Finite Group Databasebeta

← #13168327

Subgroups — #13168327

Order 1728.

1250 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 196 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
1#11723680C211yesmaximal, characteristic, complemented
2#11723503C212yesmaximal
3C4×#6289C211yesmaximal, characteristic
4C3×#63872C211yesmaximal, characteristic, complemented
5C3×#63869C212yesmaximal, complemented
6#11723016C214yesmaximal, complemented
7#11722912C214yesmaximal
8#10594482—32nomaximal
9#10593948—31nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×#6289C2212yes—
12C2×#6289C2214yes—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C2212yes—
16C32×C4×Dic3C2211yescharacteristic
17#88815C2214yes—
18#88806C2214yes—
19Dic3×C3⋊Dic3C2212yes—
20C3×#3724C2214yes—
21C3×Dic3⋊Dic3C2214yes—
22C3×Dic3⋊Dic3C2214yes—
23C4⋊C4×C3⋊D3—32no—
24C42×C3⋊D3—31no—
25C3×#890—32no—
26C3×#890—32no—
27C3×#870—32no—
28#63900—32no—
29#63900—32no—
30#63900—34no—
31#63900—62no—
32#63883—34no—
33#63883—34no—
34C4×#624—32no—
35C4×#624—61no—
36#63872—31no—
37#63872D311yescharacteristic, complemented
38#63870—34no—
39#63869—32no—
40#63869D312yescomplemented
41#63847—34no—
42#63847—34no—
43C22×C3⋊(C3⋊D3)C2311yescharacteristic
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
46C4×C3⋊(C3⋊D3)Q814yes—
47C2×C3×C3⋊Dic3C2312yes—
48C2×C3×C3⋊Dic3C2314yes—
49C2×C32×Dic3C2314yes—
50C2×C32×Dic3C2312yes—
51#6289—24no—
52#6289—28no—
53D3×#125—92no—
54D3×#125—91no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—32no—
57C2×C4×C3⋊D3—32no—
58C2×C4×C3⋊D3—34no—
59C2×C4×C3⋊D3—61no—
60C2×C4×C3⋊D3—62no—
61C2×C4×C3⋊D3—121no—
62C2×#624—32no—
63C2×#624—32no—
64C2×#624—34no—
65C2×#624—34no—
66C2×#624—62no—
67C2×#624—64no—
68C32×C4⋊C4—32no—
69C32×C42—31no—
70C4⋊(C3⋊Dic3)C2×D312yes—
71#3750—34no—
72C4×C3⋊Dic3—32no—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C3×C4⋊Dic3—22no—
75C3×C4⋊Dic3—32no—
76C3×C4⋊Dic3—32no—
77C3×C4⋊Dic3—34no—
78C3×C4⋊Dic3—34no—
79C3×C4⋊Dic3—62no—
80C3×C4⋊Dic3C2×D312yes—
81C3×C4⋊Dic3C2×D312yes—
82C3×Dic3⋊C4—34no—
83C3×Dic3⋊C4—34no—
84C3×Dic3⋊C4—34no—
85C3×Dic3⋊C4—34no—
86C3×C4×Dic3—21no—
87C3×C4×Dic3—32no—
88C3×C4×Dic3—61no—
89C3×C4×Dic3C2×D312yes—
90#3724—34no—
91#3724—34no—
92#3724—34no—
93#3724—64no—
94#3724C2×D314yes—
95Dic3⋊Dic3—34no—
96Dic3⋊Dic3—34no—
97Dic3⋊Dic3C2×D314yes—
98Dic3⋊Dic3C2×D314yes—
99Dic32—32no—
100Dic32—32no—
101Dic32—62no—
102C22×C33C2411yescharacteristic
103C2×C3⋊(C3⋊D3)C2×Q812yes—
104C2×C3⋊(C3⋊D3)#4114yes—
105C33×C4C2×Q812yes—
106C3⋊(C3⋊Dic3)C2×Q812yes—
107C3×C3⋊Dic3—22no—
108C3×C3⋊Dic3—24no—
109C32×Dic3—22no—
110C32×Dic3—24no—
111C3×#125—92no—
112C3×#125—91no—
113D3×C4⋊C4—94no—
114D3×C4⋊C4—94no—
115D3×C4⋊C4—94no—
116D3×C4⋊C4—92no—
117D3×C4⋊C4—92no—
118D3×C4⋊C4—92no—
119D3×C4⋊C4—182no—
120#890—32no—
121#890—32no—
122#890—92no—
123#889—94no—
124#889—94no—
125#889—94no—
126C42×D3—92no—
127C42×D3—91no—
128C42×D3—181no—
129#870—32no—
130#870—91no—
131C22×C3⋊D3—32no—
132C22×C3⋊D3—31no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—62no—
135C22×C3⋊D3—121no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—32no—
139C2×C32×C4—34no—
140C2×C32×C4—41no—
141C2×C32×C4C22×D312yes—
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—32no—
144C2×C3⋊Dic3—31no—
145C2×C3⋊Dic3—61no—
146C2×C3⋊Dic3—62no—
147C2×C3⋊Dic3—121no—
148C2×C3⋊Dic3C22×D312yes—
149C2×C3⋊Dic3C22×D314yes—
150C4×C3⋊D3—34no—
151C4×C3⋊D3—38no—
152C4×C3⋊D3—64no—
153C4×C3⋊D3—68no—
154C4×C3⋊D3—64no—
155C4×C3⋊D3—68no—
156C4×C3⋊D3—124no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—64no—
165C2×C3×Dic3C22×D312yes—
166C2×C3×Dic3C22×D312yes—
167C2×C3×Dic3C22×D314yes—
168C2×C3×Dic3C22×D314yes—
169#624—64no—
170#624—64no—
171#624—68no—
172#624—68no—
173#624—68no—
174#624—128no—
175C2×#125—271noSylow
176C2×C33C22×Q811yescharacteristic
177C2×C33C2×#4112yes—
178C3⋊(C3⋊D3)#12514yescomplemented
179C2×C4×D3—92no—
180C2×C4×D3—91no—
181C2×C4×D3—92no—
182C2×C4×D3—92no—
183C2×C4×D3—94no—
184C2×C4×D3—94no—
185C2×C4×D3—92no—
186C2×C4×D3—94no—
187C2×C4×D3—182no—
188C2×C4×D3—181no—
189C2×C4×D3—182no—
190C2×C4×D3—184no—
191C2×C4×D3—361no—
192C3×C4⋊C4—32no—
193C3×C4⋊C4—32no—
194C3×C4⋊C4—62no—
195C3×C4⋊C4—94no—
196C3×C4⋊C4—94no—
197C3×C4⋊C4—94no—
198C3×C4⋊C4—92no—
199C3×C42—32no—
200C3×C42—61no—
201C3×C42—91no—
202C4⋊Dic3—32no—
203C4⋊Dic3—32no—
204C4⋊Dic3—34no—
205C4⋊Dic3—34no—
206C4⋊Dic3—62no—
207C4⋊Dic3—94no—
208C4⋊Dic3D3212yes—
209Dic3⋊C4—34no—
210Dic3⋊C4—34no—
211Dic3⋊C4—34no—
212Dic3⋊C4—34no—
213Dic3⋊C4—94no—
214Dic3⋊C4—94no—
215Dic3⋊C4—94no—
216Dic3⋊C4—94no—
217Dic3⋊C4—94no—
218Dic3⋊C4—184no—
219C4×Dic3—32no—
220C4×Dic3—61no—
221C4×Dic3—92no—
222C4×Dic3—92no—
223C4×Dic3—92no—
224C4×Dic3—182no—
225C4×Dic3D3211yescharacteristic
226C22×C32—21no—
227C22×C32—22no—
228C22×C32—41no—
229C22×C32C23×D312yes—
230C22×C32C23×D311yescharacteristic
231C2×C3⋊D3—34no—
232C2×C3⋊D3—32no—
233C2×C3⋊D3—34no—
234C2×C3⋊D3—34no—
235C2×C3⋊D3—34no—
236C2×C3⋊D3—62no—
237C2×C3⋊D3—64no—
238C2×C3⋊D3—64no—
239C2×C3⋊D3—64no—
240C2×C3⋊D3—64no—
241C2×C3⋊D3—122no—
242C2×C3⋊D3—124no—
243C32×C4—22no—
244C32×C4—24no—
245C32×C4—42no—
246C32×C4—62no—
247C32×C4—64no—
248C32×C4D3×Q814yes—
249C32×C4D3×Q812yes—
250C3⋊Dic3—22no—
251C3⋊Dic3—24no—
252C3⋊Dic3—34no—
253C3⋊Dic3—32no—
254C3⋊Dic3—62no—
255C3⋊Dic3—64no—
256C3⋊Dic3—122no—
257C3×Dic3—24no—
258C3×Dic3—22no—
259C3×Dic3—22no—
260C3×Dic3—24no—
261C3×Dic3—24no—
262C3×Dic3—44no—
263C3×Dic3—64no—
264C3×Dic3—64no—
265C3×Dic3—62no—
266C3×Dic3—62no—
267C3×Dic3—64no—
268C3×Dic3—124no—
269#125—92no—
270#125—91no—
271#125—271no—
272#125—274no—
273C2×C4⋊C4—272no—
274C2×C4⋊C4—274no—
275C2×C42—271no—
276C33C2×#12511yescharacteristic, complemented, Sylow
277C22×D3—91no—
278C22×D3—92no—
279C22×D3—181no—
280C22×D3—182no—
281C22×D3—361no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—32no—
285C2×C3×C4—32no—
286C2×C3×C4—34no—
287C2×C3×C4—34no—
288C2×C3×C4—41no—
289C2×C3×C4—62no—
290C2×C3×C4—64no—
291C2×C3×C4—92no—
292C2×C3×C4—94no—
293C2×C3×C4C2×D3212yes—
294C2×C3×C4C2×D3211yescharacteristic
295C2×Dic3—32no—
296C2×Dic3—32no—
297C2×Dic3—34no—
298C2×Dic3—34no—
299C2×Dic3—62no—
300C2×Dic3—64no—
301C2×Dic3—91no—
302C2×Dic3—92no—
303C2×Dic3—181no—
304C2×Dic3—182no—
305C2×Dic3—361no—
306C2×Dic3C2×D3212yes—
307C2×Dic3C2×D3214yes—
308C4×D3—94no—
309C4×D3—98no—
310C4×D3—184no—
311C4×D3—188no—
312C4×D3—184no—
313C4×D3—188no—
314C4×D3—184no—
315C4×D3—184no—
316C4×D3—188no—
317C4×D3—188no—
318C4×D3—188no—
319C4×D3—364no—
320C4×D3—368no—
321C2×C32—21no—
322C2×C32—22no—
323C2×C32—22no—
324C2×C32—22no—
325C2×C32—22no—
326C2×C32—41no—
327C2×C32—42no—
328C2×C32C2×D3×Q812yes—
329C2×C32C2×D3×Q811yescharacteristic
330C2×C32D3×#4112yes—
331C2×C32D3×#4112yes—
332C2×C32D3×#4112yes—
333C3⋊D3—38no—
334C3⋊D3—34no—
335C3⋊D3—64no—
336C3⋊D3—68no—
337C3⋊D3—124no—
338C22×C4—272no—
339C22×C4—274no—
340C22×C4—271no—
341C4⋊C4—32no—
342C4⋊C4—94no—
343C4⋊C4—94no—
344C4⋊C4—94no—
345C4⋊C4—92no—
346C4⋊C4—274no—
347C4⋊C4—274no—
348C42—31no—
349C42—91no—
350C42—272no—
351C22×C3—21no—
352C22×C3—22no—
353C22×C3—41no—
354C22×C3C22×D3211yescharacteristic
355C22×C3C22×D3212yes—
356C2×D3—92no—
357C2×D3—94no—
358C2×D3—94no—
359C2×D3—94no—
360C2×D3—94no—
361C2×D3—184no—
362C2×D3—182no—
363C2×D3—184no—
364C2×D3—184no—
365C2×D3—184no—
366C2×D3—362no—
367C2×D3—364no—
368C3×C4—22no—
369C3×C4—24no—
370C3×C4—42no—
371C3×C4—64no—
372C3×C4—64no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—64no—
376C3×C4—124no—
377C3×C4—182no—
378C3×C4—184no—
379C3×C4#379712yes—
380C3×C4#379714yes—
381Dic3—22no—
382Dic3—24no—
383Dic3—64no—
384Dic3—64no—
385Dic3—62no—
386Dic3—62no—
387Dic3—64no—
388Dic3—92no—
389Dic3—94no—
390Dic3—124no—
391Dic3—182no—
392Dic3—184no—
393Dic3—362no—
394C32—21no—
395C32—22no—
396C32—41no—
397C32D3×#12512yescomplemented
398C32D3×#12511yescharacteristic, complemented
399C23—271no—
400C2×C4—32no—
401C2×C4—34no—
402C2×C4—92no—
403C2×C4—94no—
404C2×C4—271no—
405C2×C4—274no—
406C2×C4—544no—
407C2×C4—548no—
408C2×C4D3311yescharacteristic
409C2×C3—22no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—22no—
414C2×C3—42no—
415C2×C3—41no—
416C2×C3C2×#379711yescharacteristic
417C2×C3C2×#379712yes—
418C2×C3#6430612yes—
419C2×C3#6431212yes—
420C2×C3#6431912yes—
421D3—94no—
422D3—98no—
423D3—184no—
424D3—188no—
425D3—364no—
426C22—274no—
427C22—272no—
428C22C2×D3311yescharacteristic
429C4—62no—
430C4—64no—
431C4—182no—
432C4—184no—
433C4—272no—
434C4#8895812yes—
435C3—21no—
436C3—22no—
437C3—41no—
438C3#1059394811yescharacteristic, complemented
439C3#1059448212yescomplemented
440C2—274no—
441C2#1172437612yes—
442C2C2×#8895811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3⋊(C3⋊D3)C23
hypercenterC2×C4D33
hypocenterC33C2×#125
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723680C2semidirect
4. C3×#63872C2semidirect
5. C3×#63869C2semidirect
6. #11723016C2semidirect
37. #63872D3semidirect
40. #63869D3semidirect
178. C3⋊(C3⋊D3)#125semidirect
276. C33C2×#125semidirect
397. C32D3×#125semidirect
398. C32D3×#125semidirect
438. C3#10593948semidirect
439. C3#10594482semidirect

Maximal subgroups

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

subgroupindex
#117236802
#117235032
C4×#62892
C3×#638722
C3×#638692
#117230162
#117229122
#105944823
#105939483