Finite Group Databasebeta

← #35218023

Subgroups — #35218023

Order 20864.

596 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104708C211yesmaximal, characteristic
2C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic, complemented
3#23781594C211yesmaximal, characteristic, complemented
4#23781582C211yesmaximal, characteristic, complemented
5#23781580C211yesmaximal, characteristic, complemented
6#23781565C211yesmaximal, characteristic, complemented
7#23781560C211yesmaximal, characteristic, complemented
8#23781543C211yesmaximal, characteristic, complemented
9#23781542C211yesmaximal, characteristic, complemented
10#23781529C211yesmaximal, characteristic, complemented
11#23781527C211yesmaximal, characteristic, complemented
12#23781526C211yesmaximal, characteristic, complemented
13#23781524C211yesmaximal, characteristic, complemented
14#23781512C211yesmaximal, characteristic, complemented
15#397×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C23×Dic163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21C22×C4×D163C2211yescharacteristic
22#19104708—24no—
23(C22⋊D163)⋊C4—24no—
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×C22⋊Dic163C2211yescharacteristic
29C2×(C2×D163)⋊C4C2211yescharacteristic
30C2×(C2×D163)⋊C4C2211yescharacteristic
31C2×(C2×D163)⋊C4C2211yescharacteristic
32C2×(C2×D163)⋊C4C2211yescharacteristic
33C2×(C2×D163)⋊C4C2211yescharacteristic
34C2×(C2×D163)⋊C4C2211yescharacteristic
35C2×(C2×D163)⋊C4C2211yescharacteristic
36C2×(C2×D163)⋊C4C2211yescharacteristic
37C2×C4×Dic163C2211yescharacteristic
38C2×C4×Dic163C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C22⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C23×D163C2311yescharacteristic
54C23×D163C2311yescharacteristic
55C2×D652—22no—
56C2×D652—22no—
57C2×D652—41no—
58C2×C22⋊D163—22no—
59C2×C22⋊D163—22no—
60C2×C22⋊D163—22no—
61C2×C22⋊D163—22no—
62C2×C22⋊D163—41no—
63C2×C22⋊D163D414yes—
64C22×Dic163—22no—
65C22×Dic163—22no—
66C22×Dic163—41no—
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C22×Dic163C2311yescharacteristic
70C22×Dic163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C22×Dic163C2311yescharacteristic
73C2×C4×D163—22no—
74C2×C4×D163—22no—
75C2×C4×D163—22no—
76C2×C4×D163—22no—
77C2×C4×D163—22no—
78C2×C4×D163—41no—
79Dic163⋊C4—22no—
80Dic163⋊C4—22no—
81Dic163⋊C4—22no—
82Dic163⋊C4—41no—
83C22⋊Dic163—41no—
84(C2×D163)⋊C4—22no—
85(C2×D163)⋊C4—22no—
86(C2×D163)⋊C4—22no—
87(C2×D163)⋊C4—22no—
88(C2×D163)⋊C4—41no—
89(C2×D163)⋊C4—41no—
90(C2×D163)⋊C4—41no—
91(C2×D163)⋊C4—41no—
92C4×Dic163—22no—
93C4×Dic163—22no—
94C24×C163C2311yescharacteristic
95C22×C4×C163C2311yescharacteristic
96C22×C4×C163C2311yescharacteristic
97C22×C4×C163C2311yescharacteristic
98C22×C4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C4⋊C4×C163—41no—
102C22⋊C4×C163—22no—
103C22⋊C4×C163—22no—
104C22⋊C4×C163—41no—
105C22×D163—21no—
106C22×D163—21no—
107C22×D163—21no—
108C22×D163—21no—
109C22×D163—21no—
110C22×D163—21no—
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—21no—
114C22×D163—21no—
115C22×D163—21no—
116C22×D163—21no—
117C22×D163C2×D412yes—
118C22×D163#4112yes—
119D652—44no—
120C22⋊D163—28no—
121C22⋊D163—44no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163—21no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—44no—
137C2×Dic163C2×D412yes—
138C2×Dic163#4112yes—
139C2×Dic163#4112yes—
140C2×Dic163#4112yes—
141C4×D163—44no—
142C4×D163—44no—
143C23×C163—21no—
144C23×C163—21no—
145C23×C163—21no—
146C23×C163—21no—
147C23×C163—21no—
148C23×C163—21no—
149C23×C163C2×D412yes—
150C23×C163C2411yescharacteristic
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163—21no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163—21no—
169C22×C163—22no—
170C22×C163—22no—
171C22×C163—22no—
172C22×C163—41no—
173C22×C163—41no—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163C22×D411yescharacteristic
177C22×C163C2×#4111yescharacteristic
178C22×C163C2×#4111yescharacteristic
179C22×C163C2×#4111yescharacteristic
180C22×C163C2×#4111yescharacteristic
181C22×C163H(5,𝔽2)11yescharacteristic
182C22×C163H(5,𝔽2)11yescharacteristic
183C2×D163—22no—
184C2×D163—22no—
185C2×D163—22no—
186C2×D163—22no—
187C2×D163—22no—
188C2×D163—22no—
189C2×D163—41no—
190C2×D163—41no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—41no—
194C2×D163—41no—
195C2×D163—41no—
196C2×D163—41no—
197C4×C163—41no—
198C4×C163—41no—
199C4×C163—41no—
200C4×C163—41no—
201C4×C163—41no—
202C4×C163—41no—
203Dic163—22no—
204Dic163—41no—
205Dic163—41no—
206Dic163—41no—
207Dic163—41no—
208Dic163#12314yes—
209C2×C163—42no—
210C2×C163C2×#12311yescharacteristic
211C2×C163#52911yescharacteristic
212C2×C163#54011yescharacteristic
213C2×C163#54611yescharacteristic
214C2×C163#54611yescharacteristic
215C2×C163#55311yescharacteristic
216C2×C163#55511yescharacteristic
217D163—42no—
218D163—42no—
219C163#242011yescharacteristic, complemented, Sylow
220#2420—1631nomaximal, Sylow
221C2×#121—1631no—
222C2×C4×D4—1631no—
223#397—1631no—
224#397D16311yescharacteristic, complemented
225#394—1631no—
226#394—1631no—
227#390—1631no—
228#389—1631no—
229#386—1631no—
230#386—1631no—
231#382—1631no—
232#381—1631no—
233#381—1631no—
234#380—1631no—
235C4×C22⋊C4—1631no—
236C22×D4—1631no—
237C22×D4—1631no—
238C22×D4—1631no—
239C23×C4—1631no—
240C23×C4—1631no—
241C23×C4—1631no—
242#121—3264no—
243C4×D4—3264no—
244C2×C4⋊C4—1631no—
245C2×C4⋊C4—1631no—
246C2×C4⋊C4—1631no—
247C2×C4⋊C4—1631no—
248C2×C4⋊C4C2×D16311yescharacteristic
249C2×C22⋊C4—1631no—
250C2×C22⋊C4—1631no—
251C2×C22⋊C4—1631no—
252C2×C22⋊C4—1631no—
253C2×C22⋊C4—1631no—
254C2×C22⋊C4—1631no—
255C2×C22⋊C4—1631no—
256C2×C22⋊C4—1631no—
257C2×C22⋊C4—1631no—
258C2×C22⋊C4C2×D16311yescharacteristic
259C2×C22⋊C4C2×D16311yescharacteristic
260C2×C22⋊C4C2×D16311yescharacteristic
261C2×C42—1631no—
262C2×C42—1631no—
263#95—1631no—
264#95—1631no—
265#95—1631no—
266#95—1631no—
267#95—1631no—
268#95—1631no—
269#95—1631no—
270#95C2×D16311yescharacteristic
271#95C2×D16311yescharacteristic
272#95C2×D16311yescharacteristic
273C24—1631no—
274C24—1631no—
275C24C22×D16311yescharacteristic
276C2×D4—3262no—
277C2×D4—3262no—
278C2×D4—3262no—
279C2×D4—3262no—
280C2×D4—3262no—
281C2×D4—3262no—
282C2×D4—6521no—
283C2×D4—6521no—
284C2×D4—1634no—
285C22×C4—3262no—
286C22×C4—3262no—
287C22×C4—3262no—
288C22×C4—3262no—
289C22×C4—3262no—
290C22×C4—3262no—
291C22×C4—3262no—
292C22×C4—6521no—
293C22×C4—6521no—
294C22×C4—1631no—
295C22×C4—1631no—
296C22×C4—1631no—
297C22×C4—1631no—
298C22×C4—1631no—
299C22×C4—1631no—
300C22×C4C22×D16311yescharacteristic
301C22×C4C22×D16311yescharacteristic
302C22×C4C22×D16311yescharacteristic
303C22×C4C22×D16311yescharacteristic
304C22×C4C22×D16311yescharacteristic
305C22×C4C22×D16311yescharacteristic
306C4⋊C4—41no—
307C4⋊C4—3262no—
308C4⋊C4—3262no—
309C4⋊C4—3262no—
310C4⋊C4—6521no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—41no—
314C22⋊C4—3262no—
315C22⋊C4—3262no—
316C22⋊C4—3262no—
317C22⋊C4—3262no—
318C22⋊C4—6521no—
319C22⋊C4—6521no—
320C22⋊C4—6521no—
321C22⋊C4—6521no—
322C22⋊C4—6521no—
323C42—3262no—
324C42—3262no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—3261no—
332C23—3261no—
333C23—3261no—
334C23—3261no—
335C23—3261no—
336C23—3261no—
337C23—3261no—
338C23—3261no—
339C23—3261no—
340C23—3261no—
341C23—3261no—
342C23—3261no—
343C23—1632no—
344C23—1632no—
345C23D4×D16312yes—
346C23C23×D16311yescharacteristic
347D4—3268no—
348D4—6524no—
349D4—6524no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
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—3261no—
369C2×C4—3261no—
370C2×C4—3261no—
371C2×C4—3261no—
372C2×C4—3261no—
373C2×C4—3261no—
374C2×C4—3261no—
375C2×C4—3261no—
376C2×C4—3261no—
377C2×C4—3261no—
378C2×C4—3261no—
379C2×C4—3261no—
380C2×C4—3261no—
381C2×C4—3261no—
382C2×C4—6524no—
383C2×C4—6524no—
384C2×C4—6524no—
385C2×C4—1632no—
386C2×C4—1632no—
387C2×C4—1632no—
388C2×C4—1632no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—3262no—
397C22—3262no—
398C22—3262no—
399C22—3262no—
400C22—3262no—
401C22—3262no—
402C22—6521no—
403C22—6521no—
404C22—6521no—
405C22—6521no—
406C22—6521no—
407C22—6521no—
408C22—6521no—
409C22—6521no—
410C22C2×D4×D16311yescharacteristic
411C22#41×D16311yescharacteristic
412C22#41×D16311yescharacteristic
413C22#41×D16311yescharacteristic
414C22#41×D16311yescharacteristic
415C22#1910478011yescharacteristic
416C22#1910478111yescharacteristic
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—3262no—
424C4—6521no—
425C4—6521no—
426C4—6521no—
427C4—6521no—
428C4—1634no—
429C2—42no—
430C2—6522no—
431C2—6522no—
432C2#123×D16311yescharacteristic
433C2#2378219811yescharacteristic
434C2#2378226711yescharacteristic
435C2#2378230411yescharacteristic
436C2#2378230711yescharacteristic
437C2#2378236611yescharacteristic
438C2#2378237711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#397×C163C2
prime radicalC2×#19104708C2
hypercenter#397D163
hypocenterC163#2420
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D163)⋊C4C2semidirect
3. #23781594C2semidirect
4. #23781582C2semidirect
5. #23781580C2semidirect
6. #23781565C2semidirect
7. #23781560C2semidirect
8. #23781543C2semidirect
9. #23781542C2semidirect
10. #23781529C2semidirect
11. #23781527C2semidirect
12. #23781526C2semidirect
13. #23781524C2semidirect
14. #23781512C2semidirect
15. #397×C163C2semidirect
219. C163#2420semidirect
224. #397D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047082
C2×(C22⋊D163)⋊C42
#237815942
#237815822
#237815802
#237815652
#237815602
#237815432
#237815422
#237815292
#237815272
#237815262
#237815242
#237815122
#397×C1632
#2420163