Finite Group Databasebeta

← #13167587

Subgroups — #13167587

Order 1728.

924 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×#6294C211yesmaximal, characteristic
2C2×C3×C22⋊#157C211yesmaximal, characteristic
3C2×#88940C212yesmaximal, complemented
4#11723723C211yesmaximal, characteristic
5#11722791C212yesmaximal, complemented
6#10593485—31nomaximal
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3×#157C2212yes—
9C22×C3×C3⋊Dic3—21no—
10C2×#6294—21no—
11C2×#6294—24no—
12C2×#6294C2211yescharacteristic
13C2×#6294C2211yescharacteristic
14C2×#6294C2212yes—
15C3×C22⋊#157C414yescomplemented
16#88940—24no—
17C3×#3757—21no—
18#88814—21no—
19C22×#624—31no—
20C22×D3×Dic3—61no—
21C2×C22⋊#157D311yescharacteristic
22C2×#3773—32no—
23#63974—31no—
24D3×#217—61no—
25#63737—32no—
26C23×C33D411yescharacteristic
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3C2×C412yes—
30C22×C3×C3⋊D3D411yescharacteristic
31C22×C3×C3⋊D3Q812yes—
32C22×C3×C3⋊D3C2311yescharacteristic
33C2×C3×#157—22no—
34C2×C3×#157—22no—
35C2×C3×#157C2×C414yescomplemented
36C2×C3×C3⋊Dic3—24no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3—21no—
40#6294—24no—
41#6294—22no—
42#6294—22no—
43#6294—22no—
44#6294D414yes—
45#4950—91nomaximal
46C23×C3⋊D3C2×D311yescharacteristic
47C23×C3×D3—61no—
48C23×C3×D3—61no—
49C22×#157C2×D312yes—
50C22×C3⋊Dic3—21no—
51C22×C3×Dic3—61no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—121no—
54C2×#624—32no—
55C2×#624—31no—
56C2×#624—31no—
57C2×#624—64no—
58C2×#624—61no—
59C2×D3×Dic3—62no—
60C2×D3×Dic3—68no—
61C2×D3×Dic3—62no—
62C2×D3×Dic3—61no—
63C2×D3×Dic3—61no—
64C22⋊#157Dic314yescomplemented
65#3773—64no—
66#3757—21no—
67C3×#217—61no—
68C3×#217—61no—
69C3×#217—121no—
70#3722—61no—
71(C2×D3)⋊Dic3—61no—
72(C2×D3)⋊Dic3—61no—
73C22×C33—21no—
74C22×C33—21no—
75C22×C33C4⋊C412yescomplemented
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—24no—
80C2×C3×C3⋊D3—24no—
81C2×C3×C3⋊D3C4⋊C412yescomplemented
82C2×C3×C3⋊D3C22×C411yescharacteristic
83C2×C3×C3⋊D3C2×D412yes—
84C2×C3×C3⋊D3C2×Q811yescharacteristic
85C2×C3×C3⋊D3#4112yes—
86C3×#157—24no—
87C3×C3⋊Dic3—24no—
88C3×C3⋊Dic3—41no—
89C3×C3⋊Dic3—41no—
90C23×Dic3—91no—
91C22×C4×D3—181no—
92C2×C3×C22⋊C4—91no—
93C2×#217—91no—
94C2×C4⋊Dic3—92no—
95D3×C22⋊C4—181no—
96C3⋊#95—92no—
97C23×C32—21no—
98C23×C32—21no—
99C23×C32—41no—
100C23×C32—41no—
101C23×C32D1211yescharacteristic
102C22×C3⋊D3—22no—
103C22×C3⋊D3—22no—
104C22×C3⋊D3Dic612yes—
105C22×C3⋊D3D1211yescharacteristic
106C22×C3⋊D3C2×Dic312yes—
107C22×C3⋊D3C22×D311yescharacteristic
108C22×C3×D3—62no—
109C22×C3×D3—61no—
110C22×C3×D3—62no—
111C22×C3×D3—61no—
112C22×C3×D3—61no—
113C22×C3×D3—61no—
114C22×C3×D3—62no—
115C22×C3×D3—62no—
116C22×C3×D3—64no—
117C22×C3×D3—64no—
118C22×C3×D3—122no—
119C22×C3×D3—122no—
120C2×#157—22no—
121C2×#157—22no—
122C2×#157C2×Dic314yescomplemented
123C2×C3⋊Dic3—24no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—122no—
136C2×C3×Dic3—124no—
137C2×C3×Dic3—121no—
138C2×C3×Dic3—121no—
139#624—34no—
140#624—64no—
141#624—62no—
142#624—62no—
143#624—62no—
144D3×Dic3—64no—
145D3×Dic3—64no—
146D3×Dic3—68no—
147D3×Dic3—122no—
148D3×Dic3—122no—
149#385—271noSylow
150C2×C33—22no—
151C2×C33C2×C4⋊C411yescharacteristic
152C2×C33#12011yescharacteristic
153C2×C33#12311yescharacteristic
154C3×C3⋊D3—22no—
155C3×C3⋊D3C4×D412yes—
156C3×C3⋊D3C22⋊Q812yes—
157C24×C3—91no—
158C23×D3—61no—
159C23×D3—61no—
160C22×C3×C4—92no—
161C22×C3×C4—181no—
162C22×Dic3—61no—
163C22×Dic3—61no—
164C22×Dic3—92no—
165C22×Dic3—91no—
166C22×Dic3—91no—
167C22×Dic3—121no—
168C22×Dic3—184no—
169C22×Dic3—181no—
170C2×C4×D3—182no—
171C2×C4×D3—181no—
172C2×C4×D3—181no—
173C2×C4×D3—188no—
174C2×C4×D3—182no—
175C3×C22⋊C4—94no—
176C3×C22⋊C4—181no—
177#217—61no—
178#217—61no—
179#217—121no—
180#217—181no—
181(C2×D3)⋊C4—181no—
182(C2×D3)⋊C4—181no—
183C4⋊Dic3—184no—
184C22×C32—22no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—42no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—42no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—42no—
199C22×C32—42no—
200C22×C32—81no—
201C22×C32—81no—
202C22×C32C4⋊Dic312yescomplemented
203C22×C32C2×D1211yescharacteristic
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3—24no—
207C2×C3⋊D3—24no—
208C2×C3⋊D3C4⋊Dic312yescomplemented
209C2×C3⋊D3C2×Dic611yescharacteristic
210C2×C3⋊D3D3×D412yes—
211C2×C3⋊D3#23712yes—
212C2×C3⋊D3C22×Dic311yescharacteristic
213C2×C3×D3—62no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—62no—
217C2×C3×D3—62no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—62no—
223C2×C3×D3—62no—
224C2×C3×D3—68no—
225C2×C3×D3—68no—
226C2×C3×D3—62no—
227C2×C3×D3—124no—
228C2×C3×D3—124no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231#157—24no—
232C3⋊Dic3—24no—
233C3⋊Dic3—41no—
234C3⋊Dic3—41no—
235C3×Dic3—64no—
236C3×Dic3—64no—
237C3×Dic3—124no—
238C3×Dic3—121no—
239C3×Dic3—121no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—241no—
243C3×Dic3—241no—
244C23×C4—271no—
245C2×C4⋊C4—272no—
246C2×C22⋊C4—91no—
247C2×C22⋊C4—271no—
248#95—272no—
249C33#38511yescharacteristic, complemented, Sylow
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—91no—
255C23×C3—91no—
256C23×C3—92no—
257C23×C3—92no—
258C23×C3—182no—
259C23×C3—182no—
260C23×C3GO+(4,𝔽2)11yescharacteristic
261C22×D3—62no—
262C22×D3—61no—
263C22×D3—62no—
264C22×D3—61no—
265C22×D3—61no—
266C22×D3—62no—
267C22×D3—61no—
268C22×D3—62no—
269C22×D3—64no—
270C22×D3—64no—
271C22×D3—122no—
272C22×D3—122no—
273C2×C3×C4—94no—
274C2×C3×C4—184no—
275C2×C3×C4—182no—
276C2×C3×C4—182no—
277C2×C3×C4—182no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×Dic3—64no—
281C2×Dic3—62no—
282C2×Dic3—64no—
283C2×Dic3—62no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—94no—
289C2×Dic3—124no—
290C2×Dic3—122no—
291C2×Dic3—121no—
292C2×Dic3—121no—
293C2×Dic3—182no—
294C2×Dic3—184no—
295C2×Dic3—182no—
296C2×Dic3—182no—
297C4×D3—184no—
298C4×D3—184no—
299C4×D3—188no—
300C4×D3—362no—
301C4×D3—362no—
302C2×C32—24no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—42no—
317C2×C32—44no—
318C2×C32—82no—
319C2×C32#88211yescharacteristic
320C2×C32#88611yescharacteristic
321C2×C32C2×C4⋊Dic311yescharacteristic
322C3⋊D3—22no—
323C3⋊D3#87812yes—
324C3⋊D3D4×Dic312yes—
325C3×D3—62no—
326C3×D3—62no—
327C3×D3—62no—
328C3×D3—62no—
329C3×D3—64no—
330C3×D3—122no—
331C24—91no—
332C22×C4—92no—
333C22×C4—181no—
334C22×C4—272no—
335C22×C4—271no—
336C22×C4—271no—
337C22×C4—544no—
338C22×C4—541no—
339C4⋊C4—544no—
340C22⋊C4—94no—
341C22⋊C4—181no—
342C22⋊C4—541no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—92no—
362C22×C3—91no—
363C22×C3—91no—
364C22×C3—92no—
365C22×C3—92no—
366C22×C3—184no—
367C22×C3—184no—
368C22×C3—181no—
369C22×C3—181no—
370C22×C3#377312yescomplemented
371C22×C3C2×GO+(4,𝔽2)11yescharacteristic
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—62no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—62no—
379C2×D3—61no—
380C2×D3—61no—
381C2×D3—62no—
382C2×D3—62no—
383C2×D3—68no—
384C2×D3—68no—
385C2×D3—62no—
386C2×D3—124no—
387C2×D3—124no—
388C2×D3—121no—
389C2×D3—121no—
390C3×C4—184no—
391C3×C4—184no—
392C3×C4—361no—
393C3×C4—361no—
394Dic3—64no—
395Dic3—64no—
396Dic3—124no—
397Dic3—121no—
398Dic3—121no—
399Dic3—121no—
400Dic3—121no—
401Dic3—241no—
402Dic3—241no—
403C32—21no—
404C32—21no—
405C32—41no—
406C32—41no—
407C32#495011yescharacteristic, complemented
408C23—91no—
409C23—91no—
410C23—92no—
411C23—92no—
412C23—182no—
413C23—182no—
414C23#157⋊D311yescharacteristic
415C2×C4—94no—
416C2×C4—184no—
417C2×C4—182no—
418C2×C4—182no—
419C2×C4—182no—
420C2×C4—181no—
421C2×C4—181no—
422C2×C4—274no—
423C2×C4—542no—
424C2×C4—544no—
425C2×C4—542no—
426C2×C4—542no—
427C2×C3—24no—
428C2×C3—22no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—42no—
436C2×C3—44no—
437C2×C3—41no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3—82no—
444C2×C3—92no—
445C2×C3—92no—
446C2×C3—182no—
447C2×C3#6423311yescharacteristic
448C2×C3C2×#377311yescharacteristic
449C2×C3#6424111yescharacteristic
450D3—62no—
451D3—62no—
452D3—62no—
453D3—62no—
454D3—64no—
455D3—122no—
456C22—21no—
457C22—21no—
458C22—92no—
459C22—91no—
460C22—91no—
461C22—92no—
462C22—92no—
463C22—184no—
464C22—184no—
465C22—181no—
466C22—181no—
467C22#8894012yescomplemented
468C22C2×#157⋊D311yescharacteristic
469C4—184no—
470C4—184no—
471C4—361no—
472C4—361no—
473C3—21no—
474C3—21no—
475C3—41no—
476C3—41no—
477C3#1059348511yescharacteristic, complemented
478C2—22no—
479C2—92no—
480C2—92no—
481C2—182no—
482C2#1172434611yescharacteristic
483C2C2×#8894011yescharacteristic
484C2#1172435411yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC23#157⋊D3
hypocenterC33#385
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
3. C2×#88940C2semidirect
5. #11722791C2semidirect
15. C3×C22⋊#157C4semidirect
35. C2×C3×#157C2×C4semidirect
64. C22⋊#157Dic3semidirect
75. C22×C33C4⋊C4semidirect
81. C2×C3×C3⋊D3C4⋊C4semidirect
122. C2×#157C2×Dic3semidirect
202. C22×C32C4⋊Dic3semidirect
208. C2×C3⋊D3C4⋊Dic3semidirect
249. C33#385semidirect
370. C22×C3#3773semidirect
407. C32#4950semidirect
467. C22#88940semidirect
477. C3#10593485semidirect

Maximal subgroups

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

subgroupindex
C22×#62942
C2×C3×C22⋊#1572
C2×#889402
#117237232
#117227912
#105934853
#49509