Finite Group Databasebeta

← #35217994

Subgroups — #35217994

Order 20864.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104716C211yesmaximal, characteristic, complemented
2C2×D4×Dic163C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic163C211yesmaximal, characteristic, complemented
4#23781601C211yesmaximal, characteristic, complemented
5#23781590C211yesmaximal, characteristic, complemented
6#23781579C211yesmaximal, characteristic, complemented
7#23781579C211yesmaximal, characteristic, complemented
8#23781553C211yesmaximal, characteristic, complemented
9#23781553C211yesmaximal, characteristic, complemented
10#23781545C211yesmaximal, characteristic, complemented
11#23781545C211yesmaximal, characteristic, complemented
12#23781537C211yesmaximal, characteristic, complemented
13#23781523C211yesmaximal, characteristic, complemented
14#23781521C211yesmaximal, characteristic, complemented
15C2×#120×C163C211yesmaximal, characteristic
16C23×Dic163C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18C23×Dic163C2211yescharacteristic, complemented
19#19104716—24no—
20D4×Dic163—24no—
21C2×C4⋊Dic163C2211yescharacteristic, complemented
22C2×Dic163⋊C4C2211yescharacteristic, complemented
23C2×Dic163⋊C4C2211yescharacteristic, complemented
24C2×Dic163⋊C4C2211yescharacteristic, complemented
25C2×Dic163⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic163—24no—
27C2×C22⋊Dic163C2211yescharacteristic
28C2×C22⋊Dic163C2211yescharacteristic
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, complemented
38C2×C22⋊Dic163C2211yescharacteristic, complemented
39C2×C22⋊Dic163C2211yescharacteristic, complemented
40C2×C4×Dic163C2211yescharacteristic, complemented
41C163⋊#95C2211yescharacteristic, complemented
42C163⋊#95C2211yescharacteristic, complemented
43C163⋊#95C2211yescharacteristic, complemented
44C163⋊#95C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C25×C163C2211yescharacteristic
48C22×D4×C163C2211yescharacteristic
49C22×D4×C163C2211yescharacteristic
50C22×D4×C163C2211yescharacteristic
51#120×C163—24no—
52C2×C22⋊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—41no—
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—41no—
76Dic163⋊C4—41no—
77C22⋊Dic163—22no—
78C22⋊Dic163—22no—
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⋊Dic163—41no—
88C22⋊Dic163—41no—
89C22⋊Dic163D414yescomplemented
90C4×Dic163—22no—
91C24×C163—22no—
92C24×C163—22no—
93C24×C163—22no—
94C24×C163—22no—
95C24×C163—22no—
96C24×C163—41no—
97C24×C163—41no—
98C24×C163C2311yescharacteristic
99C24×C163C2311yescharacteristic
100C24×C163C2311yescharacteristic
101C24×C163C2311yescharacteristic
102C2×D4×C163—22no—
103C2×D4×C163—22no—
104C2×D4×C163—22no—
105C2×D4×C163—22no—
106C2×D4×C163—22no—
107C2×D4×C163—22no—
108C2×D4×C163—41no—
109C2×D4×C163—41no—
110C2×D4×C163D414yes—
111C22×C4×C163C2311yescharacteristic
112C22×C4×C163C2311yescharacteristic
113C22×C4×C163C2311yescharacteristic
114C22⋊C4×C163—22no—
115C22⋊C4×C163—22no—
116C22⋊C4×C163—22no—
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—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—44no—
140C2×Dic163—44no—
141C2×Dic163—44no—
142C2×Dic163C2×D412yes—
143C2×Dic163C2×D412yes—
144C23×C163—21no—
145C23×C163—21no—
146C23×C163—21no—
147C23×C163—21no—
148C23×C163—28no—
149C23×C163—21no—
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—44no—
168C23×C163—44no—
169C23×C163—44no—
170C23×C163—44no—
171C23×C163—44no—
172C23×C163—44no—
173C23×C163C2×D412yes—
174C23×C163C2×D412yes—
175C23×C163C2×D412yes—
176C23×C163#4112yes—
177C23×C163#4112yes—
178C23×C163#4112yes—
179C23×C163C2411yescharacteristic
180D4×C163—28no—
181D4×C163—44no—
182D4×C163—44no—
183C2×C4×C163—21no—
184C2×C4×C163—21no—
185C2×C4×C163—21no—
186C2×C4×C163—21no—
187C2×C4×C163—21no—
188C2×C4×C163—21no—
189C2×C4×C163—21no—
190C2×C4×C163—21no—
191C2×C4×C163C2×D412yes—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163—22no—
195C22×C163—22no—
196C22×C163—22no—
197C22×C163—22no—
198C22×C163—22no—
199C22×C163—22no—
200C22×C163—22no—
201C22×C163—22no—
202C22×C163—22no—
203C22×C163—22no—
204C22×C163—22no—
205C22×C163—22no—
206C22×C163—22no—
207C22×C163—416no—
208C22×C163—41no—
209C22×C163—41no—
210C22×C163—41no—
211C22×C163—41no—
212C22×C163—41no—
213C22×C163—41no—
214C22×C163—41no—
215C22×C163—41no—
216C22×C163—41no—
217C22×C163—41no—
218C22×C163—41no—
219C22×C163—41no—
220C22×C163#12314yescomplemented
221C22×C163C22×D411yescharacteristic
222C22×C163C22×D411yescharacteristic
223C22×C163C2×#4111yescharacteristic
224C22×C163C2×#4111yescharacteristic
225C22×C163C2×#4111yescharacteristic
226C22×C163H(5,𝔽2)11yescharacteristic
227C22×C163H(5,𝔽2)11yescharacteristic
228C4×C163—22no—
229C4×C163—41no—
230C4×C163—41no—
231Dic163—22no—
232Dic163—22no—
233Dic163—41no—
234Dic163—41no—
235Dic163—41no—
236Dic163—41no—
237Dic163—41no—
238Dic163—41no—
239C2×C163—24no—
240C2×C163—42no—
241C2×C163—42no—
242C2×C163—42no—
243C2×C163C2×#12311yescharacteristic
244C2×C163#52511yescharacteristic
245C2×C163#53811yescharacteristic
246C2×C163D4211yescharacteristic
247C2×C163#54611yescharacteristic
248C2×C163#54611yescharacteristic
249C2×C163#55111yescharacteristic
250C163#241811yescharacteristic, complemented, Sylow
251#2418—1631nomaximal, Sylow
252C2×#123—1631no—
253C2×#120D16311yescharacteristic
254C2×C4×D4—1631no—
255C22×C22⋊C4—1631no—
256#399—1631no—
257#397—1631no—
258#394—1631no—
259#394—1631no—
260#388—1631no—
261#388—1631no—
262#386—1631no—
263#386—1631no—
264#385—1631no—
265#380—1631no—
266#379—1631no—
267C25C2×D16311yescharacteristic
268C22×D4C2×D16311yescharacteristic
269C22×D4C2×D16311yescharacteristic
270C22×D4C2×D16311yescharacteristic
271C23×C4—1631no—
272C23×C4—1631no—
273C23×C4—1631no—
274#123—3264no—
275#120—24no—
276C4×D4—3264no—
277C2×C4⋊C4—1631no—
278C2×C4⋊C4—1631no—
279C2×C4⋊C4—1631no—
280C2×C4⋊C4—1631no—
281C2×C4⋊C4—1631no—
282C2×C22⋊C4—3264no—
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⋊C4—1631no—
291C2×C22⋊C4—1631no—
292C2×C22⋊C4—1631no—
293C2×C22⋊C4—1631no—
294C2×C22⋊C4—1631no—
295C2×C22⋊C4—1631no—
296C2×C22⋊C4C2×D16311yescharacteristic
297C2×C22⋊C4C2×D16311yescharacteristic
298C2×C22⋊C4C2×D16311yescharacteristic
299C2×C42—1631no—
300#95—1631no—
301#95—1631no—
302#95—1631no—
303#95—1631no—
304#95—1631no—
305#95—1631no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—41no—
312C24—41no—
313C24C22×D16311yescharacteristic
314C24C22×D16311yescharacteristic
315C24C22×D16311yescharacteristic
316C24C22×D16311yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4C22⋊D16314yes—
326C22×C4—3262no—
327C22×C4—3262no—
328C22×C4—3262no—
329C22×C4—3262no—
330C22×C4—3262no—
331C22×C4—3262no—
332C22×C4—6521no—
333C22×C4—6521no—
334C22×C4—6521no—
335C22×C4—1631no—
336C22×C4—1631no—
337C22×C4—1631no—
338C22×C4—1631no—
339C22×C4—1631no—
340C22×C4—1631no—
341C22×C4—1631no—
342C22×C4—1631no—
343C22×C4C22×D16311yescharacteristic
344C22×C4C22×D16311yescharacteristic
345C22×C4C22×D16311yescharacteristic
346C4⋊C4—3262no—
347C4⋊C4—3262no—
348C4⋊C4—3262no—
349C4⋊C4—6521no—
350C4⋊C4—6521no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—3262no—
355C22⋊C4—3262no—
356C22⋊C4—3262no—
357C22⋊C4—3262no—
358C22⋊C4—3262no—
359C22⋊C4—3262no—
360C22⋊C4—3262no—
361C22⋊C4—3262no—
362C22⋊C4—3262no—
363C22⋊C4—6521no—
364C22⋊C4—6521no—
365C22⋊C4—6521no—
366C22⋊C4—1634no—
367C42—3262no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—28no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23D4×D16312yes—
398C23C2×C22⋊D16312yes—
399C23C2×C22⋊D16312yes—
400C23#1457969812yes—
401C23#1457969812yes—
402C23#1457969812yes—
403C23C23×D16311yescharacteristic
404D4—28no—
405D4—44no—
406D4—44no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
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—3261no—
425C2×C4—3261no—
426C2×C4—3261no—
427C2×C4—3261no—
428C2×C4—3261no—
429C2×C4—3261no—
430C2×C4—3261no—
431C2×C4—3261no—
432C2×C4—3261no—
433C2×C4—3261no—
434C2×C4—3261no—
435C2×C4—3261no—
436C2×C4—3261no—
437C2×C4—6524no—
438C2×C4—6524no—
439C2×C4—6524no—
440C2×C4—1632no—
441C2×C4—1632no—
442C2×C4C2×C22⋊D16312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—416no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22#1910471614yescomplemented
472C22C2×D4×D16311yescharacteristic
473C22C22×C22⋊D16311yescharacteristic
474C22C2×#1457969811yescharacteristic
475C22C2×#1457969811yescharacteristic
476C22C2×#1457969811yescharacteristic
477C22#1910478011yescharacteristic
478C22#1910478011yescharacteristic
479C4—22no—
480C4—41no—
481C4—41no—
482C4—3262no—
483C4—3262no—
484C4—6521no—
485C4—6521no—
486C4—6521no—
487C4—6521no—
488C4—6521no—
489C4—6521no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—42no—
494C2C2×#1910471611yescharacteristic
495C2#2378217111yescharacteristic
496C2#2378225011yescharacteristic
497C2D4×C22⋊D16311yescharacteristic
498C2#2378230211yescharacteristic
499C2#2378230911yescharacteristic
500C2#2378234611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting, prime radicalC2×#120×C163C2
hypercenterC2×#120D163
hypocenterC163#2418
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#19104716C2semidirect
2. C2×D4×Dic163C2semidirect
3. C22×C22⋊Dic163C2semidirect
4. #23781601C2semidirect
5. #23781590C2semidirect
6. #23781579C2semidirect
7. #23781579C2semidirect
8. #23781553C2semidirect
9. #23781553C2semidirect
10. #23781545C2semidirect
11. #23781545C2semidirect
12. #23781537C2semidirect
13. #23781523C2semidirect
14. #23781521C2semidirect
18. C23×Dic163C22semidirect
21. C2×C4⋊Dic163C22semidirect
22. C2×Dic163⋊C4C22semidirect
23. C2×Dic163⋊C4C22semidirect
24. C2×Dic163⋊C4C22semidirect
25. C2×Dic163⋊C4C22semidirect
37. C2×C22⋊Dic163C22semidirect
38. C2×C22⋊Dic163C22semidirect
39. C2×C22⋊Dic163C22semidirect
40. C2×C4×Dic163C22semidirect
41. C163⋊#95C22semidirect
42. C163⋊#95C22semidirect
43. C163⋊#95C22semidirect
44. C163⋊#95C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
89. C22⋊Dic163D4semidirect
220. C22×C163#123semidirect
250. C163#2418semidirect
471. C22#19104716semidirect

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
C22×C22⋊Dic1632
#237816012
#237815902
#237815792
#237815532
#237815452
#237815372
#237815232
#237815212
C2×#120×C1632
#2418163