Finite Group Databasebeta

← #13165526

Subgroups — #13165526

Order 1728.

540 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 60 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×#88941C211yesmaximal, characteristic
2#11722938C211yesmaximal, characteristic, complemented
3#11722931C211yesmaximal, characteristic, complemented
4#11722772C211yesmaximal, characteristic, complemented
5#11722772C211yesmaximal, characteristic, complemented
6C3×#63742C211yesmaximal, characteristic, complemented
7#11722724C211yesmaximal, characteristic, complemented
8#10593419—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×#6295C2211yescharacteristic
11C2×Dic3×C3⋊D3—21no—
12C2×C3×#624—21no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88941C414yescomplemented
16#88803—21no—
17Dic3×C3⋊Dic3C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×#3724C2211yescharacteristic
20C3×#3722—21no—
21C2×#3774—31no—
22#63904—61no—
23#63902—31no—
24C4×#625—31no—
25#63881—61no—
26#63777—31no—
27#63742D311yescharacteristic, complemented
28#63739—31no—
29#63739—31no—
30C22×C3×C3⋊D3—21no—
31C22×C3×C3⋊D3—21no—
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3D411yescharacteristic
37#6295—21no—
38#6295—21no—
39#6295C2×C412yescomplemented
40#6295C2×C412yescomplemented
41Dic3×C3⋊D3—42no—
42C3×#624—42no—
43C3⋊#622—21no—
44C3×#622C2×C412yescomplemented
45#4869—91nomaximal
46C2×C4×C3⋊D3—61no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—61no—
49C2×#626—61no—
50C2×#626—61no—
51C2×#625—31no—
52C2×#625—31no—
53C2×#624—21no—
54C2×D3×Dic3—61no—
55C2×D3×Dic3—61no—
56C2×D3×Dic3—121no—
57C2×#622—31no—
58C2×#622C2×D311yescharacteristic
59#3774—34no—
60#3752—61no—
61C4×C3⋊Dic3—31no—
62C3×(C2×D3)⋊C4—61no—
63C3×(C2×D3)⋊C4—61no—
64C3×Dic3⋊C4—61no—
65C3×Dic3⋊C4—61no—
66#3724C2×D311yescharacteristic
67#3724C2×D311yescharacteristic
68#3722—21no—
69(C2×D3)⋊Dic3—61no—
70(C2×D3)⋊Dic3—61no—
71(C2×D3)⋊Dic3—121no—
72Dic32—61no—
73Dic32—61no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C3⋊(C3⋊Dic3)—21no—
82C3×C3⋊Dic3C22×C411yescharacteristic
83C3×C3⋊Dic3#4111yescharacteristic
84C32×Dic3—21no—
85C32×Dic3—41no—
86C32×Dic3—41no—
87C4×#67—181no—
88C2×D24—91no—
89#893—91no—
90#893—91no—
91#67⋊C4—181no—
92C3×#106—91no—
93#815—91no—
94#809—91no—
95#809—91no—
96C22×C3⋊D3—21no—
97C22×C3⋊D3—21no—
98C22×C3×D3—61no—
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C22×C3×D3—121no—
103C22×C3×D3—121no—
104C2×C32×C4—31no—
105C2×C32×C4—61no—
106C2×C32×C4—61no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—121no—
111C2×C3⋊Dic3—121no—
112C2×C3⋊Dic3C22×D311yescharacteristic
113C4×C3⋊D3—122no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C3×C4×D3—122no—
125C3×C4×D3—122no—
126#626—62no—
127#626—62no—
128#626—62no—
129#626—62no—
130#625—32no—
131#625—32no—
132#625—61no—
133#625—61no—
134#624—42no—
135D3×Dic3—122no—
136D3×Dic3—122no—
137D3×Dic3—124no—
138#622—61no—
139#622C4×D312yescomplemented
140#442—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33AGL(1,ℤ/8)11yescharacteristic
144C3×C3⋊D3—42no—
145C3×C3⋊D3—42no—
146C22×C3×C4—181no—
147C2×#67—181no—
148C2×#67—181no—
149C22×Dic3—181no—
150C2×D12—91no—
151C2×D12—91no—
152C2×C4×D3—61no—
153C2×C4×D3—61no—
154C2×C4×D3—181no—
155C2×C4×D3—181no—
156C2×C4×D3—361no—
157C2×C3×C8—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—91no—
160C3×C22⋊C4—181no—
161#217—181no—
162(C2×D3)⋊C4—61no—
163(C2×D3)⋊C4—61no—
164(C2×D3)⋊C4—181no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—361no—
167Dic3⋊C4—61no—
168Dic3⋊C4—61no—
169C4×Dic3—91no—
170C4×Dic3—181no—
171C4×Dic3—181no—
172C2×C3⋊C8—91no—
173D24—94no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32D3×D411yescharacteristic
179C2×C3⋊D3—22no—
180C2×C3⋊D3—22no—
181C2×C3⋊D3—41no—
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—62no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C2×C3×D3—62no—
189C2×C3×D3—122no—
190C2×C3×D3—122no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C2×C3×D3—121no—
195C2×C3×D3—122no—
196C2×C3×D3—122no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—241no—
202C2×C3×D3—241no—
203C32×C4—61no—
204C32×C4—121no—
205C32×C4—121no—
206C3⋊Dic3—62no—
207C3⋊Dic3—61no—
208C3⋊Dic3—121no—
209C3⋊Dic3—122no—
210C3⋊Dic3—241no—
211C3⋊Dic3C2×C4×D311yescharacteristic
212C3⋊Dic3#23911yescharacteristic
213C3×Dic3—22no—
214C3×Dic3—41no—
215C3×Dic3—41no—
216C3×Dic3—41no—
217C3×Dic3—42no—
218C3×Dic3—41no—
219C3×Dic3—41no—
220C3×Dic3—61no—
221C3×Dic3—61no—
222C3×Dic3—61no—
223C3×Dic3—61no—
224C3×Dic3—81no—
225C2×D8—271no—
226C4×D4—271no—
227C4×D4—271no—
228#106—91no—
229#102—271no—
230#102—271no—
231#97—271no—
232C33#44211yescharacteristic, complemented, Sylow
233C23×C3—181no—
234C23×C3—181no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—61no—
238C22×D3—61no—
239C22×D3—121no—
240C22×D3—121no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—91no—
248C2×C3×C4—121no—
249C2×C3×C4—121no—
250C2×C3×C4—362no—
251#67—182no—
252#67—182no—
253#67—182no—
254#67—182no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—91no—
260C2×Dic3—181no—
261C2×Dic3—181no—
262C2×Dic3—361no—
263C2×Dic3—361no—
264C2×Dic3—362no—
265C2×Dic3GO+(4,𝔽2)11yescharacteristic
266D12—92no—
267D12—92no—
268D12—181no—
269D12—181no—
270C4×D3—122no—
271C4×D3—122no—
272C4×D3—362no—
273C4×D3—362no—
274C4×D3—364no—
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#89311yescharacteristic
290C2×C32#91411yescharacteristic
291C2×C32#91411yescharacteristic
292C3⋊D3—42no—
293C3⋊D3—42no—
294C3×D3—122no—
295C3×D3—122no—
296C3×D3—124no—
297C3×D3—122no—
298C3×D3—122no—
299C3×D3—242no—
300C2×D4—271no—
301C2×D4—271no—
302C22×C4—181no—
303C22×C4—541no—
304D8—274no—
305C2×C8—91no—
306C2×C8—271no—
307C4⋊C4—91no—
308C4⋊C4—91no—
309C22⋊C4—181no—
310C22⋊C4—541no—
311C42—271no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—182no—
317C22×C3—182no—
318C22×C3—361no—
319C22×C3—361no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3C2×GO+(4,𝔽2)11yescharacteristic
323C2×D3—62no—
324C2×D3—62no—
325C2×D3—62no—
326C2×D3—62no—
327C2×D3—122no—
328C2×D3—122no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—122no—
334C2×D3—122no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—241no—
340C2×D3—241no—
341C3×C4—62no—
342C3×C4—91no—
343C3×C4—91no—
344C3×C4—121no—
345C3×C4—121no—
346C3×C4—121no—
347C3×C4—122no—
348C3×C4—121no—
349C3×C4—121no—
350C3×C4—241no—
351Dic3—21no—
352Dic3—41no—
353Dic3—41no—
354Dic3—61no—
355Dic3—61no—
356Dic3—61no—
357Dic3—61no—
358Dic3—182no—
359Dic3—181no—
360Dic3—361no—
361Dic3—362no—
362Dic3—721no—
363C32—21no—
364C32—21no—
365C32—41no—
366C32—41no—
367C32#486911yescharacteristic, complemented
368C23—181no—
369C23—181no—
370D4—272no—
371D4—272no—
372D4—541no—
373D4—541no—
374C2×C4—31no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—91no—
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—362no—
396C2×C3—362no—
397C2×C3#6422411yescharacteristic
398C2×C3C4×GO+(4,𝔽2)11yescharacteristic
399C2×C3#6423611yescharacteristic
400D3—122no—
401D3—122no—
402D3—124no—
403D3—122no—
404D3—122no—
405D3—242no—
406C22—182no—
407C22—182no—
408C22—361no—
409C22—361no—
410C22—361no—
411C22—361no—
412C22D3×GO+(4,𝔽2)11yescharacteristic
413C4—61no—
414C4—91no—
415C4—91no—
416C4—121no—
417C4—121no—
418C4—541no—
419C3—21no—
420C3—21no—
421C3—41no—
422C3—41no—
423C3#1059341911yescharacteristic, complemented
424C2—362no—
425C2—362no—
426C2#1172411111yescharacteristic
427C2#1172413911yescharacteristic
428C2#1172414111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC2×#88941C2
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. #11722938C2semidirect
3. #11722931C2semidirect
4. #11722772C2semidirect
5. #11722772C2semidirect
6. C3×#63742C2semidirect
7. #11722724C2semidirect
15. #88941C4semidirect
27. #63742D3semidirect
39. #6295C2×C4semidirect
40. #6295C2×C4semidirect
44. C3×#622C2×C4semidirect
139. #622C4×D3semidirect
232. C33#442semidirect
367. C32#4869semidirect
423. C3#10593419semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
#117229382
#117229312
#117227722
C3×#637422
#117227242
#105934193
#48699