Finite Group Databasebeta

← #13161974

Subgroups — #13161974

Order 1728.

1126 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1126 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
2#11723656C212yesmaximal, complemented
3C3×#63660C212yesmaximal, complemented
4#11722073C212yesmaximal, complemented
5#10591503—31nomaximal
6D3×#833—32nomaximal
7C23×C3×C3⋊D3C412yes—
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638C412yes—
10C3×D4×C3⋊D3—22no—
11C3×D4×C3⋊D3—22no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C3×#3748C2214yescomplemented
15#88809—22no—
16#88808—22no—
17C2×D4×C3⋊D3D311yescharacteristic
18C2×C3×D3×D4—32no—
19C2×C3×D3×D4—61no—
20#63818—32no—
21D3×#208—34no—
22#63660D312yescomplemented
23C3×#833—32no—
24C3×#833—32no—
25C3×#833—62no—
26#63559—32no—
27#63558—34no—
28C23×C33C2×C412yes—
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3—22no—
31C22×C3×C3⋊D3—44no—
32C22×C3×C3⋊D3C2×C411yescharacteristic
33C22×C3×C3⋊D3C2×C412yes—
34C33×D4—22no—
35C2×C33×C4C2311yescharacteristic
36C3×#638—24no—
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C3×C4⋊(C3⋊D3)—22no—
39C3×C4×C3⋊D3D412yes—
40C3×C4×C3⋊D3D412yes—
41C3×C3⋊(C3⋊C8)—24no—
42C2×#833—91no—
43D3×#100—92no—
44C23×C3⋊D3Dic312yes—
45C23×C3×D3—34no—
46C23×C3×D3—62no—
47C2×C32×D4—21no—
48C2×C32×D4—22no—
49C2×C32×D4—41no—
50C2×C32×D4C2×D312yes—
51C2×C32×D4C2×D311yescharacteristic
52C2×#638Dic312yes—
53D4×C3⋊D3—22no—
54D4×C3⋊D3—22no—
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×#67—34no—
58C2×C3×#67—62no—
59C3×D3×D4—64no—
60C3×D3×D4—64no—
61C3×D3×D4—64no—
62C3×D3×D4—64no—
63C2×C3×D12—32no—
64C2×C3×D12—61no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—61no—
67#3748C2×D314yescomplemented
68C3×#208—34no—
69C3×#208—34no—
70C3×#208—64no—
71#3712—62no—
72#3711—64no—
73#3710—62no—
74D3×C3⋊C8—64no—
75C22×C33—22no—
76C22×C33—42no—
77C22×C33C22×C411yescharacteristic
78C2×C3×C3⋊D3—22no—
79C2×C3×C3⋊D3—22no—
80C2×C3×C3⋊D3—44no—
81C2×C3×C3⋊D3—44no—
82C2×C3×C3⋊D3—42no—
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C3×C3⋊D3C22⋊C411yescharacteristic
85C33×C4C2×D412yes—
86C3×C3⋊Dic3C22⋊C412yes—
87C22×C3×D4—91no—
88C2×D3×D4—32no—
89C2×D3×D4—61no—
90C2×#208—92no—
91D3×M16—94no—
92C3×#100—92no—
93#833—32no—
94#833—32no—
95#833—62no—
96#833—92no—
97#823—94no—
98C23×C32—24no—
99C23×C32—22no—
100C23×C32—42no—
101C23×C32C4×D314yes—
102C23×C32C2×Dic312yes—
103C22×C3⋊D3—22no—
104C22×C3⋊D3—22no—
105C22×C3⋊D3—44no—
106C22×C3⋊D3C2×Dic311yescharacteristic
107C22×C3⋊D3C2×Dic312yes—
108C22×C3×D3—32no—
109C22×C3×D3—34no—
110C22×C3×D3—61no—
111C22×C3×D3—62no—
112C22×C3×D3—64no—
113C22×C3×D3—64no—
114C22×C3×D3—64no—
115C22×C3×D3—64no—
116C22×C3×D3—128no—
117C22×C3×D3—128no—
118C32×D4—22no—
119C32×D4—24no—
120C32×D4—24no—
121C32×D4—44no—
122C32×D4—44no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—41no—
126C2×C32×C4C22×D312yes—
127C2×C32×C4C22×D311yescharacteristic
128#638—24no—
129C2×C3⋊Dic3C2×Dic311yescharacteristic
130C4⋊(C3⋊D3)—22no—
131C4×C3⋊D3#6712yes—
132C4×C3⋊D3#6712yes—
133C3×#67—68no—
134C3×#67—68no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—61no—
137C3×D12—64no—
138C3×D12—64no—
139C3×C4×D3—34no—
140C3×C4×D3—34no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3⋊(C3⋊C8)—24no—
144C3×C3⋊C8—64no—
145C3×C3⋊C8—64no—
146C3×C3⋊C8—124no—
147C2×#100—271noSylow
148C2×C33—21no—
149C2×C33—42no—
150C2×C33C2×C22⋊C411yescharacteristic
151C3×C3⋊D3—21no—
152C3×C3⋊D3—42no—
153C3×C3⋊D3#10012yescomplemented
154C24×C3—92no—
155C23×D3—34no—
156C23×D3—62no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—41no—
160C2×C3×D4—92no—
161C2×C3×D4—91no—
162C2×C3×D4—182no—
163C2×C3×D4—182no—
164C2×C3×D4D3212yes—
165C2×C3×D4D3211yescharacteristic
166C22×C3×C4—91no—
167C2×#67—34no—
168C2×#67—62no—
169D3×D4—64no—
170D3×D4—64no—
171D3×D4—64no—
172D3×D4—64no—
173C2×D12—32no—
174C2×D12—61no—
175C2×C4×D3—32no—
176C2×C4×D3—61no—
177C3×M16—94no—
178#208—34no—
179#208—34no—
180#208—64no—
181#208—182no—
182C2×C3⋊C8—182no—
183ΓL(1,𝔽25)—184no—
184D3×C8—184no—
185C22×C32—22no—
186C22×C32—24no—
187C22×C32—24no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—41no—
191C22×C32—44no—
192C22×C32—44no—
193C22×C32—42no—
194C22×C32—44no—
195C22×C32—44no—
196C22×C32—84no—
197C22×C32—84no—
198C22×C32C2×C4×D312yes—
199C22×C32C22×Dic311yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—22no—
202C2×C3⋊D3—44no—
203C2×C3⋊D3—44no—
204C2×C3⋊D3—42no—
205C2×C3⋊D3#21711yescharacteristic
206C2×C3⋊D3#21711yescharacteristic
207C2×C3×D3—32no—
208C2×C3×D3—32no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—128no—
216C2×C3×D3—128no—
217C2×C3×D3—128no—
218C2×C3×D3—128no—
219C2×C3×D3—124no—
220C2×C3×D3—124no—
221C32×C4—24no—
222C32×C4—22no—
223C32×C4—42no—
224C32×C4D3×D414yes—
225C32×C4C2×#6712yes—
226C3⋊Dic3#21712yes—
227C3×Dic3—34no—
228C3×Dic3—62no—
229C22×D4—91no—
230C2×M16—272no—
231#100—92no—
232#100—272no—
233C33C2×#10011yescharacteristic, complemented, Sylow
234C23×C3—22no—
235C23×C3—24no—
236C23×C3—42no—
237C23×C3—91no—
238C23×C3—92no—
239C23×C3—182no—
240C23×C3—182no—
241C23×C3—364no—
242C23×C3D3×Dic314yes—
243C23×C3#62412yes—
244C22×D3—32no—
245C22×D3—34no—
246C22×D3—61no—
247C22×D3—62no—
248C22×D3—64no—
249C22×D3—64no—
250C22×D3—64no—
251C22×D3—64no—
252C22×D3—128no—
253C22×D3—128no—
254C3×D4—24no—
255C3×D4—24no—
256C3×D4—22no—
257C3×D4—44no—
258C3×D4—44no—
259C3×D4—184no—
260C3×D4—182no—
261C2×C3×C4—22no—
262C2×C3×C4—21no—
263C2×C3×C4—41no—
264C2×C3×C4—91no—
265C2×C3×C4—92no—
266C2×C3×C4—92no—
267C2×C3×C4C2×D3212yes—
268C2×C3×C4C2×D3211yescharacteristic
269#67—68no—
270#67—68no—
271C2×Dic3—32no—
272C2×Dic3—61no—
273D12—64no—
274D12—64no—
275C4×D3—34no—
276C4×D3—34no—
277C4×D3—62no—
278C4×D3—62no—
279C3×C8—184no—
280C3⋊C8—64no—
281C3⋊C8—64no—
282C3⋊C8—124no—
283C2×C32—21no—
284C2×C32—22no—
285C2×C32—22no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—41no—
289C2×C32—42no—
290C2×C32—42no—
291C2×C32—42no—
292C2×C32—44no—
293C2×C32—44no—
294C2×C32—84no—
295C2×C32—84no—
296C2×C32D3×C22⋊C412yes—
297C2×C32C2×#21711yescharacteristic
298C3⋊D3—21no—
299C3⋊D3—42no—
300C3⋊D3#83312yescomplemented
301C3×D3—34no—
302C3×D3—62no—
303C3×D3—62no—
304C3×D3—62no—
305C3×D3—124no—
306C3×D3—124no—
307C24—92no—
308C2×D4—92no—
309C2×D4—91no—
310C2×D4—182no—
311C2×D4—182no—
312C2×D4#111311yescharacteristic
313C22×C4—91no—
314M16—94no—
315M16—542no—
316C2×C8—542no—
317C22×C3—24no—
318C22×C3—24no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—44no—
323C22×C3—44no—
324C22×C3—44no—
325C22×C3—42no—
326C22×C3—44no—
327C22×C3—41no—
328C22×C3—84no—
329C22×C3—84no—
330C22×C3—91no—
331C22×C3—91no—
332C22×C3—182no—
333C22×C3—182no—
334C22×C3—362no—
335C22×C3—364no—
336C22×C3—364no—
337C22×C3C2×D3×Dic312yes—
338C22×C3C2×#62411yescharacteristic
339C2×D3—32no—
340C2×D3—32no—
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—64no—
344C2×D3—64no—
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—128no—
348C2×D3—128no—
349C2×D3—128no—
350C2×D3—128no—
351C2×D3—124no—
352C2×D3—124no—
353C3×C4—24no—
354C3×C4—22no—
355C3×C4—42no—
356C3×C4—92no—
357C3×C4D3×#6714yes—
358C3×C4#381112yes—
359Dic3—34no—
360Dic3—62no—
361C32—22no—
362C32—21no—
363C32—41no—
364C32D3×#10012yescomplemented
365C32C2×#83311yescharacteristic, complemented
366C23—91no—
367C23—92no—
368C23—182no—
369C23—182no—
370C23—364no—
371C23#629412yes—
372D4—22no—
373D4—184no—
374D4—182no—
375C2×C4—91no—
376C2×C4—92no—
377C2×C4—92no—
378C2×C4C2×#111311yescharacteristic
379C8—184no—
380C2×C3—22no—
381C2×C3—22no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—22no—
385C2×C3—42no—
386C2×C3—42no—
387C2×C3—44no—
388C2×C3—44no—
389C2×C3—42no—
390C2×C3—41no—
391C2×C3—84no—
392C2×C3—84no—
393C2×C3—92no—
394C2×C3—181no—
395C2×C3—362no—
396C2×C3D3×#21712yes—
397C2×C3#6397411yescharacteristic
398D3—34no—
399D3—62no—
400D3—62no—
401D3—62no—
402D3—124no—
403D3—124no—
404C22—22no—
405C22—42no—
406C22—91no—
407C22—91no—
408C22—182no—
409C22—182no—
410C22—362no—
411C22—364no—
412C22—364no—
413C22C2×#629411yescharacteristic
414C4—92no—
415C4#8904912yes—
416C3—21no—
417C3—22no—
418C3—41no—
419C3D3×#83312yescomplemented
420C3#1059150311yescharacteristic, complemented
421C2—21no—
422C2—42no—
423C2—92no—
424C2—181no—
425C2—362no—
426C2#1172372311yescharacteristic
subgroupG/H
centerC2#11723723
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C33×C4C23
hypercenterC2×D4#1113
hypocenterC33C2×#100
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723656C2semidirect
3. C3×#63660C2semidirect
4. #11722073C2semidirect
14. C3×#3748C22semidirect
22. #63660D3semidirect
67. #3748C2×D3semidirect
153. C3×C3⋊D3#100semidirect
233. C33C2×#100semidirect
300. C3⋊D3#833semidirect
364. C32D3×#100semidirect
365. C32C2×#833semidirect
419. C3D3×#833semidirect
420. C3#10591503semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
#117236562
C3×#636602
#117220732
#105915033
D3×#8333