Finite Group Databasebeta

← #13161970

Subgroups — #13161970

Order 1728.

764 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3754C212yesmaximal
3#11723678C211yesmaximal, characteristic, complemented
4#11722078C211yesmaximal, characteristic, complemented
5#11722076C212yesmaximal, complemented
6#10591497—31nomaximal
7#10591489—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C2×C3×#634C2212yes—
11C3×#3757—21no—
12C3×#3754—24no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88815—21no—
17#88815—21no—
18D4×C3⋊Dic3D311yescharacteristic
19C2×#3754D312yes—
20C3×D4×Dic3—32no—
21C3×D4×Dic3—61no—
22C2×C3×#214—32no—
23C2×C3×#214—32no—
24C2×C3×#214—62no—
25#63873—31no—
26#63866—32no—
27#63572—31no—
28Dic3⋊(C3⋊C8)—32no—
29#63567—32no—
30#63566—32no—
31#63565—32no—
32C23×C33—21no—
33C33×D4—21no—
34C33×D4D412yes—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—42no—
39C2×C3×C3⋊Dic3D411yescharacteristic
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×#634—22no—
42C3×#634—22no—
43C3×C3⋊(C3⋊C8)—22no—
44#5162—92no—
45#5019—91no—
46C2×C32×D4—21no—
47C2×C32×D4—22no—
48C2×C32×D4—41no—
49C2×C32×D4C2×D311yescharacteristic
50C2×C32×D4C2×D312yes—
51C22×C3⋊Dic3—21no—
52C2×#634C2×D312yes—
53C22×C3×Dic3—62no—
54C22×C3×Dic3—62no—
55C2×C3×Dic6—32no—
56C2×C3×Dic6—32no—
57C2×C3×Dic6—62no—
58#3757—21no—
59#3754—24no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4×C3⋊Dic3C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D312yes—
63C3×#217—62no—
64C3×#217—62no—
65C3×#214—68no—
66C3×#214—64no—
67C3×#214—64no—
68C3×C4⋊Dic3—32no—
69C3×C4⋊Dic3—61no—
70C3×C4×Dic3—32no—
71C3×C4×Dic3—61no—
72C2×C3×C3⋊C8—32no—
73C2×C3×C3⋊C8—32no—
74C2×C3×C3⋊C8—62no—
75#3724—61no—
76#3724—61no—
77Dic3⋊Dic3—62no—
78Dic3⋊Dic3—62no—
79C22×C33—22no—
80C22×C33—41no—
81C22×C33—41no—
82C22×C33C2×D411yescharacteristic
83C33×C4C2×D411yescharacteristic
84C33×C4#4111yescharacteristic
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—42no—
87C3×C3⋊Dic3—41no—
88C3×C3⋊Dic3SD1612yescomplemented
89C2×C3×SD16—92no—
90C3×C4×D4—91no—
91#944—92no—
92D4×Dic3—32no—
93D4×Dic3—61no—
94C2×#214—32no—
95C2×#214—32no—
96C2×#214—62no—
97#869—91no—
98#835—92no—
99#816—92no—
100Dic3⋊C8—92no—
101#810—92no—
102C4⋊(C3⋊C8)—91no—
103C23×C32—21no—
104C23×C32—22no—
105C23×C32—22no—
106C23×C32—42no—
107C23×C32—42no—
108C32×D4—21no—
109C32×D4—22no—
110C32×D4—22no—
111C32×D4—22no—
112C32×D4—24no—
113C32×D4—42no—
114C32×D4—42no—
115C32×D4—42no—
116C32×D4D1212yes—
117C32×D4#6714yes—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D312yes—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—42no—
126C2×C3⋊Dic3#6711yescharacteristic
127C2×C3⋊Dic3#6711yescharacteristic
128#634—22no—
129#634—22no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—122no—
139C2×C3×Dic3—124no—
140C2×C3×Dic3—124no—
141C3×Dic6—64no—
142C3×Dic6—62no—
143C3×Dic6—62no—
144C3×Dic6—62no—
145C3×Dic6—62no—
146C3×Dic6—122no—
147C3⋊(C3⋊C8)—22no—
148C3×C3⋊C8—64no—
149C3×C3⋊C8—62no—
150C3×C3⋊C8—62no—
151#461—271noSylow
152C2×C33—42no—
153C2×C33#12111yescharacteristic
154C2×C33C2×SD1611yescharacteristic
155C2×C33#13711yescharacteristic
156C2×C3×Q8—92no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—41no—
160C2×C3×D4D3211yescharacteristic
161C2×C3×D4D3212yes—
162C22×C3×C4—181no—
163C22×Dic3—62no—
164C22×Dic3—62no—
165C2×Dic6—32no—
166C2×Dic6—32no—
167C2×Dic6—62no—
168C3×SD16—184no—
169C2×C3×C8—92no—
170C3×C4⋊C4—91no—
171C3×C22⋊C4—181no—
172C3×C42—91no—
173#217—62no—
174#217—62no—
175#214—68no—
176#214—64no—
177#214—64no—
178C4⋊Dic3—32no—
179C4⋊Dic3—61no—
180C4⋊Dic3—181no—
181C4⋊Dic3—181no—
182Dic3⋊C4—182no—
183Dic3⋊C4—182no—
184C4×Dic3—32no—
185C4×Dic3—61no—
186C2×C3⋊C8—32no—
187C2×C3⋊C8—32no—
188C2×C3⋊C8—62no—
189C22×C32—22no—
190C22×C32—24no—
191C22×C32—24no—
192C22×C32—22no—
193C22×C32—21no—
194C22×C32—41no—
195C22×C32—44no—
196C22×C32—44no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—42no—
200C22×C32—42no—
201C22×C32—42no—
202C22×C32—42no—
203C22×C32—82no—
204C22×C32—82no—
205C22×C32—82no—
206C22×C32—82no—
207C22×C32D3×D412yes—
208C22×C32C2×#6711yescharacteristic
209C32×C4—22no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—22no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4C2×D1211yescharacteristic
216C32×C4#23511yescharacteristic
217C32×C4#23712yes—
218C32×C4C2×#6712yes—
219C3⋊Dic3—21no—
220C3⋊Dic3—41no—
221C3⋊Dic3—42no—
222C3⋊Dic3#21412yescomplemented
223C3×Dic3—34no—
224C3×Dic3—62no—
225C3×Dic3—62no—
226C3×Dic3—121no—
227C3×Dic3—124no—
228C3×Dic3—122no—
229C3×Dic3—122no—
230C3×Dic3—122no—
231C3×Dic3—122no—
232C2×SD16—92no—
233C4⋊Q8—271no—
234C4×D4—91no—
235C4⋊C8—271no—
236#103—272no—
237C33#46111yescharacteristic, complemented, Sylow
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—21no—
241C23×C3—42no—
242C23×C3—42no—
243C3×Q8—182no—
244C3×Q8—182no—
245C3×D4—22no—
246C3×D4—22no—
247C3×D4—21no—
248C3×D4—22no—
249C3×D4—24no—
250C3×D4—42no—
251C3×D4—42no—
252C3×D4—42no—
253C3×D4#62512yes—
254C3×D4#62614yes—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—91no—
259C2×C3×C4—91no—
260C2×C3×C4—182no—
261C2×C3×C4—181no—
262C2×C3×C4—362no—
263C2×C3×C4C2×D3212yes—
264C2×C3×C4C2×D3211yescharacteristic
265C2×Dic3—32no—
266C2×Dic3—32no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—62no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—62no—
273C2×Dic3—122no—
274C2×Dic3—124no—
275C2×Dic3—124no—
276Dic6—64no—
277Dic6—62no—
278Dic6—62no—
279Dic6—62no—
280Dic6—62no—
281Dic6—122no—
282C3×C8—182no—
283C3⋊C8—64no—
284C3⋊C8—62no—
285C3⋊C8—62no—
286C2×C32—22no—
287C2×C32—22no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—22no—
291C2×C32—21no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32—42no—
296C2×C32—44no—
297C2×C32—44no—
298C2×C32—84no—
299C2×C32—84no—
300C2×C32D3×SD1612yes—
301C2×C32#91512yes—
302C2×C32#93011yescharacteristic
303C2×C32C2×#21411yescharacteristic
304C2×C32#93912yes—
305C2×C32#95111yescharacteristic
306C2×Q8—92no—
307C2×D4#111311yescharacteristic
308C22×C4—181no—
309SD16—184no—
310C2×C8—92no—
311C4⋊C4—91no—
312C4⋊C4—541no—
313C4⋊C4—541no—
314C22⋊C4—181no—
315C42—91no—
316C22×C3—24no—
317C22×C3—24no—
318C22×C3—22no—
319C22×C3—21no—
320C22×C3—22no—
321C22×C3—44no—
322C22×C3—44no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—82no—
331C22×C3—82no—
332C22×C3—82no—
333C22×C3—82no—
334C22×C3D3×#6712yes—
335C22×C3#381111yescharacteristic
336C3×C4—22no—
337C3×C4—21no—
338C3×C4—22no—
339C3×C4—21no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4—92no—
343C3×C4—181no—
344C3×C4—361no—
345C3×C4—362no—
346C3×C4#380412yes—
347C3×C4#380511yescharacteristic
348C3×C4C2×#62511yescharacteristic
349C3×C4C2×#62612yes—
350Dic3—34no—
351Dic3—62no—
352Dic3—62no—
353Dic3—121no—
354Dic3—124no—
355Dic3—122no—
356Dic3—122no—
357Dic3—122no—
358Dic3—122no—
359C32—22no—
360C32—21no—
361C32—41no—
362C32#501911yescharacteristic, complemented
363C32#516212yescomplemented
364C23—21no—
365Q8—182no—
366Q8—182no—
367D4—21no—
368D4#629512yes—
369C2×C4—91no—
370C2×C4—91no—
371C2×C4—181no—
372C2×C4—182no—
373C2×C4—362no—
374C2×C4C2×#111311yescharacteristic
375C8—182no—
376C2×C3—21no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—22no—
381C2×C3—21no—
382C2×C3—44no—
383C2×C3—44no—
384C2×C3—42no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3—41no—
388C2×C3—84no—
389C2×C3—84no—
390C2×C3D3×#21412yes—
391C2×C3#6392912yes—
392C2×C3#6393011yescharacteristic
393C2×C3#6393111yescharacteristic
394C2×C3#6397812yes—
395C2×C3#6398011yescharacteristic
396C22—22no—
397C22—41no—
398C22—41no—
399C22#8904911yescharacteristic
400C4—92no—
401C4—181no—
402C4—361no—
403C4—362no—
404C4#8904611yescharacteristic
405C4C2×#629511yescharacteristic
406C3—21no—
407C3—22no—
408C3—41no—
409C3#1059148912yescomplemented
410C3#1059149711yescharacteristic, complemented
411C2—42no—
412C2#1172370411yescharacteristic
413C2#1172370511yescharacteristic
414C2#1172371711yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11723678C2semidirect
4. #11722078C2semidirect
5. #11722076C2semidirect
88. C3×C3⋊Dic3SD16semidirect
222. C3⋊Dic3#214semidirect
237. C33#461semidirect
362. C32#5019semidirect
363. C32#5162semidirect
409. C3#10591489semidirect
410. C3#10591497semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C2×C3×#37542
#117236782
#117220782
#117220762
#105914973
#105914893