Finite Group Databasebeta

← #35217947

Subgroups — #35217947

Order 20864.

704 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104707C211yesmaximal, characteristic, complemented
2C2×#19104707C211yesmaximal, characteristic, complemented
3C2×D4×Dic163C211yesmaximal, characteristic, complemented
4C2×#19104691C211yesmaximal, characteristic, complemented
5#23781581C211yesmaximal, characteristic, complemented
6#23781579C211yesmaximal, characteristic, complemented
7#23781573C211yesmaximal, characteristic, complemented
8#23781568C211yesmaximal, characteristic, complemented
9#23781565C211yesmaximal, characteristic, complemented
10#23781565C211yesmaximal, characteristic, complemented
11#23781562C211yesmaximal, characteristic, complemented
12#23781558C211yesmaximal, characteristic, complemented
13#23781557C211yesmaximal, characteristic, complemented
14#23781523C211yesmaximal, characteristic, complemented
15#392×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C23×Dic163C2211yescharacteristic
20C23×Dic163C2211yescharacteristic
21#19104707—24no—
22#19104707—24no—
23D4×Dic163—24no—
24#19104691—24no—
25C2×C4⋊Dic163C2211yescharacteristic
26C2×C4⋊Dic163C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×Dic163⋊C4C2211yescharacteristic
29C2×Dic163⋊C4C2211yescharacteristic
30C2×Dic163⋊C4C2211yescharacteristic
31C2×C22⋊Dic163C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×C22⋊Dic163C2211yescharacteristic
36C2×C22⋊Dic163C2211yescharacteristic
37C2×C22⋊Dic163C2211yescharacteristic
38C2×(C2×D163)⋊C4C2211yescharacteristic
39C2×(C2×D163)⋊C4C2211yescharacteristic
40C2×(C2×D163)⋊C4C2211yescharacteristic
41C2×(C2×D163)⋊C4C2211yescharacteristic
42C2×C4×Dic163C2211yescharacteristic
43C2×C4×Dic163C2211yescharacteristic
44C2×C4×Dic163C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C163⋊#95C2211yescharacteristic
48C22×D4×C163C2211yescharacteristic
49C22×D4×C163C2211yescharacteristic
50C22×D4×C163C2211yescharacteristic
51C2×C22⋊C4×C163C2211yescharacteristic
52C2×C22⋊C4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54#95×C163C2211yescharacteristic
55C23×D163C2311yescharacteristic
56C2×C22⋊D163—22no—
57C2×C22⋊D163—22no—
58C2×C22⋊D163—22no—
59C2×C22⋊D163—22no—
60C2×C22⋊D163—22no—
61C2×C22⋊D163—22no—
62C2×C22⋊D163—41no—
63C2×C22⋊D163—41no—
64C2×C22⋊D163—41no—
65C22×Dic163—22no—
66C22×Dic163—22no—
67C22×Dic163—22no—
68C22×Dic163—22no—
69C22×Dic163—22no—
70C22×Dic163D414yescomplemented
71C22×Dic163C2311yescharacteristic
72C22×Dic163C2311yescharacteristic
73C22×Dic163C2311yescharacteristic
74C22×Dic163C2311yescharacteristic
75C22×Dic163C2311yescharacteristic
76C22×Dic163C2311yescharacteristic
77C22×Dic163C2311yescharacteristic
78C4⋊Dic163—22no—
79C4⋊Dic163—41no—
80Dic163⋊C4—22no—
81Dic163⋊C4—22no—
82Dic163⋊C4—41no—
83Dic163⋊C4D414yescomplemented
84C22⋊Dic163—22no—
85C22⋊Dic163—22no—
86C22⋊Dic163—22no—
87C22⋊Dic163—22no—
88C22⋊Dic163—22no—
89C22⋊Dic163—41no—
90C22⋊Dic163—41no—
91(C2×D163)⋊C4—22no—
92(C2×D163)⋊C4—22no—
93(C2×D163)⋊C4—22no—
94(C2×D163)⋊C4—41no—
95C4×Dic163—22no—
96C4×Dic163—22no—
97C4×Dic163—22no—
98C24×C163C2311yescharacteristic
99C24×C163C2311yescharacteristic
100C24×C163C2311yescharacteristic
101C2×D4×C163—22no—
102C2×D4×C163—22no—
103C2×D4×C163—22no—
104C2×D4×C163—22no—
105C2×D4×C163—22no—
106C2×D4×C163—22no—
107C2×D4×C163—41no—
108C2×D4×C163—41no—
109C2×D4×C163—41no—
110C22×C4×C163C2311yescharacteristic
111C22×C4×C163C2311yescharacteristic
112C22×C4×C163C2311yescharacteristic
113C22×C4×C163C2311yescharacteristic
114C22⋊C4×C163—22no—
115C22⋊C4×C163—22no—
116C22⋊C4×C163—41no—
117C22×D163—21no—
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—21no—
121C22×D163—21no—
122C22×D163—21no—
123C22×D163—21no—
124C22⋊D163—44no—
125C22⋊D163—44no—
126C22⋊D163—44no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163—28no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—21no—
140C2×Dic163—21no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—44no—
145C2×Dic163C2×D412yes—
146C2×Dic163C2×D412yes—
147C2×Dic163C2×D412yes—
148C2×Dic163C2×D412yes—
149C2×Dic163#4112yes—
150C23×C163—21no—
151C23×C163—21no—
152C23×C163—21no—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163—21no—
156C23×C163—21no—
157C23×C163—21no—
158C23×C163—21no—
159C23×C163—21no—
160C23×C163—21no—
161C23×C163—21no—
162C23×C163—21no—
163C23×C163—21no—
164C23×C163—21no—
165C23×C163—21no—
166C23×C163—21no—
167C23×C163—21no—
168C23×C163—21no—
169C23×C163C2×D412yes—
170C23×C163#4112yes—
171C23×C163C2411yescharacteristic
172D4×C163—28no—
173D4×C163—44no—
174D4×C163—44no—
175C2×C4×C163—21no—
176C2×C4×C163—21no—
177C2×C4×C163—21no—
178C2×C4×C163—21no—
179C2×C4×C163—21no—
180C2×C4×C163—21no—
181C2×C4×C163—21no—
182C2×C4×C163—21no—
183C2×C4×C163—21no—
184C2×C4×C163—21no—
185C2×C4×C163C2×D412yes—
186C2×C4×C163#4112yes—
187C22×C163—22no—
188C22×C163—22no—
189C22×C163—22no—
190C22×C163—22no—
191C22×C163—22no—
192C22×C163—22no—
193C22×C163—41no—
194C22×C163—41no—
195C22×C163—41no—
196C22×C163—41no—
197C22×C163—41no—
198C22×C163—41no—
199C22×C163—41no—
200C22×C163—41no—
201C22×C163—41no—
202C22×C163—41no—
203C22×C163—41no—
204C22×C163—41no—
205C22×C163—41no—
206C22×C163—41no—
207C22×C163—41no—
208C22×C163C22×D411yescharacteristic
209C22×C163C22×D411yescharacteristic
210C22×C163C2×#4111yescharacteristic
211C22×C163C2×#4111yescharacteristic
212C22×C163C2×#4111yescharacteristic
213C22×C163H(5,𝔽2)11yescharacteristic
214C22×C163H(5,𝔽2)11yescharacteristic
215C2×D163—41no—
216C2×D163—41no—
217C2×D163—41no—
218C2×D163—41no—
219C2×D163—41no—
220C2×D163—41no—
221C2×D163—41no—
222C4×C163—22no—
223C4×C163—22no—
224C4×C163—41no—
225C4×C163—41no—
226Dic163—22no—
227Dic163—22no—
228Dic163—41no—
229Dic163—41no—
230Dic163—41no—
231Dic163—41no—
232Dic163#12114yes—
233C2×C163—42no—
234C2×C163—42no—
235C2×C163—81no—
236C2×C163C2×#12111yescharacteristic
237C2×C163#52511yescharacteristic
238C2×C163#53511yescharacteristic
239C2×C163#54611yescharacteristic
240C2×C163#55011yescharacteristic
241C2×C163#55311yescharacteristic
242C2×C163#55511yescharacteristic
243D163—81no—
244C163#241411yescharacteristic, complemented, Sylow
245#2414—1631nomaximal, Sylow
246C2×#121—1631no—
247C2×#121—1631no—
248C2×C4×D4—1631no—
249C2×#117—1631no—
250#394—1631no—
251#394—1631no—
252#392—1631no—
253#392D16311yescharacteristic, complemented
254#390—1631no—
255#390—1631no—
256#390—1631no—
257#389—1631no—
258#388—1631no—
259#388—1631no—
260#380—1631no—
261C22×D4—1631no—
262C22×D4—1631no—
263C22×D4—1631no—
264C22×D4C2×D16311yescharacteristic
265C22×D4C2×D16311yescharacteristic
266C22×D4C2×D16311yescharacteristic
267C23×C4—1631no—
268C23×C4—1631no—
269#121—3264no—
270#121—3264no—
271C4×D4—3264no—
272#117—3264no—
273C2×C4⋊C4—1631no—
274C2×C4⋊C4—1631no—
275C2×C4⋊C4—1631no—
276C2×C4⋊C4—1631no—
277C2×C4⋊C4—1631no—
278C2×C4⋊C4—1631no—
279C2×C22⋊C4—1631no—
280C2×C22⋊C4—1631no—
281C2×C22⋊C4—1631no—
282C2×C22⋊C4—1631no—
283C2×C22⋊C4—1631no—
284C2×C22⋊C4—1631no—
285C2×C22⋊C4—1631no—
286C2×C22⋊C4—1631no—
287C2×C22⋊C4—1631no—
288C2×C22⋊C4—1631no—
289C2×C22⋊C4—1631no—
290C2×C22⋊C4C2×D16311yescharacteristic
291C2×C22⋊C4C2×D16311yescharacteristic
292C2×C22⋊C4C2×D16311yescharacteristic
293C2×C42—1631no—
294C2×C42—1631no—
295C2×C42—1631no—
296#95—1631no—
297#95—1631no—
298#95—1631no—
299#95C2×D16311yescharacteristic
300C24—1631no—
301C24C22×D16311yescharacteristic
302C24C22×D16311yescharacteristic
303C24C22×D16311yescharacteristic
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—3262no—
314C2×D4—3262no—
315C2×D4—3262no—
316C2×D4—3262no—
317C2×D4—3262no—
318C2×D4—3262no—
319C2×D4—6521no—
320C2×D4—6521no—
321C2×D4—6521no—
322C22×C4—3262no—
323C22×C4—3262no—
324C22×C4—3262no—
325C22×C4—3262no—
326C22×C4—3262no—
327C22×C4—1631no—
328C22×C4—1631no—
329C22×C4—1631no—
330C22×C4—1631no—
331C22×C4—1634no—
332C22×C4—1631no—
333C22×C4—1631no—
334C22×C4—1631no—
335C22×C4C22×D16311yescharacteristic
336C22×C4C22×D16311yescharacteristic
337C22×C4C22×D16311yescharacteristic
338C22×C4C22×D16311yescharacteristic
339C4⋊C4—3262no—
340C4⋊C4—3262no—
341C4⋊C4—3262no—
342C4⋊C4—6521no—
343C4⋊C4—6521no—
344C4⋊C4—1634no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—41no—
348C22⋊C4—3262no—
349C22⋊C4—3262no—
350C22⋊C4—3262no—
351C22⋊C4—3262no—
352C22⋊C4—3262no—
353C22⋊C4—3262no—
354C22⋊C4—3262no—
355C22⋊C4—3262no—
356C22⋊C4—6521no—
357C22⋊C4—6521no—
358C22⋊C4—6521no—
359C42—3262no—
360C42—3262no—
361C42—3262no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—3261no—
382C23—3261no—
383C23—3261no—
384C23—3261no—
385C23—3261no—
386C23—3261no—
387C23—3261no—
388C23D4×D16312yes—
389C23#1457969812yes—
390C23C23×D16311yescharacteristic
391D4—28no—
392D4—44no—
393D4—44no—
394D4—6524no—
395D4—6524no—
396D4—6524no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—3261no—
408C2×C4—3261no—
409C2×C4—3261no—
410C2×C4—3261no—
411C2×C4—3268no—
412C2×C4—3261no—
413C2×C4—3261no—
414C2×C4—3261no—
415C2×C4—3261no—
416C2×C4—3261no—
417C2×C4—3261no—
418C2×C4—3261no—
419C2×C4—3261no—
420C2×C4—3261no—
421C2×C4—3261no—
422C2×C4—3261no—
423C2×C4—3261no—
424C2×C4—6524no—
425C2×C4—1632no—
426C2×C4—1632no—
427C2×C4—1632no—
428C2×C4—1632no—
429C2×C4—1632no—
430C2×C4D4×D16312yes—
431C2×C4#1457969812yes—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—6521no—
454C22—6521no—
455C22—6521no—
456C22—6521no—
457C22—6521no—
458C22—6521no—
459C22—6521no—
460C22C2×D4×D16311yescharacteristic
461C22C2×D4×D16311yescharacteristic
462C22#41×D16311yescharacteristic
463C22C2×#1457969811yescharacteristic
464C22C2×#1457969811yescharacteristic
465C22#1910478011yescharacteristic
466C22#1910478011yescharacteristic
467C4—22no—
468C4—22no—
469C4—41no—
470C4—41no—
471C4—3262no—
472C4—3262no—
473C4—6521no—
474C4—6521no—
475C4—6521no—
476C4—6521no—
477C4—1634no—
478C2—42no—
479C2—42no—
480C2—81no—
481C2—13041no—
482C2#121×D16311yescharacteristic
483C2#2378217111yescharacteristic
484C2#2378223011yescharacteristic
485C2#2378230211yescharacteristic
486C2#2378233511yescharacteristic
487C2#2378235811yescharacteristic
488C2#2378237911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#392×C163C2
hypercenter#392D163
hypocenterC163#2414
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#19104707C2semidirect
2. C2×#19104707C2semidirect
3. C2×D4×Dic163C2semidirect
4. C2×#19104691C2semidirect
5. #23781581C2semidirect
6. #23781579C2semidirect
7. #23781573C2semidirect
8. #23781568C2semidirect
9. #23781565C2semidirect
10. #23781565C2semidirect
11. #23781562C2semidirect
12. #23781558C2semidirect
13. #23781557C2semidirect
14. #23781523C2semidirect
15. #392×C163C2semidirect
70. C22×Dic163D4semidirect
83. Dic163⋊C4D4semidirect
244. C163#2414semidirect
253. #392D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047072
C2×D4×Dic1632
C2×#191046912
#237815812
#237815792
#237815732
#237815682
#237815652
#237815622
#237815582
#237815572
#237815232
#392×C1632
#2414163