Finite Group Databasebeta

← #13167056

Subgroups — #13167056

Order 1728.

602 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 79 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×#88809C211yesmaximal, characteristic, complemented
4#11722762C214yesmaximal, complemented
5#10593324—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C32×#235—21no—
8C3×#3811—21no—
9C3×#3805C412yescomplemented
10C3×#3802C411yescharacteristic, complemented
11C3×C4×D32—21no—
12C3×#3798C2212yes—
13C3×#3797C411yescharacteristic, complemented
14C3×#3796—21no—
15#88994—21no—
16C4⋊(C3⋊#157)—21no—
17C4×C3⋊#157C2212yescomplemented
18C2×C3×C3⋊(C3⋊C8)—21no—
19#88809—21no—
20#88809C2212yescomplemented
21C3×D3×#41—61no—
22#64306D311yescharacteristic
23#64286—31no—
24C2×#3712—31no—
25C2×#3711—61no—
26#63731—34no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3×D32—21no—
29C2×C3⋊#157—22no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C3×C4×C3⋊D3C2×C411yescharacteristic
33C3×C4×C3⋊D3D411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3C2311yescharacteristic
36C32×#67—22no—
37C32×D12—21no—
38C32×C4×D3—22no—
39C32×Dic6—21no—
40C3×#627C2×C412yes—
41C3×#626—22no—
42C3×#625C2×C412yes—
43C3×#624—21no—
44C3×D3×Dic3—22no—
45C3×C3⋊(C3⋊C8)—22no—
46#5236—91nomaximal
47C32×#41—61no—
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C3×#239—61no—
50C3×D3×Q8—61no—
51C3×#237—61no—
52C3×#237—62no—
53C3×D3×D4—61no—
54C3×D3×D4—62no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235—41no—
58C3×#235—62no—
59C2×C3×C4×D3—62no—
60C2×C3×C4×D3—61no—
61C2×C3×C4×D3—61no—
62#3811—21no—
63#3805Dic312yescomplemented
64#3802Dic311yescharacteristic, complemented
65C4×D32—21no—
66#3798C2×D312yes—
67#3797Dic311yescharacteristic, complemented
68#3796—21no—
69C22⋊#157—61no—
70C4⋊#157—61no—
71C4×#157—32no—
72C2×C3⋊(C3⋊C8)—21no—
73C2×C3×C3⋊C8—61no—
74C2×C3×C3⋊C8—61no—
75C2×C3×C3⋊C8—121no—
76#3712—32no—
77#3712—61no—
78#3711—62no—
79#3711—62no—
80C22×C33C22⋊C411yescharacteristic
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3C22⋊C411yescharacteristic
83C2×C3×C3⋊D3C2×D411yescharacteristic
84C2×C32×D3—22no—
85C3×D32—42no—
86C3⋊#157—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3C22⋊C411yescharacteristic
90C3×C3⋊Dic3C22×C411yescharacteristic
91C32×Dic3—22no—
92C2×C3×#41—91no—
93D3×#41—61no—
94#926—91no—
95C2×#208—91no—
96C2×ΓL(1,𝔽25)—181no—
97#837—94no—
98C22×C3⋊D3#6711yescharacteristic
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C22×C3×D3—62no—
102C2×D32—21no—
103C2×#157—62no—
104C32×Q8—61no—
105C32×D4—61no—
106C32×D4—62no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—62no—
112C2×C32×C4#6711yescharacteristic
113C2×C3⋊Dic3C2×Dic311yescharacteristic
114C4×C3⋊D3C2×Dic311yescharacteristic
115C4×C3⋊D3#6711yescharacteristic
116C4×C3⋊D3#6711yescharacteristic
117C4×C3⋊D3C22×D311yescharacteristic
118C3×#67—22no—
119C3×#67—22no—
120C3×#67—42no—
121C3×#67—62no—
122C3×#67—62no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—62no—
126C3×D12—21no—
127C3×D12—21no—
128C3×D12—41no—
129C3×D12—62no—
130C3×C4×D3—22no—
131C3×C4×D3—22no—
132C3×C4×D3—42no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
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—62no—
142C3×C4×D3—62no—
143C3×Dic6—21no—
144C3×Dic6—21no—
145C3×Dic6—41no—
146C3×Dic6—62no—
147#627C2×Dic312yes—
148#626—22no—
149#625C2×Dic312yes—
150#624—21no—
151D3×Dic3—22no—
152C3⋊(C3⋊C8)—22no—
153C3×C3⋊C8—62no—
154C3×C3⋊C8—62no—
155C3×C3⋊C8—122no—
156#421—271noSylow
157C2×C33—21no—
158C2×C33C2×C22⋊C411yescharacteristic
159C3×C3⋊D3—21no—
160C3×C3⋊D3—21no—
161C32×D3—42no—
162C3×#41—61no—
163C3×#41—61no—
164C3×#41—92no—
165C3×#41—92no—
166C3×#41—121no—
167C3×#41—181no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C2×C3×D4—181no—
171C22×C3×C4—91no—
172C22×C3×C4—181no—
173#239—61no—
174D3×Q8—61no—
175#237—62no—
176#237—61no—
177D3×D4—61no—
178D3×D4—62no—
179#235—21no—
180#235—62no—
181C2×C4×D3—61no—
182C2×C4×D3—61no—
183C2×C4×D3—62no—
184C2×C3×C8—181no—
185#217—181no—
186C4⋊Dic3—181no—
187C4×Dic3—92no—
188#208—92no—
189#208—181no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—61no—
192C2×C3⋊C8—121no—
193ΓL(1,𝔽25)—182no—
194ΓL(1,𝔽25)—182no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—62no—
200C22×C32#21711yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3#21711yescharacteristic
203C2×C3⋊D3C2×#6711yescharacteristic
204C2×C3×D3—22no—
205C2×C3×D3—22no—
206C2×C3×D3—42no—
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—124no—
212C2×C3×D3—122no—
213C2×C3×D3—122no—
214D32—42no—
215#157—122no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—62no—
225C32×C4#21711yescharacteristic
226C32×C4C2×#6711yescharacteristic
227C3⋊Dic3#21711yescharacteristic
228C3⋊Dic3C22×Dic311yescharacteristic
229C3×Dic3—22no—
230C3×Dic3—22no—
231C3×Dic3—42no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C2×#41—91no—
237C2×M16—271no—
238#117—271no—
239#104—274no—
240C33#42111yescharacteristic, complemented, Sylow
241C23×C3—91no—
242C23×C3—181no—
243C22×D3—61no—
244C22×D3—61no—
245C22×D3—62no—
246C3×Q8—61no—
247C3×Q8—61no—
248C3×Q8—92no—
249C3×Q8—121no—
250C3×D4—61no—
251C3×D4—62no—
252C3×D4—62no—
253C3×D4—61no—
254C3×D4—92no—
255C3×D4—121no—
256C3×D4—122no—
257C3×D4—182no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—91no—
265C2×C3×C4—91no—
266C2×C3×C4—91no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—122no—
270C2×C3×C4—181no—
271C2×C3×C4—182no—
272C2×C3×C4GO+(4,𝔽2)11yescharacteristic
273#67—22no—
274#67—62no—
275#67—62no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—62no—
279C2×Dic3—182no—
280D12—21no—
281D12—62no—
282C4×D3—22no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—62no—
292C4×D3—62no—
293Dic6—21no—
294Dic6—62no—
295C3×C8—182no—
296C3⋊C8—62no—
297C3⋊C8—62no—
298C3⋊C8—122no—
299C2×C32—22no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—41no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—42no—
307C2×C32—81no—
308C2×C32—122no—
309C2×C32C2×#21711yescharacteristic
310C3⋊D3—21no—
311C3⋊D3—21no—
312C3×D3—42no—
313C3×D3—42no—
314C3×D3—62no—
315C3×D3—82no—
316C3×D3—121no—
317C3×D3—121no—
318C3×D3—121no—
319#41—61no—
320#41—92no—
321#41—92no—
322#41—181no—
323C2×Q8—91no—
324C2×D4—91no—
325C2×D4—181no—
326C22×C4—91no—
327C22×C4—181no—
328M16—272no—
329M16—541no—
330C2×C8—181no—
331C4⋊C4—541no—
332C22⋊C4—541no—
333C42—272no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—62no—
339C22×C3—62no—
340C22×C3—91no—
341C22×C3—91no—
342C22×C3—122no—
343C22×C3—361no—
344C22×C3—362no—
345C22×C3D32⋊C411yescharacteristic
346C2×D3—22no—
347C2×D3—61no—
348C2×D3—61no—
349C2×D3—61no—
350C2×D3—61no—
351C2×D3—124no—
352C2×D3—122no—
353C2×D3—122no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—62no—
363C3×C4—62no—
364C3×C4—91no—
365C3×C4—91no—
366C3×C4—122no—
367C3×C4D32⋊C411yescharacteristic
368C3×C4C2×GO+(4,𝔽2)11yescharacteristic
369Dic3—22no—
370Dic3—61no—
371Dic3—61no—
372Dic3—61no—
373Dic3—61no—
374Dic3—362no—
375C32—21no—
376C32—21no—
377C32—41no—
378C32—41no—
379C32#523611yescharacteristic, complemented
380C23—91no—
381C23—181no—
382Q8—61no—
383Q8—92no—
384D4—62no—
385D4—61no—
386D4—92no—
387D4—182no—
388C2×C4—62no—
389C2×C4—91no—
390C2×C4—91no—
391C2×C4—91no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—181no—
395C2×C4—182no—
396C2×C4—542no—
397C2×C4D32⋊D311yescharacteristic
398C8—182no—
399C2×C3—22no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—42no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—81no—
408C2×C3—122no—
409C2×C3—122no—
410C2×C3—181no—
411C2×C3—181no—
412C2×C3—242no—
413C2×C3C2×D32⋊C411yescharacteristic
414D3—42no—
415D3—62no—
416D3—121no—
417D3—121no—
418D3—121no—
419C22—62no—
420C22—91no—
421C22—91no—
422C22—361no—
423C22—362no—
424C22D32⋊Dic311yescharacteristic
425C4—62no—
426C4—91no—
427C4—91no—
428C4—1082no—
429C4D32⋊Dic311yescharacteristic
430C4C2×D32⋊D311yescharacteristic
431C3—21no—
432C3—21no—
433C3—41no—
434C3—41no—
435C3#1059332411yescharacteristic, complemented
436C2—21no—
437C2—122no—
438C2—181no—
439C2—181no—
440C2C2×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
hypocenterC33#421
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×#88809C2semidirect
4. #11722762C2semidirect
9. C3×#3805C4semidirect
10. C3×#3802C4semidirect
13. C3×#3797C4semidirect
17. C4×C3⋊#157C22semidirect
20. #88809C22semidirect
63. #3805Dic3semidirect
64. #3802Dic3semidirect
67. #3797Dic3semidirect
240. C33#421semidirect
379. C32#5236semidirect
435. C3#10593324semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
#117245142
C2×#888092
#117227622
#105933243
#52369