Finite Group Databasebeta

← #13178508

Subgroups — #13178508

Order 1728.

1386 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C3×#3748C212yesmaximal, complemented
3#11723657C218yesmaximal, complemented
4#11723656C214yesmaximal, complemented
5#10594780—31nomaximal
6#10594776—32nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×C3×#638C412yes—
9C3×#3828C414yes—
10C3×#3828C2214yes—
11C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
12C2×C3×C4×C3⋊D3C2212yes—
13C2×C3×#634C411yescharacteristic
14C3×#3748C2214yes—
15C3×#3748C2214yescomplemented
16C2×C3×C3⋊(C3⋊C8)C2214yescomplemented
17#88809C2218yescomplemented
18#88808C2218yescomplemented
19C2×#3828D311yescharacteristic
20C2×C3×#235—32no—
21C2×C3×#235—61no—
22C2×#3748D312yescomplemented
23C2×C3×#208—32no—
24C2×C3×#208—32no—
25C2×C3×#208—62no—
26#63818—34no—
27#63817—38no—
28#63815—38no—
29#63814—38no—
30D3×#208—38no—
31C23×C33C2×C411yescharacteristic
32C22×C3×C3⋊D3C2×C412yes—
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C412yes—
35C2×C33×C4C2312yes—
36C2×C33×C4C2311yescharacteristic
37C3×#638C2×C418yes—
38C2×C3×C3⋊Dic3C2×C412yes—
39C3×C4⋊(C3⋊D3)C2×C414yes—
40C3×C4×C3⋊D3C2×C414yes—
41C3×C4×C3⋊D3C2314yes—
42C3×#634C2×C414yes—
43C3×C3⋊(C3⋊C8)C2318yes—
44#5819—91no—
45#5745—92no—
46C22×C32×C4—22no—
47C22×C32×C4—21no—
48C22×C32×C4—41no—
49C22×C32×C4C2×D312yes—
50C22×C32×C4C2×D311yescharacteristic
51C2×#638Dic312yes—
52#3828Dic314yes—
53#3828C2×D314yes—
54C2×C4⋊(C3⋊D3)Dic311yescharacteristic
55C2×C4×C3⋊D3C2×D312yes—
56C2×#634Dic311yescharacteristic
57C2×C3×#67—34no—
58C2×C3×#67—62no—
59C3×#235—34no—
60C3×#235—34no—
61C3×#235—34no—
62C3×#235—34no—
63C3×#235—64no—
64C3×#235—64no—
65C2×C3×D12—32no—
66C2×C3×D12—61no—
67C2×C3×C4×D3—34no—
68C2×C3×C4×D3—62no—
69C2×C3×Dic6—32no—
70C2×C3×Dic6—61no—
71#3748C2×D314yes—
72#3748C2×D314yescomplemented
73C2×C3⋊(C3⋊C8)C2×D314yescomplemented
74C3×#208—34no—
75C3×#208—34no—
76C3×#208—34no—
77C3×#208—34no—
78C3×#208—64no—
79C3×#208—64no—
80C2×C3×C3⋊C8—34no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8—64no—
83#3712—38no—
84#3711—316no—
85#3710—38no—
86D3×C3⋊C8—316no—
87C22×C33—41no—
88C22×C33C22×C411yescharacteristic
89C22×C33C22×C412yes—
90C2×C3×C3⋊D3—42no—
91C2×C3×C3⋊D3C22×C414yes—
92C33×C4C22×C412yes—
93C33×C4C22×C411yescharacteristic
94C33×C4C2411yescharacteristic
95C3×C3⋊Dic3C22×C414yes—
96C2×C3×#41—91no—
97C2×#235—32no—
98C2×#235—61no—
99C2×C3×M16—92no—
100#948—98no—
101C2×#208—32no—
102C2×#208—32no—
103C2×#208—62no—
104C2×#208—94no—
105#907—98no—
106D3×M16—98no—
107#901—98no—
108C23×C32—22no—
109C23×C32—21no—
110C23×C32—41no—
111C23×C32C4×D312yes—
112C23×C32C2×Dic311yescharacteristic
113C22×C3⋊D3C2×Dic312yes—
114C22×C3×D3—34no—
115C22×C3×D3—62no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—41no—
127C2×C32×C4—42no—
128C2×C32×C4—42no—
129C2×C32×C4—41no—
130C2×C32×C4C4×D312yes—
131C2×C32×C4C4×D312yes—
132C2×C32×C4C4×D312yes—
133C2×C32×C4C2×Dic311yescharacteristic
134C2×C32×C4C2×Dic312yes—
135C2×C32×C4C22×D312yes—
136C2×C32×C4C22×D312yes—
137C2×C32×C4C22×D312yes—
138C2×C32×C4C22×D312yes—
139C2×C32×C4C22×D311yescharacteristic
140#638C2×Dic318yes—
141C2×C3⋊Dic3C2×Dic312yes—
142C4⋊(C3⋊D3)C2×Dic314yes—
143C4×C3⋊D3C2×Dic314yes—
144C4×C3⋊D3C22×D314yes—
145#634C2×Dic314yes—
146C3×#67—38no—
147C3×#67—38no—
148C3×#67—68no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—62no—
151C3×D12—34no—
152C3×D12—34no—
153C3×D12—64no—
154C3×C4×D3—38no—
155C3×C4×D3—38no—
156C3×C4×D3—64no—
157C3×C4×D3—64no—
158C3×Dic6—34no—
159C3×Dic6—34no—
160C3×Dic6—64no—
161C3⋊(C3⋊C8)C22×D318yes—
162C3×C3⋊C8—38no—
163C3×C3⋊C8—38no—
164C3×C3⋊C8—68no—
165#568—271noSylow
166C2×C33—21no—
167C2×C33—22no—
168C2×C33C23×C411yescharacteristic
169C3×C3⋊D3—24no—
170C3×#41—94no—
171C3×#41—94no—
172C2×C3×Q8—91no—
173C2×C3×D4—92no—
174C2×C3×D4—91no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—41no—
178C22×C3×C4—92no—
179C22×C3×C4D3212yes—
180C22×C3×C4D3211yescharacteristic
181C2×#67—34no—
182C2×#67—62no—
183#235—34no—
184#235—34no—
185#235—34no—
186#235—34no—
187#235—64no—
188#235—64no—
189C2×D12—32no—
190C2×D12—61no—
191C2×C4×D3—34no—
192C2×C4×D3—62no—
193C2×Dic6—32no—
194C2×Dic6—61no—
195C3×M16—94no—
196C3×M16—94no—
197C2×C3×C8—94no—
198#208—34no—
199#208—34no—
200#208—34no—
201#208—34no—
202#208—64no—
203#208—64no—
204#208—98no—
205C2×C3⋊C8—34no—
206C2×C3⋊C8—34no—
207C2×C3⋊C8—64no—
208C2×C3⋊C8—98no—
209ΓL(1,𝔽25)—916no—
210D3×C8—916no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32—22no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—41no—
217C22×C32—42no—
218C22×C32—42no—
219C22×C32—41no—
220C22×C32—44no—
221C22×C32—42no—
222C22×C32—44no—
223C22×C32C2×C4×D312yes—
224C22×C32C2×C4×D312yes—
225C22×C32C2×C4×D312yes—
226C22×C32C22×Dic311yescharacteristic
227C22×C32C22×Dic312yes—
228C2×C3⋊D3—42no—
229C2×C3⋊D3C22×Dic314yes—
230C2×C3×D3—38no—
231C2×C3×D3—64no—
232C2×C3×D3—124no—
233C2×C3×D3—124no—
234C32×C4—22no—
235C32×C4—21no—
236C32×C4—22no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—21no—
241C32×C4—42no—
242C32×C4—41no—
243C32×C4—41no—
244C32×C4C2×C4×D312yes—
245C32×C4C2×C4×D312yes—
246C32×C4C2×C4×D312yes—
247C32×C4C22×Dic312yes—
248C32×C4C22×Dic311yescharacteristic
249C32×C4C23×D312yes—
250C32×C4C23×D311yescharacteristic
251C3⋊Dic3C22×Dic314yes—
252C3×Dic3—38no—
253C3×Dic3—64no—
254C2×#41—91no—
255#131—278no—
256C2×M16—92no—
257C2×M16—274no—
258C33#56811yescharacteristic, complemented, Sylow
259C23×C3—22no—
260C23×C3—21no—
261C23×C3—41no—
262C23×C3—92no—
263C23×C3D3×Dic312yes—
264C23×C3#62411yescharacteristic
265C22×D3—34no—
266C22×D3—62no—
267C3×Q8—94no—
268C3×D4—98no—
269C3×D4—94no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—92no—
285C2×C3×C4—94no—
286C2×C3×C4—94no—
287C2×C3×C4D3×Dic312yes—
288C2×C3×C4D3×Dic312yes—
289C2×C3×C4D3×Dic312yes—
290C2×C3×C4#62411yescharacteristic
291C2×C3×C4#62412yes—
292C2×C3×C4C2×D3212yes—
293C2×C3×C4C2×D3212yes—
294C2×C3×C4C2×D3212yes—
295C2×C3×C4C2×D3212yes—
296C2×C3×C4C2×D3211yescharacteristic
297#67—38no—
298#67—38no—
299#67—68no—
300C2×Dic3—34no—
301C2×Dic3—62no—
302D12—34no—
303D12—34no—
304D12—64no—
305C4×D3—38no—
306C4×D3—38no—
307C4×D3—64no—
308C4×D3—64no—
309Dic6—34no—
310Dic6—34no—
311Dic6—64no—
312C3×C8—98no—
313C3⋊C8—38no—
314C3⋊C8—38no—
315C3⋊C8—68no—
316C2×C32—22no—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—24no—
320C2×C32—22no—
321C2×C32—22no—
322C2×C32—22no—
323C2×C32—22no—
324C2×C32—21no—
325C2×C32—41no—
326C2×C32—42no—
327C2×C32—42no—
328C2×C32—42no—
329C2×C32—42no—
330C2×C32—44no—
331C2×C32C22×C4×D312yes—
332C2×C32C23×Dic311yescharacteristic
333C3⋊D3—24no—
334C3×D3—68no—
335C3×D3—124no—
336#41—94no—
337#41—94no—
338C2×Q8—91no—
339C2×D4—92no—
340C2×D4—91no—
341C22×C4—92no—
342C22×C4#111311yescharacteristic
343M16—94no—
344M16—94no—
345M16—278no—
346C2×C8—94no—
347C2×C8—278no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—42no—
354C22×C3—41no—
355C22×C3—42no—
356C22×C3—44no—
357C22×C3—44no—
358C22×C3—41no—
359C22×C3—42no—
360C22×C3—94no—
361C22×C3—362no—
362C22×C3C2×D3×Dic312yes—
363C22×C3C2×D3×Dic312yes—
364C22×C3C2×D3×Dic312yes—
365C22×C3C2×#62411yescharacteristic
366C22×C3C2×#62412yes—
367C2×D3—38no—
368C2×D3—64no—
369C2×D3—124no—
370C2×D3—124no—
371C3×C4—22no—
372C3×C4—21no—
373C3×C4—22no—
374C3×C4—21no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—41no—
379C3×C4—41no—
380C3×C4—42no—
381C3×C4—94no—
382C3×C4C2×D3×Dic312yes—
383C3×C4C2×D3×Dic312yes—
384C3×C4C2×D3×Dic312yes—
385C3×C4C2×#62411yescharacteristic
386C3×C4C2×#62412yes—
387C3×C4C22×D3212yes—
388C3×C4C22×D3211yescharacteristic
389Dic3—38no—
390Dic3—64no—
391C32—22no—
392C32—21no—
393C32—41no—
394C32#574512yescomplemented
395C32#581911yescharacteristic, complemented
396C23—92no—
397C23#629411yescharacteristic
398Q8—94no—
399D4—98no—
400D4—94no—
401C2×C4—92no—
402C2×C4—94no—
403C2×C4—94no—
404C2×C4#629411yescharacteristic
405C2×C4#629412yes—
406C2×C4C2×#111312yes—
407C2×C4C2×#111311yescharacteristic
408C8—98no—
409C2×C3—22no—
410C2×C3—24no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3—21no—
415C2×C3—22no—
416C2×C3—21no—
417C2×C3—22no—
418C2×C3—42no—
419C2×C3—42no—
420C2×C3—42no—
421C2×C3—42no—
422C2×C3—44no—
423C2×C3—41no—
424C2×C3—184no—
425C2×C3C22×D3×Dic312yes—
426C2×C3C22×#62411yescharacteristic
427D3—68no—
428D3—124no—
429C22—41no—
430C22—94no—
431C22—362no—
432C22C2×#629411yescharacteristic
433C22C2×#629412yes—
434C4—94no—
435C4C2×#629411yescharacteristic
436C4C2×#629412yes—
437C4C22×#111311yescharacteristic
438C3—21no—
439C3—22no—
440C3—41no—
441C3#1059477612yescomplemented
442C3#1059478011yescharacteristic, complemented
443C2—21no—
444C2—22no—
445C2—184no—
446C2C22×#629411yescharacteristic
subgroupG/H
center, FrattiniC4C22×#1113
commutator, socleC2×C33C23×C4
FittingC22×C33×C4C22
prime radicalC2×C3×#3828C2
cosocleC33×C4C24
hypercenterC22×C4#1113
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. #11723657C2semidirect
4. #11723656C2semidirect
15. C3×#3748C22semidirect
16. C2×C3×C3⋊(C3⋊C8)C22semidirect
17. #88809C22semidirect
18. #88808C22semidirect
22. C2×#3748D3semidirect
72. #3748C2×D3semidirect
73. C2×C3⋊(C3⋊C8)C2×D3semidirect
258. C33#568semidirect
394. C32#5745semidirect
395. C32#5819semidirect
441. C3#10594776semidirect
442. C3#10594780semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C3×#37482
#117236572
#117236562
#105947803
#105947763