Finite Group Databasebeta

← #11906171

Subgroups — #11906171

Order 1152.

3300 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#533C211yesmaximal, characteristic, complemented
2#10596201C212yesmaximal, complemented
3#10596198C212yesmaximal, complemented
4#10596161C214yesmaximal, complemented
5#10596105C214yesmaximal, complemented
6C3⋊D3×C22⋊Q8C214yesmaximal, complemented
7#10596050C212yesmaximal, complemented
8#10596050C214yesmaximal, complemented
9Q8×C4⋊(C3⋊D3)C212yesmaximal, complemented
10#10595992C212yesmaximal
11#10595990C212yesmaximal, complemented
12C2×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
13#10595941C211yesmaximal, characteristic, complemented
14#86280—34nomaximal
15C2×Q8×C3⋊D3C2214yescomplemented
16C22×C4⋊(C3⋊D3)C2211yescharacteristic
17C22×C4×C3⋊D3C2212yes—
18C32×C4⋊Q8C2211yescharacteristic, complemented
19C32×C4⋊Q8C2211yescharacteristic, complemented
20C32×#125C2212yescomplemented
21C32×C22⋊Q8C2212yes—
22C32×C22⋊Q8C2212yes—
23C32×C4×Q8C2212yescomplemented
24C32×C4×Q8C2212yescomplemented
25C32×#117C2212yes—
26C2×C32×C42C2211yescharacteristic
27#64155C2214yescomplemented
28#64155C2218yescomplemented
29C2×#3752C2212yes—
30#64135C2212yes—
31C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
32#64133C2212yes—
33#64133C2212yes—
34#64106C2214yescomplemented
35#64106C2218yescomplemented
36#64105C2218yescomplemented
37#64105C2218yescomplemented
38#64105C2218yescomplemented
39#64102C2214yes—
40#64102C2214yes—
41C4⋊C4×C3⋊D3C2218yescomplemented
42C4⋊C4×C3⋊D3C2214yescomplemented
43#64099C2214yescomplemented
44#64097C2214yescomplemented
45C22⋊C4×C3⋊D3C2218yes—
46#64087C2218yes—
47C4×C4⋊(C3⋊D3)C2212yes—
48C4×C4⋊(C3⋊D3)C2212yes—
49C4×C4⋊(C3⋊D3)C2212yes—
50C4×C4⋊(C3⋊D3)C2212yes—
51C4×C4⋊(C3⋊D3)C2214yes—
52C4×C4⋊(C3⋊D3)C2214yes—
53#64079C2212yescomplemented
54#64079C2214yescomplemented
55#64078C2211yescharacteristic, complemented
56#64078C2211yescharacteristic, complemented
57#64078C2214yescomplemented
58C4×#634C2212yescomplemented
59C4×#634C2212yescomplemented
60C3×#533D314yescomplemented
61#5730—38no—
62#5727—38no—
63#5690—316no—
64#5634—316no—
65D3×C22⋊Q8—316no—
66#5579—38no—
67#5579—316no—
68Q8×D12—38no—
69#5521—38no—
70#5519—38no—
71C2×C4×D12—34no—
72#5470—34no—
73C23×C3⋊D3C2312yes—
74C2×C32×Q8C2312yes—
75C2×C32×Q8C2312yes—
76C22×C32×C4C2311yescharacteristic
77C22×C32×C4C2311yescharacteristic
78C22×C32×C4C2311yescharacteristic
79C22×C3⋊Dic3C2312yes—
80Q8×C3⋊D3—48no—
81C2×C4⋊(C3⋊D3)Q818yescomplemented
82C2×C4⋊(C3⋊D3)C2312yes—
83C2×C4⋊(C3⋊D3)C2312yes—
84C2×C4×C3⋊D3—44no—
85C2×C4×C3⋊D3C2314yes—
86C2×C4×C3⋊D3C2314yes—
87C2×C4×C3⋊D3C2318yes—
88C2×C4×C3⋊D3C2318yes—
89C2×#634C2312yes—
90C2×#634C2312yes—
91C2×#634C2318yes—
92C32×C4⋊C4C2312yes—
93C32×C4⋊C4C2312yes—
94C32×C4⋊C4C2314yes—
95C32×C4⋊C4C2312yes—
96C32×C4⋊C4C2312yes—
97C32×C4⋊C4C2314yes—
98C32×C22⋊C4C2314yes—
99C32×C42C2311yescharacteristic
100C32×C42C2311yescharacteristic
101C32×C42C2311yescharacteristic
102C32×C42C2314yes—
103C32×C42C2311yescharacteristic
104#3757C2314yes—
105#3752C2314yes—
106#3752C2314yes—
107#3752C2318yes—
108#3752C2318yes—
109C4⋊(C3⋊Dic3)C2312yes—
110C4⋊(C3⋊Dic3)C2312yes—
111C4⋊(C3⋊Dic3)C2312yes—
112C4⋊(C3⋊Dic3)C2312yes—
113C4⋊(C3⋊Dic3)C2312yes—
114C4⋊(C3⋊Dic3)C2318yes—
115C4⋊(C3⋊Dic3)C2312yes—
116#3750C2314yes—
117#3750C2314yes—
118#3750C2318yes—
119#3750C2318yes—
120C4×C3⋊Dic3C2314yes—
121#3501—91noSylow
122C2×D3×Q8—316no—
123C22×D12—34no—
124C22×C4×D3—38no—
125C3×C4⋊Q8C2×D314yescomplemented
126C3×C4⋊Q8C2×D314yescomplemented
127C3×#125C2×D318yescomplemented
128C3×C22⋊Q8C2×D318yes—
129C3×C22⋊Q8C2×D318yes—
130C3×C4×Q8C2×D318yescomplemented
131C3×C4×Q8C2×D318yescomplemented
132C3×#117C2×D318yes—
133C2×C3×C42C2×D314yes—
134#946—332no—
135#946—316no—
136C2×(C2×D3)⋊C4—38no—
137#926—38no—
138C2×C4⋊Dic3—34no—
139#924—38no—
140#924—38no—
141#897—332no—
142#897—316no—
143#896—332no—
144#896—332no—
145#896—332no—
146#893—316no—
147#893—316no—
148D3×C4⋊C4—332no—
149D3×C4⋊C4—316no—
150#890—316no—
151#888—316no—
152D3×C22⋊C4—332no—
153#878—332no—
154C4×D12—38no—
155C4×D12—38no—
156C4×D12—38no—
157C4×D12—38no—
158C4×D12—316no—
159C4×D12—316no—
160#870—38no—
161#870—316no—
162#869—34no—
163#869—34no—
164#869—316no—
165C4×Dic6—38no—
166C4×Dic6—38no—
167C23×C32C2411yescharacteristic
168C22×C3⋊D3—42no—
169C22×C3⋊D3—42no—
170C22×C3⋊D3C2×Q818yescomplemented
171C22×C3⋊D3C2414yes—
172C32×Q8—24no—
173C32×Q8—42no—
174C2×C32×C4—22no—
175C2×C32×C4—41no—
176C2×C32×C4C2×Q814yescomplemented
177C2×C32×C4C2411yescharacteristic
178C2×C32×C4C2411yescharacteristic
179C2×C32×C4C2414yes—
180C2×C32×C4C2414yes—
181C2×C32×C4C2411yescharacteristic
182C2×C32×C4C2411yescharacteristic
183C2×C32×C4C2411yescharacteristic
184C2×C32×C4C2411yescharacteristic
185C2×C3⋊Dic3—42no—
186C2×C3⋊Dic3C2414yes—
187C2×C3⋊Dic3C2414yes—
188C2×C3⋊Dic3C2414yes—
189C4⋊(C3⋊D3)C2×Q818yescomplemented
190C4⋊(C3⋊D3)C2×Q818yescomplemented
191C4×C3⋊D3—28no—
192C4×C3⋊D3—44no—
193C4×C3⋊D3—48no—
194C4×C3⋊D3—48no—
195#634—24no—
196#634—42no—
197#634—48no—
198#554—92no—
199#554—94no—
200#554—92no—
201#554—94no—
202D4×Q8—92no—
203D4×Q8—92no—
204#544—92no—
205#543—94no—
206#533—91no—
207#533C3⋊D311yescharacteristic, complemented
208C2×C22⋊Q8—94no—
209#518—92no—
210C2×C4×D4—91no—
211C23×D3—38no—
212C2×C3×Q8C22×D318yes—
213C2×C3×Q8C22×D318yes—
214C22×C3×C4C22×D314yes—
215C22×C3×C4C22×D314yes—
216C22×C3×C4C22×D314yes—
217C22×Dic3—38no—
218D3×Q8—1232no—
219C2×D12—332no—
220C2×D12—38no—
221C2×D12—38no—
222C2×C4×D3—316no—
223C2×C4×D3—316no—
224C2×C4×D3—332no—
225C2×C4×D3—332no—
226C2×C4×D3—1216no—
227C2×Dic6—38no—
228C2×Dic6—38no—
229C2×Dic6—332no—
230C3×C4⋊C4C22×D318yes—
231C3×C4⋊C4C22×D318yes—
232C3×C4⋊C4C22×D3116yes—
233C3×C4⋊C4C22×D318yes—
234C3×C4⋊C4C22×D318yes—
235C3×C4⋊C4C22×D3116yes—
236C3×C22⋊C4C22×D3116yes—
237C3×C42C22×D314yes—
238C3×C42C22×D314yes—
239C3×C42C22×D314yes—
240C3×C42C22×D3116yes—
241C3×C42C22×D314yes—
242#217—316no—
243(C2×D3)⋊C4—316no—
244(C2×D3)⋊C4—316no—
245(C2×D3)⋊C4—332no—
246(C2×D3)⋊C4—332no—
247C4⋊Dic3—38no—
248C4⋊Dic3—38no—
249C4⋊Dic3—38no—
250C4⋊Dic3—38no—
251C4⋊Dic3—38no—
252C4⋊Dic3—332no—
253C4⋊Dic3—38no—
254Dic3⋊C4—316no—
255Dic3⋊C4—316no—
256Dic3⋊C4—332no—
257Dic3⋊C4—332no—
258C4×Dic3—316no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32C22×Q812yes—
262C22×C32C2511yescharacteristic
263C2×C3⋊D3—24no—
264C2×C3⋊D3—24no—
265C2×C3⋊D3—48no—
266C2×C3⋊D3C22×Q818yes—
267C32×C4—24no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—24no—
271C32×C4—21no—
272C32×C4C22×Q812yes—
273C32×C4C22×Q812yes—
274C32×C4H(5,𝔽2)12yes—
275C3⋊Dic3—24no—
276C3⋊Dic3—24no—
277C3⋊Dic3—24no—
278C22×Q8—94no—
279C22×D4—91no—
280C23×C4—92no—
281C4⋊Q8—94no—
282C4⋊Q8—91no—
283C4⋊Q8—91no—
284C4⋊Q8—94no—
285C4⋊Q8C2×C3⋊D311yescharacteristic, complemented
286C4⋊Q8C2×C3⋊D311yescharacteristic, complemented
287#125—94no—
288#125—92no—
289#125—94no—
290#125C2×C3⋊D312yescomplemented
291C22⋊Q8—94no—
292C22⋊Q8—98no—
293C22⋊Q8—92no—
294C22⋊Q8—92no—
295C22⋊Q8—94no—
296C22⋊Q8—98no—
297C22⋊Q8—98no—
298C22⋊Q8—98no—
299C22⋊Q8—98no—
300C22⋊Q8—98no—
301C22⋊Q8C2×C3⋊D312yes—
302C22⋊Q8C2×C3⋊D312yes—
303C4×Q8—92no—
304C4×Q8—92no—
305C4×Q8C2×C3⋊D312yescomplemented
306C4×Q8C2×C3⋊D312yescomplemented
307C4×D4—94no—
308C4×D4—94no—
309C4×D4—92no—
310C4×D4—92no—
311C4×D4—92no—
312C4×D4—92no—
313C4×D4—94no—
314C4×D4—94no—
315#117—92no—
316#117C2×C3⋊D312yes—
317C2×C4⋊C4—91no—
318C2×C4⋊C4—98no—
319C2×C4⋊C4—94no—
320C2×C22⋊C4—92no—
321C2×C22⋊C4—98no—
322C2×C42C2×C3⋊D311yescharacteristic
323C23×C3C23×D314yes—
324C22×D3—316no—
325C22×D3—332no—
326C22×D3—128no—
327C22×D3—128no—
328C3×Q8—216no—
329C3×Q8—48no—
330C2×C3×C4—28no—
331C2×C3×C4—44no—
332C2×C3×C4D3×Q8116yescomplemented
333C2×C3×C4C23×D314yes—
334C2×C3×C4C23×D314yes—
335C2×C3×C4C23×D3116yes—
336C2×C3×C4C23×D3116yes—
337C2×C3×C4C23×D314yes—
338C2×C3×C4C23×D314yes—
339C2×C3×C4C23×D314yes—
340C2×C3×C4C23×D314yes—
341C2×Dic3—316no—
342C2×Dic3—316no—
343C2×Dic3—316no—
344C2×Dic3—128no—
345D12—332no—
346D12—332no—
347C4×D3—632no—
348C4×D3—1232no—
349C4×D3—1216no—
350C4×D3—1232no—
351Dic6—616no—
352Dic6—128no—
353Dic6—1232no—
354C2×C32—22no—
355C2×C32C23×Q811yescharacteristic
356C2×C32C2×H(5,𝔽2)11yescharacteristic
357C2×C32#58511yescharacteristic
358C3⋊D3—28no—
359C24—92no—
360C2×Q8—92no—
361C2×Q8—92no—
362C2×Q8—98no—
363C2×Q8—368no—
364C2×Q8C22×C3⋊D312yes—
365C2×Q8C22×C3⋊D312yes—
366C2×D4—92no—
367C2×D4—92no—
368C2×D4—98no—
369C22×C4—92no—
370C22×C4—94no—
371C22×C4—98no—
372C22×C4—98no—
373C22×C4—94no—
374C22×C4—364no—
375C22×C4C22×C3⋊D311yescharacteristic
376C22×C4C22×C3⋊D311yescharacteristic
377C22×C4C22×C3⋊D311yescharacteristic
378C4⋊C4—94no—
379C4⋊C4—92no—
380C4⋊C4—92no—
381C4⋊C4—92no—
382C4⋊C4—98no—
383C4⋊C4—98no—
384C4⋊C4—98no—
385C4⋊C4—92no—
386C4⋊C4—92no—
387C4⋊C4—94no—
388C4⋊C4—92no—
389C4⋊C4C22×C3⋊D314yes—
390C4⋊C4C22×C3⋊D312yes—
391C4⋊C4C22×C3⋊D312yes—
392C4⋊C4C22×C3⋊D312yes—
393C4⋊C4C22×C3⋊D314yes—
394C4⋊C4C22×C3⋊D312yes—
395C22⋊C4—94no—
396C22⋊C4—98no—
397C22⋊C4—98no—
398C22⋊C4—94no—
399C22⋊C4—94no—
400C22⋊C4C22×C3⋊D314yes—
401C42—94no—
402C42C22×C3⋊D311yescharacteristic
403C42C22×C3⋊D311yescharacteristic
404C42C22×C3⋊D311yescharacteristic
405C42C22×C3⋊D311yescharacteristic
406C42C22×C3⋊D314yes—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3C2×D3×Q818yes—
410C22×C3C24×D314yes—
411C2×D3—332no—
412C2×D3—616no—
413C2×D3—616no—
414C2×D3—1232no—
415C3×C4—216no—
416C3×C4—24no—
417C3×C4—24no—
418C3×C4—216no—
419C3×C4—24no—
420C3×C4C2×D3×Q818yes—
421C3×C4C2×D3×Q818yes—
422C3×C4#100918yes—
423Dic3—616no—
424Dic3—616no—
425Dic3—616no—
426C32#350111yescharacteristic, complemented, Sylow
427C23—94no—
428C23—98no—
429C23—362no—
430C23—362no—
431C23C23×C3⋊D311yescharacteristic
432Q8—24no—
433Q8—42no—
434Q8—184no—
435Q8—368no—
436Q8—362no—
437D4—98no—
438D4—98no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—94no—
442C2×C4—94no—
443C2×C4—94no—
444C2×C4—188no—
445C2×C4—364no—
446C2×C4—368no—
447C2×C4—368no—
448C2×C4—362no—
449C2×C4Q8×C3⋊D314yescomplemented
450C2×C4C23×C3⋊D311yescharacteristic
451C2×C4C23×C3⋊D311yescharacteristic
452C2×C4C23×C3⋊D311yescharacteristic
453C2×C4C23×C3⋊D314yes—
454C2×C4C23×C3⋊D314yes—
455C2×C4C23×C3⋊D311yescharacteristic
456C2×C4C23×C3⋊D311yescharacteristic
457C2×C4C23×C3⋊D311yescharacteristic
458C2×C3—28no—
459C2×C3C22×D3×Q814yes—
460C2×C3C2×#100914yes—
461C2×C3#596414yes—
462D3—632no—
463C22—21no—
464C22—21no—
465C22—98no—
466C22—184no—
467C22—184no—
468C22—368no—
469C22C2×Q8×C3⋊D312yes—
470C22C24×C3⋊D311yescharacteristic
471C4—21no—
472C4—24no—
473C4—21no—
474C4—21no—
475C4—24no—
476C4—184no—
477C4—184no—
478C4—184no—
479C4C2×Q8×C3⋊D312yes—
480C4C2×Q8×C3⋊D312yes—
481C4#6436612yes—
482C3#8628014yescomplemented
483C2—22no—
484C2—188no—
485C2C22×Q8×C3⋊D311yescharacteristic
486C2C2×#6436611yescharacteristic
487C2#1059675611yescharacteristic
subgroupG/H
center, FrattiniC22C24×C3⋊D3
commutator, socle, cosocleC22×C32C25
FittingC32×#533C2
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenter#533C3⋊D3
hypocenterC32#3501
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C32×#533C2semidirect
2. #10596201C2semidirect
3. #10596198C2semidirect
4. #10596161C2semidirect
5. #10596105C2semidirect
6. C3⋊D3×C22⋊Q8C2semidirect
7. #10596050C2semidirect
8. #10596050C2semidirect
9. Q8×C4⋊(C3⋊D3)C2semidirect
11. #10595990C2semidirect
13. #10595941C2semidirect
15. C2×Q8×C3⋊D3C22semidirect
18. C32×C4⋊Q8C22semidirect
19. C32×C4⋊Q8C22semidirect
20. C32×#125C22semidirect
23. C32×C4×Q8C22semidirect
24. C32×C4×Q8C22semidirect
27. #64155C22semidirect
28. #64155C22semidirect
34. #64106C22semidirect
35. #64106C22semidirect
36. #64105C22semidirect
37. #64105C22semidirect
38. #64105C22semidirect
41. C4⋊C4×C3⋊D3C22semidirect
42. C4⋊C4×C3⋊D3C22semidirect
43. #64099C22semidirect
44. #64097C22semidirect
53. #64079C22semidirect
54. #64079C22semidirect
55. #64078C22semidirect
56. #64078C22semidirect
57. #64078C22semidirect
58. C4×#634C22semidirect
59. C4×#634C22semidirect
60. C3×#533D3semidirect
81. C2×C4⋊(C3⋊D3)Q8semidirect
125. C3×C4⋊Q8C2×D3semidirect
126. C3×C4⋊Q8C2×D3semidirect
127. C3×#125C2×D3semidirect
130. C3×C4×Q8C2×D3semidirect
131. C3×C4×Q8C2×D3semidirect
170. C22×C3⋊D3C2×Q8semidirect
176. C2×C32×C4C2×Q8semidirect
189. C4⋊(C3⋊D3)C2×Q8semidirect
190. C4⋊(C3⋊D3)C2×Q8semidirect
207. #533C3⋊D3semidirect
285. C4⋊Q8C2×C3⋊D3semidirect
286. C4⋊Q8C2×C3⋊D3semidirect
290. #125C2×C3⋊D3semidirect
305. C4×Q8C2×C3⋊D3semidirect
306. C4×Q8C2×C3⋊D3semidirect
332. C2×C3×C4D3×Q8semidirect
426. C32#3501semidirect
449. C2×C4Q8×C3⋊D3semidirect
482. C3#86280semidirect

Maximal subgroups

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

subgroupindex
C32×#5332
#105962012
#105961982
#105961612
#105961052
C3⋊D3×C22⋊Q82
#105960502
Q8×C4⋊(C3⋊D3)2
#105959922
#105959902
C2×C4×C4⋊(C3⋊D3)2
#105959412
#862803