Finite Group Databasebeta

← #13167058

Subgroups — #13167058

Order 1728.

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

normal subgroups only

This group has 632 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
2C2×C4×C3⋊#157C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4#11722762C214yesmaximal, complemented
5#10593325—31nomaximal
6C22×C3⋊#157—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C32×#235—21no—
9C3×#3811—21no—
10C3×#3805C412yescomplemented
11C3×#3802C411yescharacteristic, complemented
12C3×C4×D32—21no—
13C3×#3798C2212yes—
14C3×#3797C411yescharacteristic, complemented
15C3×#3796—21no—
16C4×C3⋊#157—21no—
17C4×C3⋊#157C2212yescomplemented
18C3×#3748—21no—
19#88809C2212yescomplemented
20#88808—21no—
21C3×D3×#41—61no—
22#64306D311yescharacteristic
23C2×C4×#157—31no—
24#63818—31no—
25D3×#208—61no—
26#63731—34no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3×D32—21no—
29C2×C3⋊#157—24no—
30C2×C3⋊#157—22no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C3×C4×C3⋊D3C2×C411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3C2311yescharacteristic
37C32×#67—22no—
38C32×D12—21no—
39C32×C4×D3—22no—
40C32×Dic6—21no—
41C3×#627C2×C412yescomplemented
42C3×#626—22no—
43C3×#625C2×C412yescomplemented
44C3×#624—21no—
45C3×D3×Dic3—22no—
46C3×C3⋊(C3⋊C8)—22no—
47C2×#837—91nomaximal
48C22×#157—61no—
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#3802Dic311yescharacteristic, complemented
67C4×D32—21no—
68#3798C2×D312yes—
69#3797Dic311yescharacteristic, complemented
70#3796—21no—
71C4×#157—32no—
72C4×#157—61no—
73#3748—21no—
74C3×#208—61no—
75C3×#208—61no—
76C3×#208—121no—
77#3712—32no—
78#3711—62no—
79#3710—61no—
80D3×C3⋊C8—62no—
81C22×C33C22⋊C411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C3×C3⋊D3C2×D411yescharacteristic
85C2×C32×D3—22no—
86C3×D32—42no—
87C3⋊#157—24no—
88C33×C4C22⋊C411yescharacteristic
89C33×C4C2×D411yescharacteristic
90C3×C3⋊Dic3C22⋊C411yescharacteristic
91C3×C3⋊Dic3C22×C411yescharacteristic
92C32×Dic3—22no—
93C2×C3×#41—91no—
94D3×#41—61no—
95C2×C4×Dic3—91no—
96C2×#208—91no—
97D3×M16—181no—
98#837—94no—
99C22×C3⋊D3#6711yescharacteristic
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C22×C3×D3—62no—
103C2×D32—21no—
104C2×#157—64no—
105C2×#157—62no—
106C32×Q8—61no—
107C32×D4—61no—
108C32×D4—62no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—41no—
112C2×C32×C4—41no—
113C2×C32×C4—62no—
114C2×C32×C4#6711yescharacteristic
115C2×C3⋊Dic3C2×Dic311yescharacteristic
116C4×C3⋊D3C2×Dic311yescharacteristic
117C4×C3⋊D3#6711yescharacteristic
118C4×C3⋊D3#6711yescharacteristic
119C4×C3⋊D3C22×D311yescharacteristic
120C3×#67—22no—
121C3×#67—22no—
122C3×#67—42no—
123C3×#67—62no—
124C3×#67—62no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—62no—
128C3×D12—21no—
129C3×D12—21no—
130C3×D12—41no—
131C3×D12—62no—
132C3×C4×D3—22no—
133C3×C4×D3—22no—
134C3×C4×D3—42no—
135C3×C4×D3—61no—
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—62no—
144C3×C4×D3—62no—
145C3×Dic6—21no—
146C3×Dic6—21no—
147C3×Dic6—41no—
148C3×Dic6—62no—
149#627C2×Dic312yescomplemented
150#626—22no—
151#625C2×Dic312yescomplemented
152#624—21no—
153D3×Dic3—22no—
154C3⋊(C3⋊C8)—22no—
155C3×C3⋊C8—62no—
156C3×C3⋊C8—62no—
157C3×C3⋊C8—122no—
158C2×#104—271noSylow
159C2×C33—21no—
160C2×C33C2×C22⋊C411yescharacteristic
161C3×C3⋊D3—21no—
162C3×C3⋊D3#10412yescomplemented
163C32×D3—42no—
164C3×#41—61no—
165C3×#41—61no—
166C3×#41—92no—
167C3×#41—92no—
168C3×#41—121no—
169C3×#41—181no—
170C2×C3×Q8—91no—
171C2×C3×D4—91no—
172C2×C3×D4—181no—
173C22×C3×C4—91no—
174C22×C3×C4—181no—
175C22×Dic3—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—
187C3×M16—181no—
188C4×Dic3—92no—
189C4×Dic3—181no—
190#208—61no—
191#208—61no—
192#208—92no—
193#208—121no—
194C2×C3⋊C8—181no—
195ΓL(1,𝔽25)—182no—
196D3×C8—182no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—62no—
202C22×C32#21711yescharacteristic
203C2×C3⋊D3—22no—
204C2×C3⋊D3#21711yescharacteristic
205C2×C3⋊D3C2×#6711yescharacteristic
206C2×C3×D3—22no—
207C2×C3×D3—22no—
208C2×C3×D3—42no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—122no—
214C2×C3×D3—122no—
215C2×C3×D3—124no—
216D32—42no—
217#157—64no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—62no—
227C32×C4#21711yescharacteristic
228C32×C4C2×#6711yescharacteristic
229C3⋊Dic3#21711yescharacteristic
230C3⋊Dic3C22×Dic311yescharacteristic
231C3×Dic3—22no—
232C3×Dic3—22no—
233C3×Dic3—42no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C2×#41—91no—
239C2×M16—271no—
240C2×C42—271no—
241#104—274no—
242C33C2×#10411yescharacteristic, complemented, Sylow
243C23×C3—91no—
244C23×C3—181no—
245C22×D3—61no—
246C22×D3—61no—
247C22×D3—62no—
248C3×Q8—61no—
249C3×Q8—61no—
250C3×Q8—92no—
251C3×Q8—121no—
252C3×D4—61no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—61no—
256C3×D4—92no—
257C3×D4—121no—
258C3×D4—122no—
259C3×D4—182no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—91no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—122no—
272C2×C3×C4—181no—
273C2×C3×C4—182no—
274C2×C3×C4GO+(4,𝔽2)11yescharacteristic
275#67—22no—
276#67—62no—
277#67—62no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—62no—
281C2×Dic3—182no—
282C2×Dic3—184no—
283D12—21no—
284D12—62no—
285C4×D3—22no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—61no—
294C4×D3—62no—
295C4×D3—62no—
296Dic6—21no—
297Dic6—62no—
298C3×C8—182no—
299C3⋊C8—62no—
300C3⋊C8—62no—
301C3⋊C8—122no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—41no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32—81no—
310C2×C32—81no—
311C2×C32—122no—
312C2×C32C2×#21711yescharacteristic
313C3⋊D3—21no—
314C3⋊D3#83712yescomplemented
315C3×D3—42no—
316C3×D3—42no—
317C3×D3—62no—
318C3×D3—62no—
319C3×D3—82no—
320C3×D3—121no—
321C3×D3—121no—
322#41—61no—
323#41—92no—
324#41—92no—
325#41—181no—
326C2×Q8—91no—
327C2×D4—91no—
328C2×D4—181no—
329C22×C4—91no—
330C22×C4—181no—
331C22×C4—541no—
332M16—181no—
333M16—272no—
334C2×C8—541no—
335C42—272no—
336C42—541no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—62no—
342C22×C3—62no—
343C22×C3—91no—
344C22×C3—91no—
345C22×C3—122no—
346C22×C3—182no—
347C22×C3—362no—
348C22×C3D32⋊C411yescharacteristic
349C2×D3—22no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—124no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—62no—
366C3×C4—62no—
367C3×C4—91no—
368C3×C4—91no—
369C3×C4—122no—
370C3×C4D32⋊C411yescharacteristic
371C3×C4C2×GO+(4,𝔽2)11yescharacteristic
372Dic3—22no—
373Dic3—61no—
374Dic3—61no—
375Dic3—61no—
376Dic3—61no—
377Dic3—184no—
378C32—21no—
379C32—21no—
380C32—41no—
381C32—41no—
382C32C2×#83711yescharacteristic, complemented
383C23—91no—
384C23—181no—
385Q8—61no—
386Q8—92no—
387D4—62no—
388D4—61no—
389D4—92no—
390D4—182no—
391C2×C4—62no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—181no—
398C2×C4—182no—
399C2×C4—542no—
400C2×C4—544no—
401C2×C4D32⋊D311yescharacteristic
402C8—182no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—81no—
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—121no—
422D3—121no—
423C22—62no—
424C22—91no—
425C22—91no—
426C22—182no—
427C22—362no—
428C22D32⋊Dic311yescharacteristic
429C4—62no—
430C4—91no—
431C4—91no—
432C4—544no—
433C4D32⋊Dic311yescharacteristic
434C4C2×D32⋊D311yescharacteristic
435C3—21no—
436C3—21no—
437C3—41no—
438C3—41no—
439C3#1059332511yescharacteristic, complemented
440C2—21no—
441C2—92no—
442C2—122no—
443C2—181no—
444C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC4C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC3×#64306C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#104
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. C2×C4×C3⋊#157C2semidirect
3. #11723656C2semidirect
4. #11722762C2semidirect
10. C3×#3805C4semidirect
11. C3×#3802C4semidirect
14. C3×#3797C4semidirect
17. C4×C3⋊#157C22semidirect
19. #88809C22semidirect
41. C3×#627C2×C4semidirect
43. C3×#625C2×C4semidirect
65. #3805Dic3semidirect
66. #3802Dic3semidirect
69. #3797Dic3semidirect
149. #627C2×Dic3semidirect
151. #625C2×Dic3semidirect
162. C3×C3⋊D3#104semidirect
242. C33C2×#104semidirect
314. C3⋊D3#837semidirect
382. C32C2×#837semidirect
439. C3#10593325semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
C2×C4×C3⋊#1572
#117236562
#117227622
#105933253
C2×#8379