Finite Group Databasebeta

← #13177496

Subgroups — #13177496

Order 1728.

2140 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#237C211yesmaximal, characteristic
2D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#67⋊(C3⋊Dic3)C212yesmaximal, complemented
5#67×C3⋊Dic3C212yesmaximal, complemented
6C2×Dic3⋊(C3⋊Dic3)C212yesmaximal, complemented
7#11723594C212yesmaximal, complemented
8D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
9#11723423C211yesmaximal, characteristic, complemented
10#11723397C211yesmaximal, characteristic, complemented
11Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
12#10596298—31nomaximal
13#10595222—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C22×C3×C3⋊Dic3C2212yes—
17C2×C32×#67C2212yes—
18C22×C32×Dic3C2212yes—
19C32×#237C418yescomplemented
20C2×C32×C4×D3C2211yescharacteristic
21C2×C32×Dic6C2211yescharacteristic
22C2×D3×C3⋊Dic3C2212yes—
23#88864C2212yes—
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
26C3×#3757C2212yes—
27C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C3×C4×C3⋊Dic3C2211yescharacteristic
29Dic3⋊(C3⋊Dic3)C2211yescharacteristic
30Dic3⋊(C3⋊Dic3)C2212yes—
31Dic3⋊(C3⋊Dic3)C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2212yes—
33Dic3⋊(C3⋊Dic3)C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
35(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37Dic3×C3⋊Dic3C2212yes—
38C2×C32×#41—31no—
39C2×C3×#237D314yes—
40Q8×C3⋊Dic3—31no—
41Q8×C3⋊Dic3—31no—
42D4×C3⋊Dic3—31no—
43D4×C3⋊Dic3—32no—
44D4×C3⋊Dic3—32no—
45D4×C3⋊Dic3—34no—
46D4×C3⋊Dic3—64no—
47D4×C3⋊Dic3D311yescharacteristic, complemented
48#64135—32no—
49#64135—31no—
50C2×C4⋊(C3⋊Dic3)—32no—
51C2×C4⋊(C3⋊Dic3)—31no—
52C3×D4×Dic3—34no—
53#67⋊Dic3—38no—
54Dic3×#67—38no—
55C2×Dic3⋊Dic3—38no—
56#63955—38no—
57D3×C4⋊Dic3—34no—
58#63855—34no—
59#63843—34no—
60Dic3×Dic6—34no—
61C23×C33C2312yes—
62C22×C32×D3C2311yescharacteristic
63C33×D4C2×C414yescomplemented
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)—42no—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3×C3⋊Dic3—42no—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2312yes—
73C32×#67C2×C418yescomplemented
74C2×C32×Dic3C2×C418yescomplemented
75C2×C32×Dic3C2312yes—
76C2×C32×Dic3C2311yescharacteristic
77C32×C4×D3C2×C414yescomplemented
78C32×Dic6C2×C414yescomplemented
79D3×C3⋊Dic3—42no—
80#5827—94no—
81#5546—91no—
82C32×#41—38no—
83C2×C32×Q8—31no—
84C2×C32×D4—24no—
85C2×C32×D4—32no—
86C2×C32×D4C2×D311yescharacteristic
87C2×C32×D4C2×D314yes—
88C22×C32×C4—32no—
89C22×C32×C4—31no—
90C22×C3⋊Dic3—32no—
91C22×C3⋊Dic3—38no—
92C22×C3⋊Dic3—32no—
93C22×C3⋊Dic3—68no—
94C22×C3⋊Dic3C2×D312yes—
95C2×C3×#67C2×D318yes—
96C22×C3×Dic3—38no—
97C22×C3×Dic3C2×D318yes—
98C3×#237Dic3132yescomplemented
99C2×C3×C4×D3C2×D314yes—
100C2×C3×Dic6C2×D314yes—
101C2×D3×Dic3—38no—
102#3757—32no—
103#3757—38no—
104#3757—31no—
105#3757—31no—
106#3757—68no—
107#3757C2×D312yes—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—34no—
110C4⋊(C3⋊Dic3)—32no—
111C4⋊(C3⋊Dic3)—31no—
112C4⋊(C3⋊Dic3)—32no—
113C4⋊(C3⋊Dic3)—32no—
114C4⋊(C3⋊Dic3)—31no—
115C4⋊(C3⋊Dic3)—64no—
116C4⋊(C3⋊Dic3)C2×D311yescharacteristic
117C4×C3⋊Dic3—31no—
118C4×C3⋊Dic3—34no—
119C4×C3⋊Dic3—32no—
120C4×C3⋊Dic3—32no—
121C4×C3⋊Dic3—64no—
122C4×C3⋊Dic3C2×D311yescharacteristic
123C3×#217—38no—
124C3×C4⋊Dic3—34no—
125C3×C4×Dic3—34no—
126Dic3⋊Dic3—38no—
127Dic3⋊Dic3—38no—
128Dic3⋊Dic3—34no—
129Dic3⋊Dic3—34no—
130Dic3⋊Dic3—38no—
131(C2×D3)⋊Dic3—34no—
132(C2×D3)⋊Dic3—34no—
133Dic32—38no—
134Dic32—38no—
135C22×C33—22no—
136C22×C33—22no—
137C22×C33C22×C414yes—
138C22×C33C2411yescharacteristic
139C2×C32×D3—21no—
140C2×C32×D3—21no—
141C2×C32×D3C22×C412yes—
142C33×C4C22×C412yes—
143C3⋊(C3⋊Dic3)—21no—
144C3⋊(C3⋊Dic3)—22no—
145C3⋊(C3⋊Dic3)—21no—
146C3×C3⋊Dic3—22no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—21no—
149C32×Dic3C22×C414yes—
150C32×Dic3C22×C412yes—
151C2×C3×#41—34no—
152C2×#237C3⋊D311yescharacteristic
153C3×C4×D4—91no—
154Q8×Dic3—94no—
155Q8×Dic3—94no—
156D4×Dic3—34no—
157D4×Dic3—98no—
158D4×Dic3—98no—
159D4×Dic3—94no—
160D4×Dic3—91no—
161D4×Dic3—184no—
162C4×#67—92no—
163#926—94no—
164#926—98no—
165C2×C4⋊Dic3—94no—
166C2×C4⋊Dic3—98no—
167C2×Dic3⋊C4—92no—
168D3×C4⋊C4—91no—
169#887—91no—
170#67⋊C4—92no—
171#877—92no—
172#872—91no—
173C4×Dic6—91no—
174C23×C32—28no—
175C23×C32—31no—
176C23×C32C22×D318yes—
177C23×C32C22×D312yes—
178C22×C3×D3C22×D314yes—
179C32×Q8—34no—
180C32×D4—216no—
181C32×D4—38no—
182C32×D4C4×D314yescomplemented
183C32×D4C2×Dic3116yescomplemented
184C2×C32×C4—24no—
185C2×C32×C4—38no—
186C2×C32×C4—32no—
187C2×C32×C4—31no—
188C2×C32×C4—34no—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—32no—
194C2×C3⋊Dic3—34no—
195C2×C3⋊Dic3—38no—
196C2×C3⋊Dic3—34no—
197C2×C3⋊Dic3—42no—
198C2×C3⋊Dic3—64no—
199C2×C3⋊Dic3—64no—
200C2×C3⋊Dic3—68no—
201C2×C3⋊Dic3—122no—
202C2×C3⋊Dic3—122no—
203C2×C3⋊Dic3—128no—
204C2×C3⋊Dic3—1216no—
205C2×C3⋊Dic3C22×D311yescharacteristic
206C2×C3⋊Dic3C22×D312yes—
207C2×C3⋊Dic3C22×D311yescharacteristic
208C3×#67C2×Dic3132yescomplemented
209C2×C3×Dic3—34no—
210C2×C3×Dic3—38no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—128no—
213C2×C3×Dic3C2×Dic3132yescomplemented
214C2×C3×Dic3C22×D318yes—
215C2×C3×Dic3C22×D314yes—
216C3×C4×D3C2×Dic3116yescomplemented
217C3×Dic6C2×Dic3116yescomplemented
218D3×Dic3—128no—
219#520—271noSylow
220C2×C33—24no—
221C2×C33C23×C411yescharacteristic
222C2×C33H(5,𝔽2)11yescharacteristic
223C2×C332−1+411yescharacteristic
224C32×D3—22no—
225C3×#41—332no—
226C2×C3×Q8—34no—
227C2×C3×D4—24no—
228C2×C3×D4—38no—
229C2×C3×D4D3214yes—
230C2×C3×D4C2×C3⋊D311yescharacteristic
231C22×C3×C4—38no—
232C22×C3×C4—34no—
233C22×C3×C4—92no—
234C2×#67C2×C3⋊D312yes—
235C22×Dic3—38no—
236C22×Dic3—98no—
237C22×Dic3—98no—
238C22×Dic3—92no—
239C22×Dic3—188no—
240C22×Dic3C2×C3⋊D312yes—
241#237C3⋊Dic318yescomplemented
242C2×C4×D3—92no—
243C2×C4×D3C2×C3⋊D311yescharacteristic
244C2×Dic6C2×C3⋊D311yescharacteristic
245C3×C4⋊C4—91no—
246C3×C22⋊C4—92no—
247C3×C42—91no—
248#217—38no—
249#217—94no—
250#217—94no—
251#217—98no—
252#217—92no—
253#217—188no—
254(C2×D3)⋊C4—91no—
255(C2×D3)⋊C4—91no—
256C4⋊Dic3—34no—
257C4⋊Dic3—94no—
258C4⋊Dic3—98no—
259C4⋊Dic3—98no—
260C4⋊Dic3—98no—
261C4⋊Dic3—94no—
262C4⋊Dic3—94no—
263C4⋊Dic3—91no—
264C4⋊Dic3—184no—
265Dic3⋊C4—92no—
266Dic3⋊C4—91no—
267Dic3⋊C4—91no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270C4×Dic3—34no—
271C4×Dic3—98no—
272C4×Dic3—98no—
273C4×Dic3—94no—
274C4×Dic3—91no—
275C4×Dic3—92no—
276C4×Dic3—92no—
277C4×Dic3—184no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—24no—
281C22×C32—216no—
282C22×C32—28no—
283C22×C32—28no—
284C22×C32—32no—
285C22×C32—48no—
286C22×C32—48no—
287C22×C32—61no—
288C22×C32—61no—
289C22×C32C2×C4×D314yes—
290C22×C32C22×Dic3116yes—
291C22×C32C23×D314yes—
292C22×C32C23×D311yescharacteristic
293C2×C3×D3—24no—
294C2×C3×D3—24no—
295C2×C3×D3C22×Dic318yes—
296C32×C4—28no—
297C32×C4—32no—
298C32×C4—34no—
299C32×C4C2×C4×D312yes—
300C32×C4C22×Dic318yes—
301C3⋊Dic3—21no—
302C3⋊Dic3—22no—
303C3⋊Dic3—21no—
304C3⋊Dic3—68no—
305C3⋊Dic3—61no—
306C3⋊Dic3—62no—
307C3⋊Dic3—64no—
308C3⋊Dic3—61no—
309C3⋊Dic3—68no—
310C3⋊Dic3—64no—
311C3⋊Dic3—124no—
312C3⋊Dic3—128no—
313C3×Dic3—68no—
314C3×Dic3—64no—
315C3×Dic3—64no—
316C3×Dic3C22×Dic3116yes—
317C3×Dic3C22×Dic318yes—
318C2×#41—31no—
319C4×Q8—271no—
320C4×Q8—271no—
321C4×D4—91no—
322C4×D4—272no—
323C4×D4—272no—
324C4×D4—271no—
325#117—271no—
326#117—272no—
327C2×C4⋊C4—271no—
328C2×C4⋊C4—272no—
329C33#52011yescharacteristic, complemented, Sylow
330C23×C3—28no—
331C23×C3—34no—
332C23×C3C2×D3218yes—
333C23×C3C22×C3⋊D312yes—
334C22×D3C22×C3⋊D311yescharacteristic
335C3×Q8—316no—
336C3×D4—216no—
337C3×D4—332no—
338C3×D4D3×Dic3116yescomplemented
339C3×D4C2×C3⋊Dic314yescomplemented
340C2×C3×C4—24no—
341C2×C3×C4—38no—
342C2×C3×C4—34no—
343C2×C3×C4—332no—
344C2×C3×C4—316no—
345C2×C3×C4—91no—
346C2×C3×C4—92no—
347C2×C3×C4—91no—
348C2×C3×C4—362no—
349C2×C3×C4C2×D3214yes—
350C2×C3×C4C22×C3⋊D311yescharacteristic
351#67C2×C3⋊Dic318yescomplemented
352C2×Dic3—34no—
353C2×Dic3—34no—
354C2×Dic3—38no—
355C2×Dic3—94no—
356C2×Dic3—94no—
357C2×Dic3—98no—
358C2×Dic3—91no—
359C2×Dic3—91no—
360C2×Dic3—92no—
361C2×Dic3—128no—
362C2×Dic3—184no—
363C2×Dic3—184no—
364C2×Dic3—188no—
365C2×Dic3—368no—
366C2×Dic3—368no—
367C2×Dic3—362no—
368C2×Dic3—3616no—
369C2×Dic3C2×C3⋊Dic318yescomplemented
370C2×Dic3C22×C3⋊D312yes—
371C2×Dic3C22×C3⋊D311yescharacteristic
372C4×D3—362no—
373C4×D3C2×C3⋊Dic314yescomplemented
374Dic6C2×C3⋊Dic314yescomplemented
375C2×C32—24no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—216no—
380C2×C32—416no—
381C2×C32—62no—
382C2×C32C22×C4×D311yescharacteristic
383C2×C32#100411yescharacteristic
384C2×C32#100914yes—
385C2×C32#101014yes—
386C2×C32#101011yescharacteristic
387C2×C32C23×Dic314yes—
388C3×D3—28no—
389#41—38no—
390C2×Q8—31no—
391C2×D4—32no—
392C2×D4D3×C3⋊D311yescharacteristic
393C22×C4—32no—
394C22×C4—31no—
395C22×C4—92no—
396C22×C4—272no—
397C22×C4—272no—
398C4⋊C4—91no—
399C4⋊C4—271no—
400C4⋊C4—272no—
401C4⋊C4—272no—
402C4⋊C4—272no—
403C4⋊C4—271no—
404C4⋊C4—271no—
405C22⋊C4—92no—
406C22⋊C4—271no—
407C22⋊C4—271no—
408C22⋊C4—272no—
409C42—91no—
410C42—272no—
411C42—272no—
412C42—271no—
413C22×C3—28no—
414C22×C3—28no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—24no—
418C22×C3—216no—
419C22×C3—38no—
420C22×C3—48no—
421C22×C3—48no—
422C22×C3—64no—
423C22×C3—64no—
424C22×C3C2×D3×Dic3116yes—
425C22×C3C22×C3⋊Dic314yes—
426C22×C3C22×D3214yes—
427C22×C3C23×C3⋊D311yescharacteristic
428C2×D3—21no—
429C2×D3—21no—
430C2×D3C22×C3⋊Dic312yes—
431C3×C4—28no—
432C3×C4—38no—
433C3×C4—316no—
434C3×C4—182no—
435C3×C4—181no—
436C3×C4—181no—
437C3×C4C2×D3×Dic318yes—
438C3×C4C22×C3⋊Dic312yes—
439Dic3—64no—
440Dic3—64no—
441Dic3—68no—
442Dic3—188no—
443Dic3—184no—
444Dic3—184no—
445Dic3—188no—
446Dic3—182no—
447Dic3—181no—
448Dic3—181no—
449Dic3—368no—
450Dic3—364no—
451Dic3C22×C3⋊Dic314yes—
452Dic3C22×C3⋊Dic312yes—
453C32—24no—
454C32#554611yescharacteristic, complemented
455C32#582714yescomplemented
456C23—31no—
457C23C2×D3×C3⋊D312yes—
458Q8—34no—
459D4—38no—
460D4D3×C3⋊Dic314yescomplemented
461C2×C4—32no—
462C2×C4—31no—
463C2×C4—38no—
464C2×C4—34no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—92no—
468C2×C4—271no—
469C2×C4—271no—
470C2×C4—272no—
471C2×C4—362no—
472C2×C4—1082no—
473C2×C4—1082no—
474C2×C4C2×D3×C3⋊D311yescharacteristic
475C2×C3—216no—
476C2×C3—24no—
477C2×C3—24no—
478C2×C3—24no—
479C2×C3—24no—
480C2×C3—416no—
481C2×C3—68no—
482C2×C3#6430914yes—
483C2×C3#6431414yes—
484C2×C3C22×D3×Dic314yes—
485C2×C3#6436611yescharacteristic
486C2×C3#6436711yescharacteristic
487C2×C3C23×C3⋊Dic311yescharacteristic
488D3—22no—
489C22—22no—
490C22—22no—
491C22—32no—
492C22—61no—
493C22—61no—
494C22C2×D3×C3⋊Dic314yes—
495C22C22×D3×C3⋊D311yescharacteristic
496C4—32no—
497C4—34no—
498C4—181no—
499C4—181no—
500C4—182no—
501C4—541no—
502C4—541no—
503C4—542no—
504C4C2×D3×C3⋊Dic312yes—
505C3—24no—
506C3#1059522214yescomplemented
507C3#1059629811yescharacteristic, complemented
508C2—24no—
509C2—62no—
510C2#1172458211yescharacteristic
511C2#1172458911yescharacteristic
512C2C22×D3×C3⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC2×C33×D4C22
socle, cosocleC22×C33C24
prime radicalC2×C32×#237C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D4×C3⋊(C3⋊Dic3)C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. #67⋊(C3⋊Dic3)C2semidirect
5. #67×C3⋊Dic3C2semidirect
6. C2×Dic3⋊(C3⋊Dic3)C2semidirect
7. #11723594C2semidirect
8. D3×C4⋊(C3⋊Dic3)C2semidirect
9. #11723423C2semidirect
10. #11723397C2semidirect
11. Dic6×C3⋊Dic3C2semidirect
19. C32×#237C4semidirect
47. D4×C3⋊Dic3D3semidirect
63. C33×D4C2×C4semidirect
73. C32×#67C2×C4semidirect
74. C2×C32×Dic3C2×C4semidirect
77. C32×C4×D3C2×C4semidirect
78. C32×Dic6C2×C4semidirect
98. C3×#237Dic3semidirect
182. C32×D4C4×D3semidirect
183. C32×D4C2×Dic3semidirect
208. C3×#67C2×Dic3semidirect
213. C2×C3×Dic3C2×Dic3semidirect
216. C3×C4×D3C2×Dic3semidirect
217. C3×Dic6C2×Dic3semidirect
241. #237C3⋊Dic3semidirect
329. C33#520semidirect
338. C3×D4D3×Dic3semidirect
339. C3×D4C2×C3⋊Dic3semidirect
351. #67C2×C3⋊Dic3semidirect
369. C2×Dic3C2×C3⋊Dic3semidirect
373. C4×D3C2×C3⋊Dic3semidirect
374. Dic6C2×C3⋊Dic3semidirect
454. C32#5546semidirect
455. C32#5827semidirect
460. D4D3×C3⋊Dic3semidirect
506. C3#10595222semidirect
507. C3#10596298semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
D4×C3⋊(C3⋊Dic3)2
C3×D4×C3⋊Dic32
#67⋊(C3⋊Dic3)2
#67×C3⋊Dic32
C2×Dic3⋊(C3⋊Dic3)2
#117235942
D3×C4⋊(C3⋊Dic3)2
#117234232
#117233972
Dic6×C3⋊Dic32
#105962983
#105952223