Finite Group Databasebeta

← #13165523

Subgroups — #13165523

Order 1728.

540 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 68 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×C3×#3774C211yesmaximal, characteristic
2#11722925C211yesmaximal, characteristic, complemented
3Dic3×#625C211yesmaximal, characteristic, complemented
4#11722796C211yesmaximal, characteristic, complemented
5#11722773C211yesmaximal, characteristic, complemented
6#11722773C211yesmaximal, characteristic, complemented
7#11722724C211yesmaximal, characteristic, complemented
8#10593419—31nomaximal
9C2×D3×C3⋊Dic3—21no—
10C2×C3×#625C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C2×C3×D3×Dic3—21no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C3×#3774C414yescomplemented
16#88815C2211yescharacteristic
17#88815C2211yescharacteristic
18(C2×D3)⋊(C3⋊Dic3)—21no—
19Dic3×C3⋊Dic3C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C2×#3774D311yescharacteristic
22#67⋊Dic3—61no—
23Dic3×#67—61no—
24#63902—31no—
25C4×#625—31no—
26#63777—31no—
27#63742—31no—
28#63739—31no—
29#63739—31no—
30C22×C32×D3—21no—
31C22×C32×D3—21no—
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3D411yescharacteristic
37D3×C3⋊Dic3—42no—
38C3×#625—21no—
39C3×#625—21no—
40C3×#625C2×C412yescomplemented
41C3×#625C2×C412yescomplemented
42C3×D3×Dic3—42no—
43C3⋊#622—21no—
44C3×#622C2×C412yescomplemented
45#5152—91nomaximal
46C22×C3⋊Dic3—61no—
47C2×C3×#67—61no—
48C2×C3×#67—61no—
49C22×C3×Dic3—61no—
50C2×C3×C4×D3—61no—
51C2×#625C2×D311yescharacteristic
52C2×#625C2×D311yescharacteristic
53C2×D3×Dic3—21no—
54C2×D3×Dic3—61no—
55C2×D3×Dic3—61no—
56C2×D3×Dic3—121no—
57C2×#622—31no—
58C2×#622C2×D311yescharacteristic
59#3774Dic314yescomplemented
60#3757—61no—
61C4×C3⋊Dic3—31no—
62C3×#217—61no—
63C3×(C2×D3)⋊C4—61no—
64#3724—31no—
65#3724—31no—
66Dic3⋊Dic3—61no—
67Dic3⋊Dic3—61no—
68(C2×D3)⋊Dic3—21no—
69(C2×D3)⋊Dic3—61no—
70(C2×D3)⋊Dic3—61no—
71(C2×D3)⋊Dic3—121no—
72Dic32—61no—
73Dic32—61no—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—22no—
76C2×C32×D3—22no—
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C3⋊(C3⋊Dic3)—21no—
82C3×C3⋊Dic3—41no—
83C3×C3⋊Dic3—41no—
84C3×C3⋊Dic3C22×C411yescharacteristic
85C3×C3⋊Dic3#4111yescharacteristic
86C32×Dic3—21no—
87C2×C3×D8—91no—
88D4×Dic3—91no—
89D4×Dic3—91no—
90C4×#67—181no—
91#67⋊C4—181no—
92#832—91no—
93#832—91no—
94#817—91no—
95#815—91no—
96C23×C32—61no—
97C23×C32—61no—
98C22×C3×D3—21no—
99C22×C3×D3—21no—
100C22×C3×D3—21no—
101C22×C3×D3—21no—
102C22×C3×D3—41no—
103C22×C3×D3—41no—
104C2×C32×C4—31no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—121no—
111C2×C3⋊Dic3—121no—
112C2×C3⋊Dic3—122no—
113C2×C3⋊Dic3C22×D311yescharacteristic
114C3×#67—62no—
115C3×#67—62no—
116C3×#67—62no—
117C3×#67—62no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—121no—
127C2×C3×Dic3—121no—
128C2×C3×Dic3—122no—
129C3×C4×D3—122no—
130#625—21no—
131#625—21no—
132#625C2×Dic312yescomplemented
133#625C2×Dic312yescomplemented
134D3×Dic3—42no—
135D3×Dic3—122no—
136D3×Dic3—122no—
137D3×Dic3—124no—
138#622—61no—
139#622C2×Dic312yescomplemented
140#442—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33AGL(1,ℤ/8)11yescharacteristic
144C32×D3—42no—
145C32×D3—42no—
146C2×C3×D4—91no—
147C2×C3×D4—91no—
148C22×C3×C4—181no—
149C2×#67—61no—
150C2×#67—61no—
151C22×Dic3—61no—
152C22×Dic3—181no—
153C22×Dic3—181no—
154C22×Dic3—361no—
155C2×C4×D3—61no—
156C2×C4×D3—181no—
157C3×D8—94no—
158C2×C3×C8—91no—
159C3×C22⋊C4—181no—
160#217—61no—
161#217—181no—
162#217—181no—
163#217—361no—
164(C2×D3)⋊C4—61no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—91no—
167C4⋊Dic3—91no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170C4×Dic3—91no—
171C4×Dic3—181no—
172C4×Dic3—181no—
173C2×C3⋊C8—91no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—62no—
179C22×C32—62no—
180C22×C32—121no—
181C22×C32—121no—
182C22×C32—121no—
183C22×C32—121no—
184C22×C32D3×D411yescharacteristic
185C2×C3×D3—22no—
186C2×C3×D3—22no—
187C2×C3×D3—22no—
188C2×C3×D3—22no—
189C2×C3×D3—42no—
190C2×C3×D3—42no—
191C2×C3×D3—41no—
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—42no—
196C2×C3×D3—42no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C2×C3×D3—41no—
201C2×C3×D3—81no—
202C2×C3×D3—81no—
203C32×C4—61no—
204C3⋊Dic3—41no—
205C3⋊Dic3—41no—
206C3⋊Dic3—62no—
207C3⋊Dic3—61no—
208C3⋊Dic3—121no—
209C3⋊Dic3—122no—
210C3⋊Dic3—241no—
211C3⋊Dic3#23711yescharacteristic
212C3⋊Dic3C22×Dic311yescharacteristic
213C3×Dic3—22no—
214C3×Dic3—41no—
215C3×Dic3—61no—
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C3×Dic3—122no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C3×Dic3—241no—
225C2×D8—91no—
226C4×D4—271no—
227C4×D4—271no—
228#106—271no—
229#102—271no—
230#102—271no—
231#97—271no—
232C33#44211yescharacteristic, complemented, Sylow
233C23×C3—61no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—61no—
237C23×C3—121no—
238C23×C3—121no—
239C22×D3—21no—
240C22×D3—21no—
241C3×D4—92no—
242C3×D4—92no—
243C3×D4—181no—
244C3×D4—181no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—91no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—362no—
251#67—62no—
252#67—62no—
253#67—62no—
254#67—62no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—91no—
262C2×Dic3—121no—
263C2×Dic3—121no—
264C2×Dic3—122no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267C2×Dic3—361no—
268C2×Dic3—361no—
269C2×Dic3—362no—
270C2×Dic3—362no—
271C2×Dic3—364no—
272C2×Dic3GO+(4,𝔽2)11yescharacteristic
273C4×D3—122no—
274C4×D3—362no—
275C3×C8—92no—
276C3⋊C8—181no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—122no—
290C2×C32—122no—
291C2×C32#91111yescharacteristic
292C2×C32#91111yescharacteristic
293C2×C32D4×Dic311yescharacteristic
294C3×D3—42no—
295C3×D3—42no—
296C3×D3—44no—
297C3×D3—42no—
298C3×D3—42no—
299C3×D3—82no—
300C2×D4—91no—
301C2×D4—91no—
302C22×C4—181no—
303C22×C4—541no—
304D8—94no—
305C2×C8—91no—
306C2×C8—271no—
307C4⋊C4—271no—
308C4⋊C4—271no—
309C22⋊C4—181no—
310C22⋊C4—541no—
311C42—271no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—62no—
317C22×C3—62no—
318C22×C3—62no—
319C22×C3—62no—
320C22×C3—122no—
321C22×C3—122no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—122no—
327C22×C3—122no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3—241no—
333C22×C3—241no—
334C22×C3C2×GO+(4,𝔽2)11yescharacteristic
335C2×D3—22no—
336C2×D3—22no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—41no—
341C3×C4—62no—
342C3×C4—91no—
343C3×C4—91no—
344C3×C4—121no—
345C3×C4—361no—
346C3×C4—361no—
347Dic3—21no—
348Dic3—61no—
349Dic3—61no—
350Dic3—61no—
351Dic3—61no—
352Dic3—121no—
353Dic3—121no—
354Dic3—122no—
355Dic3—121no—
356Dic3—121no—
357Dic3—182no—
358Dic3—181no—
359Dic3—241no—
360Dic3—361no—
361Dic3—362no—
362Dic3—721no—
363C32—21no—
364C32—21no—
365C32—41no—
366C32—41no—
367C32#515211yescharacteristic, complemented
368C23—61no—
369C23—61no—
370D4—92no—
371D4—92no—
372D4—181no—
373D4—181no—
374C2×C4—31no—
375C2×C4—91no—
376C2×C4—181no—
377C2×C4—181no—
378C2×C4—271no—
379C2×C4—362no—
380C2×C4—1082no—
381C8—92no—
382C8—541no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—122no—
396C2×C3—122no—
397C2×C3—124no—
398C2×C3—122no—
399C2×C3—122no—
400C2×C3—242no—
401C2×C3#6422411yescharacteristic
402C2×C3C4×GO+(4,𝔽2)11yescharacteristic
403C2×C3#6423611yescharacteristic
404D3—42no—
405D3—42no—
406C22—62no—
407C22—62no—
408C22—121no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22D3×GO+(4,𝔽2)11yescharacteristic
413C4—61no—
414C4—91no—
415C4—91no—
416C4—361no—
417C4—361no—
418C4—541no—
419C3—21no—
420C3—21no—
421C3—41no—
422C3—41no—
423C3#1059341911yescharacteristic, complemented
424C2—122no—
425C2—122no—
426C2Dic3×GO+(4,𝔽2)11yescharacteristic
427C2#1172413811yescharacteristic
428C2#1172414011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC2×C3×#3774C2
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33#442
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #11722925C2semidirect
3. Dic3×#625C2semidirect
4. #11722796C2semidirect
5. #11722773C2semidirect
6. #11722773C2semidirect
7. #11722724C2semidirect
15. C3×#3774C4semidirect
40. C3×#625C2×C4semidirect
41. C3×#625C2×C4semidirect
44. C3×#622C2×C4semidirect
59. #3774Dic3semidirect
132. #625C2×Dic3semidirect
133. #625C2×Dic3semidirect
139. #622C2×Dic3semidirect
232. C33#442semidirect
367. C32#5152semidirect
423. C3#10593419semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37742
#117229252
Dic3×#6252
#117227962
#117227732
#117227242
#105934193
#51529