Finite Group Databasebeta

← #13167495

Subgroups — #13167495

Order 1728.

486 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 53 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×#88942C211yesmaximal, characteristic, complemented
2C2×#88941C211yesmaximal, characteristic
3#11723685C211yesmaximal, characteristic, complemented
4#11723672C211yesmaximal, characteristic, complemented
5C3×C4⋊#622C211yesmaximal, characteristic, complemented
6#11722794C211yesmaximal, characteristic, complemented
7#11722793C211yesmaximal, characteristic, complemented
8#10593428—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×#6295C2211yescharacteristic
11C2×#6295C2211yescharacteristic
12C2×#6294—21no—
13C2×C3×#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88942—22no—
16#88941—22no—
17C3×#3752—21no—
18C3×#3752—21no—
19C3×C4×C3⋊Dic3C2211yescharacteristic
20#88815C2211yescharacteristic
21#88814—21no—
22C2×#3775—31no—
23C2×#3774—31no—
24#63915—31no—
25#63902—31no—
26#63880—61no—
27#63852—61no—
28C4⋊#622D311yescharacteristic, complemented
29#63740—31no—
30#63739—31no—
31C22×C3×C3⋊D3—21no—
32C22×C3×C3⋊D3—21no—
33C2×C33×C4D411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3C2311yescharacteristic
38#6296—21no—
39#6296—21no—
40#6295—21no—
41#6295—21no—
42#6295—21no—
43#6295D412yes—
44#6294—42no—
45C3×#622—21no—
46C3×#622—21no—
47#4844—91nomaximal
48C2×#627—31no—
49C2×#627—61no—
50C2×#626—61no—
51C2×#626—61no—
52C2×#625—31no—
53C2×#625—31no—
54C2×#624—61no—
55C2×D3×Dic3—61no—
56C2×D3×Dic3—61no—
57C2×#622C2×D311yescharacteristic
58C2×#622C2×D311yescharacteristic
59#3775—62no—
60#3774—62no—
61#3752—21no—
62#3752—21no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—61no—
66C3×(C2×D3)⋊C4—61no—
67C3×(C2×D3)⋊C4—61no—
68C3×(C2×D3)⋊C4—121no—
69C3×(C2×D3)⋊C4—121no—
70C3×C4×Dic3—61no—
71C3×C4×Dic3—61no—
72#3724—31no—
73Dic3⋊Dic3—61no—
74#3722—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—22no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3—41no—
88C3×C3⋊Dic3C2×D411yescharacteristic
89C3×C3⋊Dic3#4111yescharacteristic
90C2×D24—91no—
91C2×#204—91no—
92#885—181no—
93#67⋊C4—181no—
94#875—91no—
95C4×D12—91no—
96C3×C4⋊C8—91no—
97#821—91no—
98#816—91no—
99C22×C3⋊D3—21no—
100C22×C3⋊D3—21no—
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—61no—
105C22×C3×D3—121no—
106C22×C3×D3—121no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4D1211yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3D1211yescharacteristic
115C2×C3⋊Dic3C22×D311yescharacteristic
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—121no—
125C2×C3×Dic3—121no—
126#627—61no—
127#627—61no—
128#627—121no—
129#627—121no—
130#626—62no—
131#626—62no—
132#626—121no—
133#626—121no—
134#625—32no—
135#625—61no—
136#625—61no—
137#625—61no—
138#624—122no—
139D3×Dic3—122no—
140D3×Dic3—122no—
141#622—21no—
142#622—21no—
143#463—271noSylow
144C2×C33#12111yescharacteristic
145C2×C33#13511yescharacteristic
146C2×C33AGL(1,ℤ/8)11yescharacteristic
147C3×C3⋊D3—42no—
148C3×C3⋊D3—81no—
149C2×#67—181no—
150C2×#67—181no—
151C22×Dic3—181no—
152C2×D12—91no—
153C2×D12—91no—
154C2×C4×D3—181no—
155C2×C4×D3—181no—
156C2×Dic6—91no—
157C2×Dic6—181no—
158C2×C3×C8—91no—
159C2×C3×C8—91no—
160C3×C22⋊C4—181no—
161C3×C22⋊C4—181no—
162C3×C42—91no—
163#217—181no—
164(C2×D3)⋊C4—61no—
165(C2×D3)⋊C4—61no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—121no—
169(C2×D3)⋊C4—121no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172C4⋊Dic3—91no—
173Dic3⋊C4—181no—
174C4×Dic3—61no—
175C4×Dic3—61no—
176D24—182no—
177#204—182no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32C2×D1211yescharacteristic
183C2×C3⋊D3—22no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3×D3—62no—
190C2×C3×D3—62no—
191C2×C3×D3—122no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C2×C3×D3—121no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—122no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C2×C3×D3—121no—
203C2×C3×D3—241no—
204C2×C3×D3—241no—
205C2×C3×D3—241no—
206C2×C3×D3—241no—
207C32×C4—21no—
208C32×C4—41no—
209C32×C4—41no—
210C32×C4—81no—
211C32×C4—81no—
212C3⋊Dic3—21no—
213C3⋊Dic3—41no—
214C3⋊Dic3—41no—
215C3⋊Dic3D3×D411yescharacteristic
216C3⋊Dic3#23911yescharacteristic
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—62no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—241no—
228C3×Dic3—241no—
229C2×SD16—271no—
230C2×D8—271no—
231#124—271no—
232C4×D4—271no—
233C4⋊C8—91no—
234#103—271no—
235#102—271no—
236C33#46311yescharacteristic, complemented, Sylow
237C23×C3—181no—
238C23×C3—181no—
239C22×D3—61no—
240C22×D3—61no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—121no—
244C22×D3—121no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—91no—
250C2×C3×C4—91no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4GO+(4,𝔽2)11yescharacteristic
254#67—182no—
255#67—182no—
256#67—361no—
257#67—361no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—121no—
267C2×Dic3—121no—
268C2×Dic3—362no—
269D12—92no—
270D12—181no—
271D12—181no—
272D12—181no—
273C4×D3—362no—
274C4×D3—362no—
275Dic6—181no—
276Dic6—181no—
277Dic6—361no—
278Dic6—361no—
279C3×C8—181no—
280C3×C8—181no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32#89511yescharacteristic
294C2×C32#90411yescharacteristic
295C2×C32#90811yescharacteristic
296C3⋊D3—42no—
297C3⋊D3—81no—
298C3×D3—122no—
299C3×D3—122no—
300C3×D3—242no—
301C3×D3—241no—
302C3×D3—242no—
303C3×D3—241no—
304C2×Q8—271no—
305C2×D4—271no—
306C2×D4—271no—
307C22×C4—541no—
308SD16—542no—
309D8—542no—
310C2×C8—91no—
311C2×C8—91no—
312C4⋊C4—271no—
313C22⋊C4—181no—
314C22⋊C4—181no—
315C22⋊C4—541no—
316C42—91no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—182no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3—361no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3C2×GO+(4,𝔽2)11yescharacteristic
328C2×D3—62no—
329C2×D3—62no—
330C2×D3—122no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—122no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—241no—
343C2×D3—241no—
344C2×D3—241no—
345C2×D3—241no—
346C3×C4—21no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—81no—
350C3×C4—81no—
351C3×C4—91no—
352C3×C4—91no—
353C3×C4—181no—
354C3×C4—361no—
355C3×C4—361no—
356Dic3—61no—
357Dic3—61no—
358Dic3—61no—
359Dic3—61no—
360Dic3—62no—
361Dic3—121no—
362Dic3—121no—
363Dic3—121no—
364Dic3—121no—
365Dic3—121no—
366Dic3—241no—
367Dic3—241no—
368C32—21no—
369C32—21no—
370C32—41no—
371C32—41no—
372C32#484411yescharacteristic, complemented
373C23—181no—
374C23—181no—
375Q8—541no—
376Q8—541no—
377D4—272no—
378D4—541no—
379D4—541no—
380D4—541no—
381C2×C4—91no—
382C2×C4—91no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4—1082no—
386C2×C4#157⋊D311yescharacteristic
387C8—181no—
388C8—181no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—362no—
402C2×C3—721no—
403C2×C3#6422411yescharacteristic
404C2×C3#6423011yescharacteristic
405C2×C3#6423711yescharacteristic
406D3—122no—
407D3—122no—
408D3—242no—
409D3—241no—
410D3—242no—
411D3—241no—
412C22—182no—
413C22—361no—
414C22—361no—
415C22—361no—
416C22—361no—
417C22—361no—
418C22C2×#157⋊D311yescharacteristic
419C4—21no—
420C4—91no—
421C4—91no—
422C4—181no—
423C4—361no—
424C4—361no—
425C3—21no—
426C3—21no—
427C3—41no—
428C3—41no—
429C3#1059342811yescharacteristic, complemented
430C2—362no—
431C2—721no—
432C2#1172434011yescharacteristic
433C2#1172434311yescharacteristic
434C2#1172435111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#88941C2
hypercenterC2×C4#157⋊D3
hypocenterC33#463
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88942C2semidirect
3. #11723685C2semidirect
4. #11723672C2semidirect
5. C3×C4⋊#622C2semidirect
6. #11722794C2semidirect
7. #11722793C2semidirect
28. C4⋊#622D3semidirect
236. C33#463semidirect
372. C32#4844semidirect
429. C3#10593428semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
C2×#889412
#117236852
#117236722
C3×C4⋊#6222
#117227942
#117227932
#105934283
#48449