Finite Group Databasebeta

← #13155046

Subgroups — #13155046

Order 1728.

636 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C2×#88994C211yesmaximal, characteristic
2C2×#88814C211yesmaximal, characteristic
3C3×#63974C211yesmaximal, characteristic
4#10590076—31nomaximal
5C23×C3×C3⋊D3C2211yescharacteristic
6C22×C3⋊#157—21no—
7C22×C3×C3⋊Dic3—21no—
8C2×C3×#624C412yescomplemented
9#88994—21no—
10#88994C412yescomplemented
11C32×#217—21no—
12#88814—21no—
13#88814C412yescomplemented
14C3×#3722—21no—
15C2×C22⋊#157—31no—
16C3×D3×C22⋊C4—61no—
17#63974D311yescharacteristic
18C2×#3722—31no—
19C2×(C2×D3)⋊Dic3—61no—
20C23×C33D411yescharacteristic
21C22×C3×C3⋊D3—42no—
22C22×C3×C3⋊D3C2×C411yescharacteristic
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C22×C3×C3⋊D3C2×C411yescharacteristic
25C22×C3×C3⋊D3D411yescharacteristic
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C3×C3⋊D3Q811yescharacteristic
28C2×C3⋊#157—22no—
29C2×C3⋊#157—42no—
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3—42no—
32C2×C32×Dic3—22no—
33C3×#624—42no—
34#4526—91nomaximal
35C23×C3⋊D3C2×D311yescharacteristic
36C23×C3×D3—61no—
37C23×C3×D3—61no—
38C22×#157—61no—
39C22×C3⋊Dic3—21no—
40C22×C3×Dic3—61no—
41C22×C3×Dic3—61no—
42C22×C3×Dic3—121no—
43C2×C3×C4×D3—62no—
44C2×#624Dic312yescomplemented
45C22⋊#157—32no—
46C22⋊#157—61no—
47C32×C22⋊C4—61no—
48C3×#217—21no—
49C3×#217—21no—
50C3×#217—41no—
51C3×(C2×D3)⋊C4—62no—
52#3722—21no—
53#3722—32no—
54#3722—61no—
55(C2×D3)⋊Dic3—62no—
56(C2×D3)⋊Dic3—62no—
57C22×C33—22no—
58C22×C33C22⋊C411yescharacteristic
59C22×C33C22⋊C411yescharacteristic
60C22×C33C4⋊C411yescharacteristic
61C2×C3×C3⋊D3—22no—
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3—41no—
64C2×C3×C3⋊D3—41no—
65C2×C3×C3⋊D3—41no—
66C2×C3×C3⋊D3—41no—
67C2×C3×C3⋊D3C4211yescharacteristic
68C2×C3×C3⋊D3C22⋊C411yescharacteristic
69C2×C3×C3⋊D3C4⋊C411yescharacteristic
70C2×C3×C3⋊D3C4⋊C411yescharacteristic
71C3⋊#157—42no—
72C3×C3⋊Dic3—42no—
73C32×Dic3—42no—
74C2×C3×C22⋊C4—91no—
75C2×#217—91no—
76C2×#217—91no—
77C2×(C2×D3)⋊C4—181no—
78D3×C22⋊C4—61no—
79C23×C32—21no—
80C23×C32—21no—
81C23×C32—41no—
82C23×C32—41no—
83C23×C32#6711yescharacteristic
84C22×C3⋊D3—42no—
85C22×C3⋊D3Dic611yescharacteristic
86C22×C3⋊D3C4×D311yescharacteristic
87C22×C3⋊D3C4×D311yescharacteristic
88C22×C3⋊D3D1211yescharacteristic
89C22×C3⋊D3C2×Dic311yescharacteristic
90C22×C3⋊D3#6711yescharacteristic
91C22×C3×D3—61no—
92C22×C3×D3—61no—
93C22×C3×D3—61no—
94C22×C3×D3—61no—
95C22×C3×D3—61no—
96C22×C3×D3—61no—
97C22×C3×D3—61no—
98C22×C3×D3—61no—
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C22×C3×D3—124no—
104C22×C3×D3—122no—
105C22×C3×D3—122no—
106C2×#157—62no—
107C2×#157—122no—
108C2×C32×C4—62no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3—42no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—42no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—122no—
117C2×C3×Dic3—122no—
118C2×C3×Dic3—122no—
119C2×C3×Dic3—124no—
120C3×C4×D3—124no—
121#624—42no—
122#342—271noSylow
123C2×C33—21no—
124C2×C33—21no—
125C2×C33#9511yescharacteristic
126C2×C33#9912yes—
127C3×C3⋊D3—21no—
128C3×C3⋊D3—21no—
129C3×C3⋊D3—21no—
130C3×C3⋊D3—21no—
131C24×C3—91no—
132C23×D3—61no—
133C23×D3—61no—
134C22×C3×C4—92no—
135C22×C3×C4—181no—
136C22×Dic3—61no—
137C22×Dic3—61no—
138C22×Dic3—121no—
139C22×Dic3—181no—
140C2×C4×D3—62no—
141C3×C22⋊C4—61no—
142C3×C22⋊C4—61no—
143C3×C22⋊C4—121no—
144C3×C22⋊C4—181no—
145#217—21no—
146#217—92no—
147#217—92no—
148#217—181no—
149#217—181no—
150(C2×D3)⋊C4—62no—
151(C2×D3)⋊C4—182no—
152(C2×D3)⋊C4—182no—
153C22×C32—22no—
154C22×C32—22no—
155C22×C32—22no—
156C22×C32—21no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—41no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32—41no—
168C22×C32—42no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—82no—
172C22×C32Dic3⋊C411yescharacteristic
173C22×C32(C2×D3)⋊C411yescharacteristic
174C22×C32#21711yescharacteristic
175C2×C3⋊D3—22no—
176C2×C3⋊D3—22no—
177C2×C3⋊D3—41no—
178C2×C3⋊D3—41no—
179C2×C3⋊D3—41no—
180C2×C3⋊D3—41no—
181C2×C3⋊D3C4×Dic311yescharacteristic
182C2×C3⋊D3Dic3⋊C411yescharacteristic
183C2×C3⋊D3C4⋊Dic311yescharacteristic
184C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C2×C3×D3—64no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C2×C3×D3—122no—
197C2×C3×D3—122no—
198C2×C3×D3—122no—
199C2×C3×D3—122no—
200C2×C3×D3—122no—
201C2×C3×D3—121no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—122no—
206C2×C3×D3—122no—
207#157—122no—
208C32×C4—122no—
209C3⋊Dic3—42no—
210C3×Dic3—42no—
211C3×Dic3—42no—
212C3×Dic3—82no—
213C3×Dic3—122no—
214C3×Dic3—122no—
215C3×Dic3—124no—
216C2×C22⋊C4—91no—
217C2×C22⋊C4—271no—
218C2×C22⋊C4—271no—
219C33#34211yescharacteristic, complemented, Sylow
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—91no—
225C23×C3—91no—
226C23×C3—91no—
227C23×C3—91no—
228C23×C3—91no—
229C23×C3—91no—
230C23×C3—362no—
231C23×C3GO+(4,𝔽2)11yescharacteristic
232C22×D3—61no—
233C22×D3—61no—
234C22×D3—61no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—61no—
238C22×D3—61no—
239C22×D3—61no—
240C22×D3—61no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—124no—
245C22×D3—122no—
246C22×D3—122no—
247C2×C3×C4—62no—
248C2×C3×C4—62no—
249C2×C3×C4—122no—
250C2×C3×C4—182no—
251C2×C3×C4—362no—
252C2×C3×C4—362no—
253C2×Dic3—22no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—122no—
257C2×Dic3—122no—
258C2×Dic3—122no—
259C2×Dic3—124no—
260C2×Dic3—182no—
261C2×Dic3—362no—
262C4×D3—124no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—22no—
271C2×C32—22no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—42no—
275C2×C32—42no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—42no—
281C2×C32—81no—
282C2×C32—81no—
283C2×C32C3⋊#9511yescharacteristic
284C2×C32#83412yes—
285C3⋊D3—21no—
286C3⋊D3—21no—
287C3⋊D3—21no—
288C3⋊D3—21no—
289C3×D3—62no—
290C3×D3—62no—
291C3×D3—121no—
292C3×D3—121no—
293C3×D3—121no—
294C3×D3—121no—
295C3×D3—121no—
296C3×D3—121no—
297C24—91no—
298C22×C4—92no—
299C22×C4—181no—
300C22×C4—541no—
301C22⋊C4—61no—
302C22⋊C4—181no—
303C22⋊C4—272no—
304C22⋊C4—272no—
305C22⋊C4—541no—
306C22⋊C4—541no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—82no—
326C22×C3—91no—
327C22×C3—91no—
328C22×C3—91no—
329C22×C3—91no—
330C22×C3—182no—
331C22×C3—182no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3D32⋊C411yescharacteristic
337C22×C3D32⋊C411yescharacteristic
338C22×C3#377311yescharacteristic
339C2×D3—61no—
340C2×D3—61no—
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—64no—
348C2×D3—122no—
349C2×D3—122no—
350C2×D3—122no—
351C2×D3—122no—
352C2×D3—122no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—121no—
358C2×D3—121no—
359C2×D3—122no—
360C2×D3—122no—
361C3×C4—122no—
362C3×C4—122no—
363C3×C4—242no—
364C3×C4—362no—
365Dic3—42no—
366Dic3—122no—
367Dic3—122no—
368Dic3—124no—
369Dic3—362no—
370C32—21no—
371C32—21no—
372C32—41no—
373C32—41no—
374C32#452611yescharacteristic, complemented
375C23—91no—
376C23—91no—
377C23—91no—
378C23—91no—
379C23—91no—
380C23—91no—
381C23—362no—
382C23D32⋊D311yescharacteristic
383C2×C4—62no—
384C2×C4—182no—
385C2×C4—362no—
386C2×C4—362no—
387C2×C4—542no—
388C2×C4—1082no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—22no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—22no—
397C2×C3—22no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—41no—
402C2×C3—42no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3—42no—
406C2×C3—42no—
407C2×C3—81no—
408C2×C3—81no—
409C2×C3—181no—
410C2×C3—181no—
411C2×C3—181no—
412C2×C3—181no—
413C2×C3#6373711yescharacteristic
414C2×C3#6373812yes—
415D3—62no—
416D3—62no—
417D3—121no—
418D3—121no—
419D3—121no—
420D3—121no—
421D3—121no—
422D3—121no—
423C22—22no—
424C22—91no—
425C22—91no—
426C22—91no—
427C22—91no—
428C22—182no—
429C22—182no—
430C22—361no—
431C22—361no—
432C22—361no—
433C22—361no—
434C22#8893211yescharacteristic
435C22D32⋊Dic311yescharacteristic
436C22#8893411yescharacteristic
437C4—122no—
438C4—362no—
439C4—1082no—
440C3—21no—
441C3—21no—
442C3—41no—
443C3—41no—
444C3#1059007611yescharacteristic, complemented
445C2—21no—
446C2—21no—
447C2—181no—
448C2—181no—
449C2—181no—
450C2—181no—
451C2#1172276911yescharacteristic
452C2#1172277112yes—
subgroupG/H
centerC22D32⋊Dic3
commutatorC2×C3×C3⋊D3C42
Frattini, hypercenterC23D32⋊D3
FittingC23×C33D4
socleC22×C33C22⋊C4
prime radical, cosocleC23×C3×C3⋊D3C22
hypocenterC33#342
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
8. C2×C3×#624C4semidirect
10. #88994C4semidirect
13. #88814C4semidirect
44. C2×#624Dic3semidirect
219. C33#342semidirect
374. C32#4526semidirect
444. C3#10590076semidirect

Maximal subgroups

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

subgroupindex
C2×#889942
C2×#888142
C3×#639742
#105900763
#45269