Finite Group Databasebeta

← #13167025

Subgroups — #13167025

Order 1728.

618 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 83 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×#64306C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3C2×#88808C211yesmaximal, characteristic, complemented
4#11722760C212yesmaximal, complemented
5#11722758C212yesmaximal, complemented
6#10593317—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C32×#235—21no—
9C3×#3811—21no—
10C3×#3805C412yescomplemented
11C3×#3802C2211yescharacteristic
12C3×C4×D32—21no—
13C3×#3798C412yescomplemented
14C3×#3797C2211yescharacteristic
15C3×#3796—21no—
16#88994—21no—
17C4⋊(C3⋊#157)C2212yes—
18C4×C3⋊#157—21no—
19C2×C3×C3⋊(C3⋊C8)—21no—
20#88808—21no—
21#88808C2212yes—
22C3×D3×#41—61no—
23#64306D311yescharacteristic
24#64286—31no—
25C2×#3710—31no—
26C2×D3×C3⋊C8—61no—
27#63730—32no—
28#63729—32no—
29C22×C3×C3⋊D3D411yescharacteristic
30C2×C3×D32—21no—
31C2×C3⋊#157—22no—
32C2×C33×C4D411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4×C3⋊D3C2×C411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3C2311yescharacteristic
38C32×#67—22no—
39C32×D12—21no—
40C32×C4×D3—22no—
41C32×Dic6—21no—
42C3×#627C2×C412yes—
43C3×#626—22no—
44C3×#625C2×C412yes—
45C3×#624—21no—
46C3×D3×Dic3—22no—
47C3×C3⋊(C3⋊C8)—22no—
48#5233—91nomaximal
49C32×#41—61no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C3×#239—61no—
52C3×D3×Q8—61no—
53C3×#237—61no—
54C3×#237—62no—
55C3×D3×D4—61no—
56C3×D3×D4—62no—
57C3×#235—21no—
58C3×#235—21no—
59C3×#235—41no—
60C3×#235—62no—
61C2×C3×C4×D3—62no—
62C2×C3×C4×D3—61no—
63C2×C3×C4×D3—61no—
64#3811—21no—
65#3805Dic312yescomplemented
66#3802C2×D311yescharacteristic
67C4×D32—21no—
68#3798Dic312yescomplemented
69#3797C2×D311yescharacteristic
70#3796—21no—
71C22⋊#157—61no—
72C4⋊#157—32no—
73C4×#157—61no—
74C2×C3⋊(C3⋊C8)—21no—
75C2×C3×C3⋊C8—61no—
76C2×C3×C3⋊C8—61no—
77C2×C3×C3⋊C8—121no—
78#3710—32no—
79#3710—61no—
80D3×C3⋊C8—62no—
81D3×C3⋊C8—62no—
82C22×C33C22⋊C411yescharacteristic
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3C22⋊C411yescharacteristic
85C2×C3×C3⋊D3C2×D411yescharacteristic
86C2×C32×D3—22no—
87C3×D32—42no—
88C3⋊#157—42no—
89C33×C4C22⋊C411yescharacteristic
90C33×C4C2×D411yescharacteristic
91C3×C3⋊Dic3C22⋊C411yescharacteristic
92C3×C3⋊Dic3C22×C411yescharacteristic
93C32×Dic3—22no—
94C2×C3×#41—91no—
95D3×#41—61no—
96#926—91no—
97C22×C3⋊C8—91no—
98C2×D3×C8—181no—
99#835—92no—
100#832—92no—
101C22×C3⋊D3#6711yescharacteristic
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—62no—
105C2×D32—21no—
106C2×#157—62no—
107C32×Q8—61no—
108C32×D4—61no—
109C32×D4—62no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—41no—
113C2×C32×C4—41no—
114C2×C32×C4—62no—
115C2×C32×C4#6711yescharacteristic
116C2×C3⋊Dic3C2×Dic311yescharacteristic
117C4×C3⋊D3C2×Dic311yescharacteristic
118C4×C3⋊D3#6711yescharacteristic
119C4×C3⋊D3#6711yescharacteristic
120C4×C3⋊D3C22×D311yescharacteristic
121C3×#67—22no—
122C3×#67—22no—
123C3×#67—42no—
124C3×#67—62no—
125C3×#67—62no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—62no—
129C3×D12—21no—
130C3×D12—21no—
131C3×D12—41no—
132C3×D12—62no—
133C3×C4×D3—22no—
134C3×C4×D3—22no—
135C3×C4×D3—42no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—62no—
145C3×C4×D3—62no—
146C3×Dic6—21no—
147C3×Dic6—21no—
148C3×Dic6—41no—
149C3×Dic6—62no—
150#627C2×Dic312yes—
151#626—22no—
152#625C2×Dic312yes—
153#624—21no—
154D3×Dic3—22no—
155C3⋊(C3⋊C8)—22no—
156C3×C3⋊C8—62no—
157C3×C3⋊C8—62no—
158C3×C3⋊C8—122no—
159#416—271noSylow
160C2×C33—21no—
161C2×C33C2×C22⋊C411yescharacteristic
162C3×C3⋊D3—21no—
163C3×C3⋊D3#13512yes—
164C32×D3—42no—
165C3×#41—61no—
166C3×#41—61no—
167C3×#41—92no—
168C3×#41—92no—
169C3×#41—121no—
170C3×#41—181no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C22×C3×C4—91no—
175C22×C3×C4—181no—
176#239—61no—
177D3×Q8—61no—
178#237—62no—
179#237—61no—
180D3×D4—61no—
181D3×D4—62no—
182#235—21no—
183#235—62no—
184C2×C4×D3—61no—
185C2×C4×D3—61no—
186C2×C4×D3—62no—
187C2×C3×C8—181no—
188#217—181no—
189C4⋊Dic3—92no—
190C4×Dic3—181no—
191C2×C3⋊C8—61no—
192C2×C3⋊C8—61no—
193C2×C3⋊C8—92no—
194C2×C3⋊C8—121no—
195C2×C3⋊C8—181no—
196D3×C8—182no—
197D3×C8—182no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—62no—
203C22×C32#21711yescharacteristic
204C2×C3⋊D3—22no—
205C2×C3⋊D3#21711yescharacteristic
206C2×C3⋊D3C2×#6711yescharacteristic
207C2×C3×D3—22no—
208C2×C3×D3—22no—
209C2×C3×D3—42no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—64no—
215C2×C3×D3—122no—
216C2×C3×D3—124no—
217D32—42no—
218#157—122no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4—62no—
228C32×C4#21711yescharacteristic
229C32×C4C2×#6711yescharacteristic
230C3⋊Dic3#21711yescharacteristic
231C3⋊Dic3C22×Dic311yescharacteristic
232C3×Dic3—22no—
233C3×Dic3—22no—
234C3×Dic3—42no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C2×#41—91no—
240C22×C8—271no—
241#117—271no—
242#103—272no—
243#102—272no—
244C33#41611yescharacteristic, complemented, Sylow
245C23×C3—91no—
246C23×C3—181no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—62no—
250C3×Q8—61no—
251C3×Q8—61no—
252C3×Q8—92no—
253C3×Q8—121no—
254C3×D4—61no—
255C3×D4—62no—
256C3×D4—62no—
257C3×D4—61no—
258C3×D4—92no—
259C3×D4—121no—
260C3×D4—122no—
261C3×D4—182no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—122no—
274C2×C3×C4—181no—
275C2×C3×C4—182no—
276C2×C3×C4GO+(4,𝔽2)11yescharacteristic
277#67—22no—
278#67—62no—
279#67—62no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—62no—
283C2×Dic3—182no—
284D12—21no—
285D12—62no—
286C4×D3—22no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—61no—
294C4×D3—61no—
295C4×D3—62no—
296C4×D3—62no—
297Dic6—21no—
298Dic6—62no—
299C3×C8—182no—
300C3⋊C8—62no—
301C3⋊C8—62no—
302C3⋊C8—122no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—42no—
311C2×C32—81no—
312C2×C32—122no—
313C2×C32C2×#21711yescharacteristic
314C3⋊D3—21no—
315C3⋊D3#95012yes—
316C3×D3—42no—
317C3×D3—42no—
318C3×D3—62no—
319C3×D3—62no—
320C3×D3—62no—
321C3×D3—82no—
322C3×D3—121no—
323#41—61no—
324#41—92no—
325#41—92no—
326#41—181no—
327C2×Q8—91no—
328C2×D4—91no—
329C2×D4—181no—
330C22×C4—91no—
331C22×C4—181no—
332C2×C8—181no—
333C2×C8—272no—
334C2×C8—541no—
335C4⋊C4—272no—
336C22⋊C4—541no—
337C42—541no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—62no—
343C22×C3—62no—
344C22×C3—91no—
345C22×C3—91no—
346C22×C3—122no—
347C22×C3—182no—
348C22×C3—362no—
349C22×C3D32⋊C411yescharacteristic
350C2×D3—22no—
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—61no—
355C2×D3—64no—
356C2×D3—122no—
357C2×D3—124no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—62no—
367C3×C4—62no—
368C3×C4—91no—
369C3×C4—91no—
370C3×C4—122no—
371C3×C4D32⋊C411yescharacteristic
372C3×C4C2×GO+(4,𝔽2)11yescharacteristic
373Dic3—22no—
374Dic3—61no—
375Dic3—61no—
376Dic3—61no—
377Dic3—61no—
378Dic3—362no—
379C32—21no—
380C32—21no—
381C32—41no—
382C32—41no—
383C32#523311yescharacteristic, complemented
384C23—91no—
385C23—181no—
386Q8—61no—
387Q8—92no—
388D4—62no—
389D4—61no—
390D4—92no—
391D4—182no—
392C2×C4—62no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—181no—
399C2×C4—182no—
400C2×C4—542no—
401C2×C4D32⋊D311yescharacteristic
402C8—182no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—41no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—81no—
412C2×C3—92no—
413C2×C3—122no—
414C2×C3—122no—
415C2×C3—181no—
416C2×C3—242no—
417C2×C3C2×D32⋊C411yescharacteristic
418D3—42no—
419D3—62no—
420D3—62no—
421D3—62no—
422D3—121no—
423C22—62no—
424C22—91no—
425C22—91no—
426C22—182no—
427C22—362no—
428C22D32⋊Dic311yescharacteristic
429C4—62no—
430C4—91no—
431C4—91no—
432C4—1082no—
433C4D32⋊Dic311yescharacteristic
434C4C2×D32⋊D311yescharacteristic
435C3—21no—
436C3—21no—
437C3—41no—
438C3—41no—
439C3#1059331711yescharacteristic, complemented
440C2—21no—
441C2—92no—
442C2—122no—
443C2—181no—
444C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
centerC4D32⋊Dic3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC3×#64306C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D32⋊D3
hypocenterC33#416
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. C2×#88808C2semidirect
4. #11722760C2semidirect
5. #11722758C2semidirect
10. C3×#3805C4semidirect
13. C3×#3798C4semidirect
65. #3805Dic3semidirect
68. #3798Dic3semidirect
244. C33#416semidirect
383. C32#5233semidirect
439. C3#10593317semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
#117245142
C2×#888082
#117227602
#117227582
#105933173
#52339