Finite Group Databasebeta

← #35218093

Subgroups — #35218093

Order 20864.

666 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104716C211yesmaximal, characteristic, complemented
2C2×#19104716C211yesmaximal, characteristic, complemented
3C2×D4×Dic163C211yesmaximal, characteristic, complemented
4#23781598C211yesmaximal, characteristic, complemented
5#23781593C211yesmaximal, characteristic, complemented
6#23781588C211yesmaximal, characteristic, complemented
7#23781579C211yesmaximal, characteristic, complemented
8#23781560C211yesmaximal, characteristic, complemented
9#23781559C211yesmaximal, characteristic, complemented
10#23781545C211yesmaximal, characteristic, complemented
11#23781545C211yesmaximal, characteristic, complemented
12#23781538C211yesmaximal, characteristic, complemented
13#23781523C211yesmaximal, characteristic, complemented
14C2×C163⋊#95C211yesmaximal, characteristic, complemented
15C2×#121×C163C211yesmaximal, characteristic
16C23×Dic163C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18C23×Dic163C2211yescharacteristic
19#19104716—24no—
20#19104716—24no—
21D4×Dic163—24no—
22C2×C4⋊Dic163C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×Dic163⋊C4C2211yescharacteristic
29C2×C22⋊Dic163C2211yescharacteristic
30C2×C22⋊Dic163C2211yescharacteristic
31C2×C22⋊Dic163C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×C22⋊Dic163C2211yescharacteristic
36C2×C22⋊Dic163C2211yescharacteristic
37C2×C22⋊Dic163C2211yescharacteristic
38C2×C4×Dic163C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C22×D4×C163C2211yescharacteristic
48C22×D4×C163C2211yescharacteristic
49C22×D4×C163C2211yescharacteristic
50C23×C4×C163C2211yescharacteristic
51#121×C163—24no—
52C2×C4⋊C4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54C2×C22⋊C4×C163C2211yescharacteristic
55C22×Dic163—22no—
56C22×Dic163—22no—
57C22×Dic163—22no—
58C22×Dic163—22no—
59C22×Dic163—22no—
60C22×Dic163—22no—
61C22×Dic163—22no—
62C22×Dic163—41no—
63C22×Dic163—41no—
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C22×Dic163C2311yescharacteristic
70C22×Dic163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C4⋊Dic163—22no—
73Dic163⋊C4—22no—
74Dic163⋊C4—22no—
75Dic163⋊C4—22no—
76Dic163⋊C4—22no—
77Dic163⋊C4—22no—
78Dic163⋊C4—41no—
79C22⋊Dic163—22no—
80C22⋊Dic163—22no—
81C22⋊Dic163—22no—
82C22⋊Dic163—22no—
83C22⋊Dic163—22no—
84C22⋊Dic163—22no—
85C22⋊Dic163—22no—
86C22⋊Dic163—41no—
87C22⋊Dic163D414yescomplemented
88C4×Dic163—22no—
89C24×C163C2311yescharacteristic
90C24×C163C2311yescharacteristic
91C24×C163C2311yescharacteristic
92C2×D4×C163—22no—
93C2×D4×C163—22no—
94C2×D4×C163—22no—
95C2×D4×C163—22no—
96C2×D4×C163—22no—
97C2×D4×C163—22no—
98C2×D4×C163—22no—
99C2×D4×C163—41no—
100C2×D4×C163D414yes—
101C22×C4×C163—22no—
102C22×C4×C163—22no—
103C22×C4×C163—41no—
104C22×C4×C163C2311yescharacteristic
105C22×C4×C163C2311yescharacteristic
106C22×C4×C163C2311yescharacteristic
107C22×C4×C163C2311yescharacteristic
108C4⋊C4×C163—22no—
109C22⋊C4×C163—22no—
110C22⋊C4×C163—22no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
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—44no—
133C2×Dic163—44no—
134C2×Dic163—44no—
135C2×Dic163C2×D412yes—
136C2×Dic163C2×D412yes—
137C2×Dic163#4112yes—
138C23×C163—21no—
139C23×C163—21no—
140C23×C163—21no—
141C23×C163—21no—
142C23×C163—21no—
143C23×C163—21no—
144C23×C163—21no—
145C23×C163—21no—
146C23×C163—21no—
147C23×C163—21no—
148C23×C163—21no—
149C23×C163—21no—
150C23×C163—21no—
151C23×C163—21no—
152C23×C163—21no—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163C2×D412yes—
156C23×C163C2×D412yes—
157C23×C163C2×D412yes—
158C23×C163#4112yes—
159C23×C163C2411yescharacteristic
160D4×C163—28no—
161D4×C163—44no—
162D4×C163—44no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163—21no—
169C2×C4×C163—21no—
170C2×C4×C163—21no—
171C2×C4×C163—21no—
172C2×C4×C163—21no—
173C2×C4×C163—44no—
174C2×C4×C163C2×D412yes—
175C2×C4×C163#4112yes—
176C22×C163—22no—
177C22×C163—22no—
178C22×C163—22no—
179C22×C163—22no—
180C22×C163—22no—
181C22×C163—22no—
182C22×C163—22no—
183C22×C163—22no—
184C22×C163—22no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163—41no—
188C22×C163—41no—
189C22×C163—41no—
190C22×C163—41no—
191C22×C163—41no—
192C22×C163—41no—
193C22×C163—41no—
194C22×C163—41no—
195C22×C163—41no—
196C22×C163#12114yes—
197C22×C163C22×D411yescharacteristic
198C22×C163C22×D411yescharacteristic
199C22×C163C2×#4111yescharacteristic
200C22×C163C2×#4111yescharacteristic
201C22×C163C2×#4111yescharacteristic
202C22×C163H(5,𝔽2)11yescharacteristic
203C22×C1632−1+411yescharacteristic
204C4×C163—22no—
205C4×C163—22no—
206C4×C163—41no—
207C4×C163—41no—
208Dic163—22no—
209Dic163—22no—
210Dic163—22no—
211Dic163—41no—
212Dic163—41no—
213Dic163—41no—
214Dic163—41no—
215Dic163—41no—
216C2×C163—24no—
217C2×C163—42no—
218C2×C163—81no—
219C2×C163C2×#12111yescharacteristic
220C2×C163#52911yescharacteristic
221C2×C163#53911yescharacteristic
222C2×C163#54611yescharacteristic
223C2×C163#54611yescharacteristic
224C2×C163#54711yescharacteristic
225C2×C163#54711yescharacteristic
226C163#242511yescharacteristic, complemented, Sylow
227#2425—1631nomaximal, Sylow
228C2×#123—1631no—
229C2×#123—1631no—
230C2×#121D16311yescharacteristic
231C2×C4×D4—1631no—
232#398—1631no—
233#397—1631no—
234#396—1631no—
235#394—1631no—
236#389—1631no—
237#388—1631no—
238#386—1631no—
239#386—1631no—
240#385—1631no—
241#380—1631no—
242C2×#95—1631no—
243C22×D4C2×D16311yescharacteristic
244C22×D4C2×D16311yescharacteristic
245C22×D4C2×D16311yescharacteristic
246C23×C4—1631no—
247C23×C4—1631no—
248C23×C4—1631no—
249C23×C4C2×D16311yescharacteristic
250#123—3264no—
251#123—3264no—
252#121—24no—
253C4×D4—3264no—
254C2×C4⋊C4—1631no—
255C2×C4⋊C4—1631no—
256C2×C4⋊C4—1631no—
257C2×C4⋊C4—1631no—
258C2×C4⋊C4—1631no—
259C2×C4⋊C4—1631no—
260C2×C4⋊C4—1631no—
261C2×C4⋊C4C2×D16311yescharacteristic
262C2×C22⋊C4—1631no—
263C2×C22⋊C4—1631no—
264C2×C22⋊C4—1631no—
265C2×C22⋊C4—1631no—
266C2×C22⋊C4—1631no—
267C2×C22⋊C4—1631no—
268C2×C22⋊C4—1631no—
269C2×C22⋊C4—1631no—
270C2×C22⋊C4—1631no—
271C2×C22⋊C4C2×D16311yescharacteristic
272C2×C22⋊C4C2×D16311yescharacteristic
273C2×C42—1631no—
274#95—1631no—
275#95—1631no—
276#95—1631no—
277#95—1631no—
278#95—1631no—
279#95—1631no—
280#95—1631no—
281#95—1631no—
282C24C22×D16311yescharacteristic
283C24C22×D16311yescharacteristic
284C24C22×D16311yescharacteristic
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—41no—
293C2×D4C22⋊D16314yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—3262no—
298C22×C4—3262no—
299C22×C4—3262no—
300C22×C4—3262no—
301C22×C4—3262no—
302C22×C4—3262no—
303C22×C4—3262no—
304C22×C4—6521no—
305C22×C4—6521no—
306C22×C4—1631no—
307C22×C4—1631no—
308C22×C4—1631no—
309C22×C4—1631no—
310C22×C4—1631no—
311C22×C4—1631no—
312C22×C4—1631no—
313C22×C4—1631no—
314C22×C4C22×D16311yescharacteristic
315C22×C4C22×D16311yescharacteristic
316C22×C4C22×D16311yescharacteristic
317C22×C4C22×D16311yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—3262no—
320C4⋊C4—3262no—
321C4⋊C4—3262no—
322C4⋊C4—3262no—
323C4⋊C4—3262no—
324C4⋊C4—3262no—
325C4⋊C4—6521no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—3262no—
329C22⋊C4—3262no—
330C22⋊C4—3262no—
331C22⋊C4—3262no—
332C22⋊C4—3262no—
333C22⋊C4—3262no—
334C22⋊C4—3262no—
335C22⋊C4—6521no—
336C22⋊C4—1634no—
337C42—3262no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23D4×D16312yes—
356C23C2×C22⋊D16312yes—
357C23C2×C22⋊D16312yes—
358C23#1457969812yes—
359C23C23×D16311yescharacteristic
360D4—28no—
361D4—44no—
362D4—44no—
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—44no—
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—3261no—
383C2×C4—3261no—
384C2×C4—3261no—
385C2×C4—3261no—
386C2×C4—3261no—
387C2×C4—3261no—
388C2×C4—3261no—
389C2×C4—3261no—
390C2×C4—3261no—
391C2×C4—3261no—
392C2×C4—3261no—
393C2×C4—3261no—
394C2×C4—3261no—
395C2×C4—6524no—
396C2×C4—6524no—
397C2×C4—6524no—
398C2×C4—1632no—
399C2×C4—1632no—
400C2×C4—1632no—
401C2×C4C2×C22⋊D16312yes—
402C2×C4#1457969812yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22#1910470714yes—
424C22C2×D4×D16311yescharacteristic
425C22C22×C22⋊D16311yescharacteristic
426C22#41×D16311yescharacteristic
427C22C2×#1457969811yescharacteristic
428C22C2×#1457969811yescharacteristic
429C22#1910478011yescharacteristic
430C22#1910478211yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—3262no—
436C4—3262no—
437C4—3262no—
438C4—6521no—
439C4—6521no—
440C4—6521no—
441C4—6521no—
442C4—6521no—
443C2—24no—
444C2—42no—
445C2—81no—
446C2C2×#1910470711yescharacteristic
447C2#2378219611yescharacteristic
448C2#2378226011yescharacteristic
449C2#2378230411yescharacteristic
450C2#2378230911yescharacteristic
451C2#2378231411yescharacteristic
452C2#2378231511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting, prime radicalC2×#121×C163C2
hypercenterC2×#121D163
hypocenterC163#2425
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#19104716C2semidirect
2. C2×#19104716C2semidirect
3. C2×D4×Dic163C2semidirect
4. #23781598C2semidirect
5. #23781593C2semidirect
6. #23781588C2semidirect
7. #23781579C2semidirect
8. #23781560C2semidirect
9. #23781559C2semidirect
10. #23781545C2semidirect
11. #23781545C2semidirect
12. #23781538C2semidirect
13. #23781523C2semidirect
14. C2×C163⋊#95C2semidirect
87. C22⋊Dic163D4semidirect
226. C163#2425semidirect

Maximal subgroups

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

subgroupindex
C2×#191047162
C2×D4×Dic1632
#237815982
#237815932
#237815882
#237815792
#237815602
#237815592
#237815452
#237815382
#237815232
C2×C163⋊#952
C2×#121×C1632
#2425163