Finite Group Databasebeta

← #13155864

Subgroups — #13155864

Order 1728.

824 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 118 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×#3797C211yesmaximal, characteristic
2#11723431C211yesmaximal, characteristic
3C2×#88799C212yesmaximal, complemented
4#11722072C211yesmaximal, characteristic
5C3×#63565C212yesmaximal, complemented
6#10591595—32nomaximal
7#10590605—31nomaximal
8C2×#6308C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×Dic6C2212yes—
11C2×C3×#627—21no—
12C2×C3×#624—21no—
13C3×#3797—24no—
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15C32×C4⋊Dic3C2211yescharacteristic
16Dic3⋊(C3⋊Dic3)—21no—
17#88803—21no—
18#88799—24no—
19C2×C3×D3×Q8—32no—
20C2×#3797D311yescharacteristic
21C2×#3756—32no—
22#64106—31no—
23#63857—32no—
24#63857—61no—
25C2×#3719—32no—
26C2×#3719—32no—
27C2×#3719—62no—
28C3×#835—32no—
29C3×#810—32no—
30#63565D312yescomplemented
31#63557—31no—
32(C2×D3)⋊(C3⋊C8)—32no—
33C22×C3×C3⋊D3D411yescharacteristic
34C2×C33×C4C2311yescharacteristic
35C2×C3⋊(C3⋊Dic3)—21no—
36#6308—22no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4×C3⋊D3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3—21no—
41C32×Dic6—22no—
42C32×Dic6D414yes—
43C3×#627—24no—
44C3×#624—24no—
45C3×C3⋊(C3⋊C8)—22no—
46#5228—91no—
47#4809—92no—
48C2×C32×Q8—32no—
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×#634—31no—
51C2×#634—32no—
52C2×#634—62no—
53C2×#634—61no—
54C2×#634—121no—
55C3×D3×Q8—64no—
56C3×D3×Q8—64no—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—62no—
59C2×C3×C4×D3—61no—
60C2×C3×Dic6—22no—
61C2×C3×Dic6—62no—
62C2×C3×Dic6C2×D312yes—
63C2×C3×Dic6C2×D312yes—
64C2×#627—21no—
65C2×#624—21no—
66#3797—24no—
67C32×C4⋊C4—31no—
68#3756—64no—
69#3752—61no—
70C4⋊(C3⋊Dic3)—61no—
71C2×C3⋊(C3⋊C8)C2×D312yes—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3C2×D312yes—
74C2×C3×C3⋊C8—32no—
75C2×C3×C3⋊C8—32no—
76C2×C3×C3⋊C8—62no—
77Dic3⋊Dic3—62no—
78Dic3⋊Dic3—62no—
79(C2×D3)⋊Dic3—62no—
80(C2×D3)⋊Dic3—62no—
81#3719—68no—
82#3719—64no—
83#3719—64no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3Q1612yescomplemented
88C33×C4C2×D412yes—
89C3⋊(C3⋊Dic3)—41no—
90C3×C3⋊Dic3—21no—
91C32×Dic3—24no—
92C32×Dic3—41no—
93C22×C3×Q8—91no—
94C2×D3×Q8—32no—
95C2×#216—92no—
96C2×#216—92no—
97C2×#216—182no—
98#924—91no—
99C2×Dic12—92no—
100#897—92no—
101#897—181no—
102C3×#103—92no—
103#835—32no—
104#830—91no—
105(C2×D3)⋊C8—92no—
106#810—32no—
107C22×C3⋊D3#6711yescharacteristic
108C22×C3×D3—32no—
109C22×C3×D3—61no—
110C32×Q8—34no—
111C32×Q8—62no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4—61no—
116C2×C32×C4—62no—
117C2×C32×C4C22×D312yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—62no—
121C2×C3⋊Dic3—62no—
122C2×C3⋊Dic3—122no—
123C2×C3⋊Dic3—122no—
124C2×C3⋊Dic3#6711yescharacteristic
125C4×C3⋊D3—22no—
126#634—62no—
127#634—64no—
128#634—62no—
129#634—62no—
130#634—122no—
131#634—122no—
132#634—124no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—42no—
139C2×C3×Dic3—61no—
140C3×C4×D3—68no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×C4×D3—122no—
144C3×Dic6—22no—
145C3×Dic6—22no—
146C3×Dic6—24no—
147C3×Dic6—42no—
148C3×Dic6—68no—
149C3×Dic6D1214yes—
150C3×Dic6#6714yes—
151#627—24no—
152#624—24no—
153C3⋊(C3⋊C8)—22no—
154C3×C3⋊C8—64no—
155C3×C3⋊C8—62no—
156C3×C3⋊C8—62no—
157#451—271noSylow
158C2×C33#12011yescharacteristic
159C2×C33C2×Q1611yescharacteristic
160C2×C33#13711yescharacteristic
161C3×C3⋊D3—22no—
162C2×C3×Q8—32no—
163C2×C3×Q8—32no—
164C2×C3×Q8—62no—
165C2×C3×Q8—181no—
166C2×C3×Q8—184no—
167C22×C3×C4—91no—
168C22×C3×C4—181no—
169D3×Q8—64no—
170D3×Q8—64no—
171C2×C4×D3—32no—
172C2×C4×D3—61no—
173C2×C4×D3—62no—
174C2×Dic6—62no—
175C2×Dic6—92no—
176C2×Dic6—91no—
177C2×Dic6—182no—
178C2×Dic6—181no—
179C2×Dic6—361no—
180C2×Dic6D3212yes—
181C2×C3×C8—92no—
182C3×C4⋊C4—32no—
183C3×C4⋊C4—61no—
184#217—181no—
185#216—184no—
186#216—184no—
187#216—188no—
188(C2×D3)⋊C4—182no—
189(C2×D3)⋊C4—182no—
190C4⋊Dic3—182no—
191C4⋊Dic3—182no—
192C4⋊Dic3D3211yescharacteristic
193Dic3⋊C4—181no—
194C2×C3⋊C8—32no—
195C2×C3⋊C8—32no—
196C2×C3⋊C8—62no—
197Dic12—184no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32D3×D412yes—
202C22×C32C2×#6711yescharacteristic
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3#21612yescomplemented
206C2×C3×D3—34no—
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C32×C4—22no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—42no—
216C32×C4—64no—
217C32×C4—121no—
218C32×C4C2×D1212yes—
219C32×C4D3×D412yes—
220C32×C4C2×#6712yes—
221C3⋊Dic3—21no—
222C3⋊Dic3—124no—
223C3⋊Dic3—122no—
224C3⋊Dic3—124no—
225C3⋊Dic3—122no—
226C3⋊Dic3—121no—
227C3×Dic3—24no—
228C3×Dic3—24no—
229C3×Dic3—44no—
230C3×Dic3—42no—
231C3×Dic3—42no—
232C3×Dic3—62no—
233C3×Dic3—121no—
234C22×Q8—91no—
235C2×Q16—272no—
236C22⋊Q8—271no—
237#103—92no—
238C22⋊C8—271no—
239C33#45111yescharacteristic, complemented, Sylow
240C23×C3—91no—
241C22×D3—32no—
242C22×D3—61no—
243C3×Q8—34no—
244C3×Q8—34no—
245C3×Q8—64no—
246C3×Q8—62no—
247C3×Q8—62no—
248C3×Q8—122no—
249C3×Q8—184no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—91no—
258C2×C3×C4—122no—
259C2×C3×C4—184no—
260C2×C3×C4—182no—
261C2×C3×C4C2×D3211yescharacteristic
262C2×C3×C4C2×D3212yes—
263C2×Dic3—22no—
264C2×Dic3—21no—
265C2×Dic3—32no—
266C2×Dic3—61no—
267C2×Dic3—182no—
268C2×Dic3—182no—
269C2×Dic3—181no—
270C2×Dic3—362no—
271C2×Dic3—362no—
272C4×D3—68no—
273C4×D3—62no—
274C4×D3—62no—
275C4×D3—122no—
276Dic6—22no—
277Dic6—68no—
278Dic6—184no—
279Dic6—182no—
280Dic6—182no—
281Dic6—182no—
282Dic6—362no—
283Dic6—362no—
284Dic6—364no—
285Dic6#62614yes—
286C3×C8—182no—
287C3⋊C8—64no—
288C3⋊C8—62no—
289C3⋊C8—62no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—22no—
293C2×C32—21no—
294C2×C32—22no—
295C2×C32—22no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32#88212yes—
300C2×C32#91512yes—
301C2×C32D3×Q1612yes—
302C2×C32#94211yescharacteristic
303C2×C32C2×#21611yescharacteristic
304C2×C32#95311yescharacteristic
305C3⋊D3—22no—
306C3×D3—64no—
307C3×D3—122no—
308C2×Q8—32no—
309C2×Q8—181no—
310C2×Q8—184no—
311C2×Q8—271no—
312C22×C4—91no—
313C22×C4—181no—
314Q16—544no—
315C2×C8—92no—
316C4⋊C4—31no—
317C4⋊C4—541no—
318C22⋊C4—541no—
319C22×C3—21no—
320C22×C3—22no—
321C22×C3—41no—
322C22×C3—92no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3D3×#6712yes—
326C22×C3#381111yescharacteristic
327C2×D3—34no—
328C2×D3—62no—
329C2×D3—62no—
330C2×D3—62no—
331C2×D3—121no—
332C2×D3—121no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—42no—
337C3×C4—64no—
338C3×C4—64no—
339C3×C4—124no—
340C3×C4—122no—
341C3×C4—122no—
342C3×C4—181no—
343C3×C4D3×D1212yes—
344C3×C4C2×#62612yes—
345C3×C4D3×#6712yes—
346Dic3—24no—
347Dic3—41no—
348Dic3—62no—
349Dic3—121no—
350Dic3—364no—
351Dic3—362no—
352Dic3—364no—
353Dic3—361no—
354Dic3—362no—
355C32—22no—
356C32—21no—
357C32—41no—
358C32#480912yescomplemented
359C32#522811yescharacteristic, complemented
360C23—91no—
361Q8—34no—
362Q8—62no—
363Q8—184no—
364Q8—542no—
365C2×C4—62no—
366C2×C4—61no—
367C2×C4—91no—
368C2×C4—184no—
369C2×C4—182no—
370C2×C4—541no—
371C2×C4D3311yescharacteristic
372C8—182no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—182no—
383C2×C3#6392311yescharacteristic
384C2×C3#6393111yescharacteristic
385C2×C3D3×#21612yes—
386C2×C3#6394411yescharacteristic
387C2×C3#6394612yes—
388C2×C3#6397712yes—
389D3—64no—
390D3—122no—
391C22—92no—
392C22—181no—
393C22—181no—
394C22#8895211yescharacteristic
395C4—64no—
396C4—121no—
397C4—181no—
398C4—1081no—
399C4D3×#62612yes—
400C3—21no—
401C3—22no—
402C3—41no—
403C3#1059060511yescharacteristic, complemented
404C3#1059159512yescomplemented
405C2—182no—
406C2#1172284611yescharacteristic
407C2#1172288911yescharacteristic
408C2#1172294811yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#451
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C2×#88799C2semidirect
5. C3×#63565C2semidirect
30. #63565D3semidirect
87. C2×C3×C3⋊D3Q16semidirect
205. C2×C3⋊D3#216semidirect
239. C33#451semidirect
358. C32#4809semidirect
359. C32#5228semidirect
403. C3#10590605semidirect
404. C3#10591595semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
#117234312
C2×#887992
#117220722
C3×#635652
#105915953
#105906053