2×C4C23×D16314yes—
408C2×C4C23×D16311yescharacteristic
409C2×C4C23×D16311yescharacteristic
410C2×C4C23×D16311yescharacteristic
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—3261no—
417C22—3261no—
418C22—3261no—
419C22C2×#1457969814yes—
420C22C24×D16311yescharacteristic
421C4—22no—
422C4—22no—
423C4—24no—
424C4—21no—
425C4—3261no—
426C4—3264no—
427C4—3261no—
428C4—3262no—
429C4—3262no—
430C4—3262no—
431C4—1634no—
432C4—1632no—
433C4#41×D16312yes—
434C4C2×#1457969812yes—
435C2—24no—
436C2—42no—
437C2—6521no—
438C2C2×#41×D16311yescharacteristic
439C2C22×#1457969811yescharacteristic
440C22−1+4×D16311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#547×C163C2
hypercenter#547D163
hypocenterC163#3482
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #23782390C2semidirect
2. #23782386C2semidirect
3. #23782385C2semidirect
4. #23782352C2semidirect
5. #23782350C2semidirect
6. D4×Dic326C2semidirect
7. #23782321C2semidirect
8. #23782314C2semidirect
9. #23782220C2semidirect
10. #23782217C2semidirect
11. #23782196C2semidirect
12. #23782192C2semidirect
13. #41×Dic163C2semidirect
14. C4×#14579698C2semidirect
15. C4×#14579698C2semidirect
16. C2×#19104700C2semidirect
17. #547×C163C2semidirect
156. Dic326#41semidirect
180. D4×C163#41semidirect
220. C163#3482semidirect
230. #547D163semidirect
385. D4#14579698semidirect

Maximal subgroups

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

subgroupindex
#237823902
#237823862
#237823852
#237823522
#237823502
D4×Dic3262
#237823212
#237823142
#237822202
#237822172
#237821962
#237821922
#41×Dic1632
C4×#145796982
C2×#191047002
#547×C1632
#3482163