Finite Group Databasebeta

← #13167504

Subgroups — #13167504

Order 1728.

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

normal subgroups only

This group has 468 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×#88942C211yesmaximal, characteristic
2#11723685C211yesmaximal, characteristic
3#11723665C211yesmaximal, characteristic, complemented
4C3×C4×#622C211yesmaximal, characteristic, complemented
5#11722796C211yesmaximal, characteristic, complemented
6#11722794C211yesmaximal, characteristic, complemented
7#11722793C211yesmaximal, characteristic
8#10593433—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×#6295C2211yescharacteristic
11C2×#6294—21no—
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88942C414yescomplemented
15C3×#3752—21no—
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18#88815C2211yescharacteristic
19#88815C2211yescharacteristic
20#88813—21no—
21C2×#3775—31no—
22#63902—31no—
23#63880—61no—
24#63838—31no—
25#63837—61no—
26C4×#622D311yescharacteristic, complemented
27#63742—31no—
28#63740—31no—
29#63739—31no—
30C22×C3×C3⋊D3—21no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37#6296—21no—
38#6296C2×C412yescomplemented
39#6295—21no—
40#6295C2×C412yes—
41#6294—42no—
42C3×#622—21no—
43C3×#622C2×C412yescomplemented
44C4×#204—91nomaximal
45C2×#627—31no—
46C2×#627—61no—
47C2×#626—61no—
48C2×#625—31no—
49C2×#624—61no—
50C2×D3×Dic3—61no—
51C2×D3×Dic3—61no—
52C2×#622C2×D311yescharacteristic
53C2×#622C2×D311yescharacteristic
54#3775—34no—
55#3752—21no—
56#3750—21no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C3×(C2×D3)⋊C4—61no—
59C3×(C2×D3)⋊C4—61no—
60C3×(C2×D3)⋊C4—121no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C3×Dic3⋊C4—121no—
64C3×C4×Dic3—61no—
65C3×C4×Dic3—61no—
66#3724—31no—
67#3724—31no—
68Dic3⋊Dic3—61no—
69Dic3⋊Dic3—61no—
70Dic32—61no—
71Dic32—61no—
72Dic32—61no—
73C22×C33C2×D411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—41no—
77C33×C4—21no—
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3SD1612yescomplemented
82C3×C3⋊Dic3C22×C411yescharacteristic
83C3×C3⋊Dic3#4111yescharacteristic
84C2×#204—91no—
85#887—181no—
86#67⋊C4—181no—
87C4×D12—91no—
88C4×Dic6—91no—
89C3×C4×C8—91no—
90#821—91no—
91#817—91no—
92#816—91no—
93C22×C3⋊D3—21no—
94C22×C3×D3—61no—
95C22×C3×D3—61no—
96C22×C3×D3—121no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4D1211yescharacteristic
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3D1211yescharacteristic
106C2×C3⋊Dic3C22×D311yescharacteristic
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—121no—
118C2×C3×Dic3—121no—
119C2×C3×Dic3—121no—
120#627—32no—
121#627—62no—
122#627—62no—
123#627—61no—
124#626—62no—
125#626—62no—
126#625—32no—
127#625—61no—
128#624—122no—
129D3×Dic3—122no—
130D3×Dic3—122no—
131#622—21no—
132#622C4×D312yescomplemented
133C4×SD16—271noSylow
134C2×C33C4×D411yescharacteristic
135C2×C33C2×SD1611yescharacteristic
136C2×C33#13511yescharacteristic
137C3×C3⋊D3—42no—
138C2×#67—181no—
139C22×Dic3—181no—
140C2×D12—91no—
141C2×C4×D3—181no—
142C2×C4×D3—181no—
143C2×Dic6—91no—
144C2×Dic6—181no—
145C2×C3×C8—91no—
146C2×C3×C8—91no—
147C3×C4⋊C4—181no—
148C3×C22⋊C4—181no—
149C3×C42—91no—
150(C2×D3)⋊C4—61no—
151(C2×D3)⋊C4—61no—
152(C2×D3)⋊C4—121no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155Dic3⋊C4—61no—
156Dic3⋊C4—61no—
157Dic3⋊C4—121no—
158Dic3⋊C4—181no—
159Dic3⋊C4—181no—
160C4×Dic3—61no—
161C4×Dic3—61no—
162C4×Dic3—181no—
163C4×Dic3—181no—
164C4×Dic3—181no—
165#204—94no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32C2×D1211yescharacteristic
171C2×C3⋊D3—22no—
172C2×C3⋊D3—41no—
173C2×C3⋊D3—41no—
174C2×C3×D3—62no—
175C2×C3×D3—62no—
176C2×C3×D3—122no—
177C2×C3×D3—121no—
178C2×C3×D3—121no—
179C2×C3×D3—121no—
180C2×C3×D3—121no—
181C2×C3×D3—241no—
182C2×C3×D3—241no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4—41no—
186C32×C4—81no—
187C32×C4—81no—
188C3⋊Dic3—22no—
189C3⋊Dic3—41no—
190C3⋊Dic3—41no—
191C3⋊Dic3#20412yescomplemented
192C3⋊Dic3C2×C4×D311yescharacteristic
193C3⋊Dic3#23511yescharacteristic
194C3×Dic3—62no—
195C3×Dic3—62no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—61no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—122no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—241no—
208C3×Dic3—241no—
209C2×SD16—271no—
210C4×Q8—271no—
211C4×D4—271no—
212#106—271no—
213#103—271no—
214#102—271no—
215C4×C8—91no—
216C33C4×SD1611yescharacteristic, complemented, Sylow
217C23×C3—181no—
218C22×D3—61no—
219C22×D3—61no—
220C22×D3—121no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4GO+(4,𝔽2)11yescharacteristic
231#67—182no—
232#67—182no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—121no—
244C2×Dic3—121no—
245C2×Dic3—121no—
246C2×Dic3—362no—
247D12—92no—
248D12—181no—
249C4×D3—362no—
250C4×D3—362no—
251Dic6—92no—
252Dic6—182no—
253Dic6—181no—
254Dic6—182no—
255C3×C8—92no—
256C3×C8—181no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32C4×D1211yescharacteristic
270C2×C32C2×#20411yescharacteristic
271C2×C32#90411yescharacteristic
272C3⋊D3—42no—
273C3×D3—122no—
274C3×D3—122no—
275C3×D3—242no—
276C2×Q8—271no—
277C2×D4—271no—
278C22×C4—541no—
279SD16—274no—
280C2×C8—91no—
281C2×C8—91no—
282C4⋊C4—181no—
283C4⋊C4—271no—
284C4⋊C4—271no—
285C22⋊C4—181no—
286C42—91no—
287C42—541no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—182no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3C2×GO+(4,𝔽2)11yescharacteristic
296C2×D3—62no—
297C2×D3—62no—
298C2×D3—122no—
299C2×D3—121no—
300C2×D3—121no—
301C2×D3—121no—
302C2×D3—121no—
303C2×D3—241no—
304C2×D3—241no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—81no—
309C3×C4—81no—
310C3×C4—92no—
311C3×C4—91no—
312C3×C4—91no—
313C3×C4—182no—
314C3×C4—361no—
315C3×C4—361no—
316Dic3—61no—
317Dic3—61no—
318Dic3—62no—
319Dic3—61no—
320Dic3—61no—
321Dic3—62no—
322Dic3—62no—
323Dic3—62no—
324Dic3—122no—
325Dic3—121no—
326Dic3—121no—
327Dic3—121no—
328Dic3—121no—
329Dic3—241no—
330Dic3—241no—
331C32—21no—
332C32—21no—
333C32—41no—
334C32—41no—
335C32C4×#20411yescharacteristic, complemented
336C23—181no—
337Q8—272no—
338Q8—541no—
339D4—272no—
340D4—541no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—1082no—
347C2×C4#157⋊D311yescharacteristic
348C8—92no—
349C8—181no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—362no—
363C2×C3#6422711yescharacteristic
364C2×C3C4×GO+(4,𝔽2)11yescharacteristic
365C2×C3#6423711yescharacteristic
366D3—122no—
367D3—122no—
368D3—242no—
369C22—182no—
370C22—361no—
371C22—361no—
372C22C2×#157⋊D311yescharacteristic
373C4—21no—
374C4—92no—
375C4—91no—
376C4—91no—
377C4—182no—
378C4—361no—
379C4—361no—
380C3—21no—
381C3—21no—
382C3—41no—
383C3—41no—
384C3#1059343311yescharacteristic, complemented
385C2—362no—
386C2#1172433911yescharacteristic
387C2C4×#157⋊D311yescharacteristic
388C2#1172435111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#6295C22
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4#157⋊D3
hypocenterC33C4×SD16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723665C2semidirect
4. C3×C4×#622C2semidirect
5. #11722796C2semidirect
6. #11722794C2semidirect
14. #88942C4semidirect
26. C4×#622D3semidirect
38. #6296C2×C4semidirect
43. C3×#622C2×C4semidirect
81. C3×C3⋊Dic3SD16semidirect
132. #622C4×D3semidirect
191. C3⋊Dic3#204semidirect
216. C33C4×SD16semidirect
335. C32C4×#204semidirect
384. C3#10593433semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
#117236852
#117236652
C3×C4×#6222
#117227962
#117227942
#117227932
#105934333
C4×#2049