Finite Group Databasebeta

← #13177253

Subgroups — #13177253

Order 1728.

2132 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×C3×#3828C211yesmaximal, characteristic, complemented
2C2×C32×#235C211yesmaximal, characteristic, complemented
3C2×C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
4#11723637C214yesmaximal, complemented
5#11723522C212yesmaximal, complemented
6#11723515C212yesmaximal, complemented
7#11723435C212yesmaximal, complemented
8#11723425C212yesmaximal, complemented
9#10596304—31nomaximal
10#10595044—34nomaximal
11C22×C33×C4C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2212yes—
13C2×C3×#638C2212yes—
14C3×#3828—24no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×#634C2211yescharacteristic
18C2×C32×#67C2212yes—
19C32×#235—24no—
20C2×C32×D12C2211yescharacteristic
21C2×C32×C4×D3C2212yes—
22C2×C32×Dic6C2211yescharacteristic
23C2×#6290C2214yes—
24C4⋊(C3⋊(C3⋊Dic3))C2212yes—
25C4⋊(C3⋊(C3⋊Dic3))C2212yes—
26#88806C2214yes—
27#88803C2214yes—
28(C2×D3)⋊(C3⋊Dic3)C2214yes—
29C2×C32×#41—31no—
30C2×#3828D311yescharacteristic, complemented
31C2×C3×#235—34no—
32C2×C3×#235D314yescomplemented
33#64155—32no—
34#64148—34no—
35#64147—32no—
36#64139—32no—
37C2×C4⋊(C3⋊Dic3)—31no—
38C2×C4⋊(C3⋊Dic3)—34no—
39C2×C4⋊(C3⋊Dic3)—64no—
40#64133—32no—
41#63980—316no—
42#63908—38no—
43#63908—38no—
44#63856—38no—
45#63856—38no—
46C23×C33C2311yescharacteristic
47C22×C3×C3⋊D3C2312yes—
48C22×C32×D3C2312yes—
49C2×C33×C4D414yescomplemented
50C2×C33×C4C2311yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)—42no—
53C2×C3⋊(C3⋊Dic3)C2314yes—
54C3×#638—24no—
55C2×C3×C3⋊Dic3C2312yes—
56C3×C4⋊(C3⋊D3)—22no—
57C3×C4×C3⋊D3—24no—
58C3×#634—22no—
59C32×#67—24no—
60C2×C32×Dic3C2312yes—
61C32×D12—22no—
62C32×C4×D3—24no—
63C32×Dic6—22no—
64#6290—44no—
65#5833—94no—
66#5833—91no—
67C32×#41—64no—
68C2×C32×Q8—31no—
69C2×C32×D4—32no—
70C2×C32×D4—31no—
71C22×C32×C4—24no—
72C22×C32×C4—32no—
73C22×C32×C4C2×D314yes—
74C22×C32×C4C2×D311yescharacteristic
75C2×#638—34no—
76C2×#638C2×D312yes—
77C22×C3⋊Dic3—32no—
78C22×C3⋊Dic3—38no—
79C22×C3⋊Dic3—68no—
80#3828—24no—
81C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D312yes—
83C2×#634C2×D311yescharacteristic
84C2×C3×#67—38no—
85C2×C3×#67C2×D318yes—
86C3×#235—216no—
87C3×#235—616no—
88C2×C3×D12—34no—
89C2×C3×D12C2×D314yes—
90C2×C3×C4×D3—38no—
91C2×C3×C4×D3C2×D318yes—
92C2×C3×Dic6—34no—
93C2×C3×Dic6C2×D314yes—
94C2×#625—316no—
95#3757—34no—
96#3752—34no—
97C4⋊(C3⋊Dic3)—32no—
98C4⋊(C3⋊Dic3)—32no—
99C4⋊(C3⋊Dic3)—38no—
100C4⋊(C3⋊Dic3)—38no—
101C4⋊(C3⋊Dic3)—68no—
102C4⋊(C3⋊Dic3)—68no—
103#3750—34no—
104#3724—316no—
105(C2×D3)⋊Dic3—316no—
106(C2×D3)⋊Dic3—316no—
107C22×C33—21no—
108C22×C33—21no—
109C22×C33C2×D412yes—
110C22×C33C2411yescharacteristic
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—22no—
114C2×C32×D3—22no—
115C2×C32×D3—22no—
116C2×C32×D3—22no—
117C33×C4C2×D412yes—
118C33×C4C2×D412yes—
119C3⋊(C3⋊Dic3)—24no—
120C3×C3⋊Dic3—22no—
121C32×Dic3—22no—
122C2×C3×#41—34no—
123C2×C3×#41—91no—
124C2×#235—34no—
125C2×#235C3⋊D311yescharacteristic, complemented
126#946—98no—
127#946—92no—
128#939—916no—
129#939—94no—
130#938—98no—
131#938—92no—
132#930—98no—
133#930—92no—
134C2×C4⋊Dic3—94no—
135C2×C4⋊Dic3—91no—
136C2×C4⋊Dic3—184no—
137#924—98no—
138#924—92no—
139C23×C32—24no—
140C23×C32—32no—
141C23×C32C22×D314yes—
142C23×C32C22×D311yescharacteristic
143C22×C3⋊D3C22×D312yes—
144C22×C3×D3—38no—
145C22×C3×D3C22×D318yes—
146C32×Q8—62no—
147C32×D4—64no—
148C32×D4—62no—
149C2×C32×C4—216no—
150C2×C32×C4—24no—
151C2×C32×C4—24no—
152C2×C32×C4—32no—
153C2×C32×C4—64no—
154C2×C32×C4#67116yescomplemented
155C2×C32×C4#6714yescomplemented
156C2×C32×C4C22×D314yes—
157C2×C32×C4C22×D314yes—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160#638—24no—
161#638—124no—
162C2×C3⋊Dic3—34no—
163C2×C3⋊Dic3—316no—
164C2×C3⋊Dic3—616no—
165C2×C3⋊Dic3—122no—
166C2×C3⋊Dic3—128no—
167C2×C3⋊Dic3—1216no—
168C2×C3⋊Dic3C22×D312yes—
169C4⋊(C3⋊D3)—22no—
170C4×C3⋊D3—24no—
171#634—22no—
172C3×#67—216no—
173C3×#67—616no—
174C2×C3×Dic3—38no—
175C2×C3×Dic3C22×D318yes—
176C3×D12—28no—
177C3×D12—68no—
178C3×C4×D3—216no—
179C3×C4×D3—616no—
180C3×Dic6—28no—
181C3×Dic6—68no—
182#625—1216no—
183#537—271noSylow
184C2×C33—22no—
185C2×C33C22×D411yescharacteristic
186C2×C33H(5,𝔽2)11yescharacteristic
187C2×C332−1+411yescharacteristic
188C3×C3⋊D3—42no—
189C32×D3—42no—
190C3×#41—616no—
191C3×#41—184no—
192C2×C3×Q8—34no—
193C2×C3×Q8—91no—
194C2×C3×D4—38no—
195C2×C3×D4—34no—
196C2×C3×D4—92no—
197C2×C3×D4—91no—
198C22×C3×C4—24no—
199C22×C3×C4—38no—
200C22×C3×C4—92no—
201C22×C3×C4D3214yes—
202C22×C3×C4C2×C3⋊D311yescharacteristic
203C2×#67—38no—
204C2×#67—916no—
205C2×#67—94no—
206C2×#67C2×C3⋊D312yes—
207C22×Dic3—98no—
208C22×Dic3—92no—
209C22×Dic3—188no—
210#235—24no—
211#235—616no—
212C2×D12—34no—
213C2×D12C2×C3⋊D311yescharacteristic
214C2×C4×D3—38no—
215C2×C4×D3C2×C3⋊D312yes—
216C2×Dic6—34no—
217C2×Dic6C2×C3⋊D311yescharacteristic
218#217—916no—
219#217—94no—
220(C2×D3)⋊C4—916no—
221(C2×D3)⋊C4—94no—
222C4⋊Dic3—98no—
223C4⋊Dic3—98no—
224C4⋊Dic3—92no—
225C4⋊Dic3—92no—
226C4⋊Dic3—188no—
227C4⋊Dic3—188no—
228Dic3⋊C4—916no—
229Dic3⋊C4—94no—
230C22×C32—28no—
231C22×C32—28no—
232C22×C32—24no—
233C22×C32—21no—
234C22×C32—24no—
235C22×C32—21no—
236C22×C32—24no—
237C22×C32—28no—
238C22×C32—62no—
239C22×C32—62no—
240C22×C32—62no—
241C22×C32C2×#6718yes—
242C22×C32C2×#6712yes—
243C22×C32C23×D314yes—
244C22×C32C23×D311yescharacteristic
245C2×C3⋊D3—22no—
246C2×C3⋊D3—22no—
247C2×C3⋊D3—22no—
248C2×C3×D3—28no—
249C2×C3×D3—28no—
250C2×C3×D3—28no—
251C2×C3×D3—68no—
252C2×C3×D3—68no—
253C2×C3×D3—68no—
254C32×C4—28no—
255C32×C4—28no—
256C32×C4—62no—
257C32×C4C2×#6718yes—
258C32×C4C2×#6718yes—
259C32×C4C2×#6712yes—
260C32×C4C2×#6712yes—
261C3⋊Dic3—22no—
262C3⋊Dic3—64no—
263C3⋊Dic3—616no—
264C3⋊Dic3—1216no—
265C3×Dic3—28no—
266C3×Dic3—68no—
267C2×#41—31no—
268C2×#41—91no—
269C22⋊Q8—272no—
270C22⋊Q8—272no—
271#121—272no—
272#121—272no—
273#121—274no—
274C2×C4⋊C4—271no—
275C33#53711yescharacteristic, complemented, Sylow
276C23×C3—24no—
277C23×C3—38no—
278C23×C3—92no—
279C23×C3C2×D3214yes—
280C23×C3C22×C3⋊D311yescharacteristic
281C22×D3—38no—
282C22×D3C22×C3⋊D312yes—
283C3×Q8—68no—
284C3×Q8—182no—
285C3×D4—616no—
286C3×D4—68no—
287C3×D4—184no—
288C3×D4—182no—
289C2×C3×C4—216no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—38no—
293C2×C3×C4—616no—
294C2×C3×C4—92no—
295C2×C3×C4—184no—
296C2×C3×C4#625116yescomplemented
297C2×C3×C4#63814yescomplemented
298C2×C3×C4C2×D3214yes—
299C2×C3×C4C2×D3214yes—
300C2×C3×C4C22×C3⋊D311yescharacteristic
301C2×C3×C4C22×C3⋊D311yescharacteristic
302#67—24no—
303#67—616no—
304#67—364no—
305#67—3616no—
306C2×Dic3—38no—
307C2×Dic3—916no—
308C2×Dic3—94no—
309C2×Dic3—1816no—
310C2×Dic3—368no—
311C2×Dic3—362no—
312C2×Dic3—3616no—
313C2×Dic3C22×C3⋊D312yes—
314D12—22no—
315D12—68no—
316C4×D3—24no—
317C4×D3—616no—
318Dic6—22no—
319Dic6—68no—
320C2×C32—216no—
321C2×C32—28no—
322C2×C32—22no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—24no—
326C2×C32—122no—
327C2×C32#100914yes—
328C2×C32#100911yescharacteristic
329C2×C32#101014yes—
330C2×C32#101011yescharacteristic
331C2×C32C22×#6714yes—
332C2×C32C22×#6711yescharacteristic
333C3⋊D3—42no—
334C3×D3—48no—
335C3×D3—128no—
336#41—64no—
337#41—184no—
338C2×Q8—31no—
339C2×Q8—91no—
340C2×D4—32no—
341C2×D4—31no—
342C2×D4—92no—
343C2×D4—91no—
344C2×D4—274no—
345C22×C4—32no—
346C22×C4—92no—
347C22×C4—272no—
348C22×C4D3×C3⋊D311yescharacteristic
349C4⋊C4—274no—
350C4⋊C4—272no—
351C4⋊C4—272no—
352C22⋊C4—274no—
353C22⋊C4—274no—
354C22×C3—28no—
355C22×C3—28no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—24no—
361C22×C3—28no—
362C22×C3—68no—
363C22×C3—68no—
364C22×C3—68no—
365C22×C3—182no—
366C22×C3—182no—
367C22×C3—182no—
368C22×C3C2×#62518yes—
369C22×C3C2×#63812yes—
370C22×C3C22×D3214yes—
371C22×C3C23×C3⋊D311yescharacteristic
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—22no—
375C2×D3—68no—
376C2×D3—68no—
377C2×D3—68no—
378C3×C4—28no—
379C3×C4—28no—
380C3×C4—68no—
381C3×C4—182no—
382C3×C4C2×#62518yes—
383C3×C4C2×#62518yes—
384C3×C4C2×#63812yes—
385C3×C4C2×#63812yes—
386Dic3—22no—
387Dic3—68no—
388Dic3—1816no—
389Dic3—184no—
390Dic3—3616no—
391C32—24no—
392C32#583314yescomplemented
393C32#583311yescharacteristic, complemented
394C23—32no—
395C23—92no—
396C23C2×D3×C3⋊D311yescharacteristic
397Q8—62no—
398Q8—182no—
399D4—64no—
400D4—62no—
401D4—184no—
402D4—182no—
403D4—1084no—
404C2×C4—32no—
405C2×C4—64no—
406C2×C4—92no—
407C2×C4—184no—
408C2×C4—274no—
409C2×C4—1082no—
410C2×C4#629014yescomplemented
411C2×C4C2×D3×C3⋊D311yescharacteristic
412C2×C4C2×D3×C3⋊D311yescharacteristic
413C2×C3—216no—
414C2×C3—22no—
415C2×C3—28no—
416C2×C3—24no—
417C2×C3—24no—
418C2×C3—24no—
419C2×C3—128no—
420C2×C3—362no—
421C2×C3#6429814yes—
422C2×C3#6430814yes—
423C2×C3C22×#62514yes—
424C2×C3#6436611yescharacteristic
425C2×C3#6436711yescharacteristic
426C2×C3C22×#63811yescharacteristic
427D3—42no—
428D3—128no—
429C22—21no—
430C22—21no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—182no—
435C22—182no—
436C22—182no—
437C22C2×#629012yes—
438C22C22×D3×C3⋊D311yescharacteristic
439C4—62no—
440C4—182no—
441C4—544no—
442C4C2×#629012yes—
443C4C2×#629012yes—
444C3—24no—
445C3#1059504414yescomplemented
446C3#1059630411yescharacteristic, complemented
447C2—22no—
448C2—122no—
449C2—362no—
450C2#1172456311yescharacteristic
451C2#1172458111yescharacteristic
452C2C22×#629011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#537
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C2×C32×#235C2semidirect
3. C2×C4⋊(C3⋊(C3⋊Dic3))C2semidirect
4. #11723637C2semidirect
5. #11723522C2semidirect
6. #11723515C2semidirect
7. #11723435C2semidirect
8. #11723425C2semidirect
30. C2×#3828D3semidirect
32. C2×C3×#235D3semidirect
49. C2×C33×C4D4semidirect
125. C2×#235C3⋊D3semidirect
154. C2×C32×C4#67semidirect
155. C2×C32×C4#67semidirect
275. C33#537semidirect
296. C2×C3×C4#625semidirect
297. C2×C3×C4#638semidirect
392. C32#5833semidirect
393. C32#5833semidirect
410. C2×C4#6290semidirect
445. C3#10595044semidirect
446. C3#10596304semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C32×#2352
C2×C4⋊(C3⋊(C3⋊Dic3))2
#117236372
#117235222
#117235152
#117234352
#117234252
#105963043
#105950443