Finite Group Databasebeta

← #11722949

Subgroups — #11722949

Order 864.

774 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 84 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×D3×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#6291C211yesmaximal, characteristic, complemented
3C2×C3×#626C211yesmaximal, characteristic, complemented
4C2×C3×#625C211yesmaximal, characteristic, complemented
5#88814C211yesmaximal, characteristic, complemented
6#88804C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
8#63983—31nomaximal
9#63981—31nomaximal
10#63922—31nomaximal
11C22×C3⋊(C3⋊D3)C2211yescharacteristic
12C22×C3×C3⋊D3C2211yescharacteristic
13C22×C32×D3C2211yescharacteristic
14C22×C32×D3C2211yescharacteristic, complemented
15C2×D3×C3⋊D3—22no—
16C2×D3×C3⋊D3—22no—
17C2×D3×C3⋊D3—22no—
18C2×C3×C3⋊Dic3C2211yescharacteristic, complemented
19C2×C3×C3⋊Dic3C2211yescharacteristic, complemented
20C2×C32×Dic3C2211yescharacteristic, complemented
21#6291—22no—
22C3×#626—22no—
23C3×#625—22no—
24C23×C3⋊D3—31no—
25C22×D32—31no—
26C22×D32—31no—
27C22×D32—61no—
28C2×#638—31no—
29C2×C3×#67—31no—
30C2×C3×#67—31no—
31C2×C3×#67—31no—
32C2×C3×D12—31no—
33C2×#626—31no—
34C2×#626—31no—
35C2×#626—61no—
36C2×#626D311yescharacteristic, complemented
37C2×#625D311yescharacteristic, complemented
38#3752—31no—
39C3×#217—31no—
40C3×(C2×D3)⋊C4—31no—
41#3722—31no—
42#3722—31no—
43#3722—31no—
44#3722—61no—
45(C2×D3)⋊Dic3—31no—
46(C2×D3)⋊Dic3—31no—
47(C2×D3)⋊Dic3D311yescharacteristic, complemented
48C22×C33C2311yescharacteristic
49C2×C3⋊(C3⋊D3)—21no—
50C2×C3⋊(C3⋊D3)—21no—
51C2×C3⋊(C3⋊D3)D412yescomplemented
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3D412yescomplemented
55C2×C32×D3—21no—
56C2×C32×D3—21no—
57C2×C32×D3—21no—
58C2×C32×D3—21no—
59C2×C32×D3—21no—
60C2×C32×D3D412yescomplemented
61D3×C3⋊D3—28no—
62C3×C3⋊Dic3—21no—
63C3×C3⋊Dic3—21no—
64C32×Dic3—21no—
65#937—91no—
66#937—91no—
67#882—91no—
68C23×C32—31no—
69C23×C32—31no—
70C22×C3⋊D3—31no—
71C22×C3⋊D3—31no—
72C22×C3⋊D3—31no—
73C22×C3⋊D3—61no—
74C22×C3⋊D3—61no—
75C22×C3⋊D3—61no—
76C22×C3⋊D3—62no—
77C22×C3⋊D3—62no—
78C22×C3⋊D3—62no—
79C22×C3⋊D3—121no—
80C22×C3⋊D3C2×D311yescharacteristic
81C22×C3×D3—21no—
82C22×C3×D3—21no—
83C22×C3×D3—31no—
84C22×C3×D3—31no—
85C22×C3×D3—61no—
86C22×C3×D3C2×D311yescharacteristic
87C22×C3×D3C2×D311yescharacteristic
88C22×C3×D3C2×D311yescharacteristic, complemented
89C22×C3×D3C2×D311yescharacteristic, complemented
90C2×D32—62no—
91C2×D32—62no—
92C2×D32—62no—
93C2×D32—62no—
94C2×D32—64no—
95C2×D32—64no—
96C2×D32—64no—
97C2×D32—62no—
98C2×D32—62no—
99C2×C32×C4—31no—
100#638—62no—
101C2×C3⋊Dic3C2×D311yescharacteristic, complemented
102C2×C3⋊Dic3C2×D311yescharacteristic, complemented
103C3×#67—62no—
104C3×#67—62no—
105C3×#67—62no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—31no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3C2×D311yescharacteristic, complemented
114C2×C3×Dic3C2×D311yescharacteristic, complemented
115C3×D12—62no—
116#626—22no—
117#626—62no—
118#626—62no—
119#626—64no—
120#625—22no—
121C2×C33C2×D411yescharacteristic
122C2×C33C2×D411yescharacteristic
123C2×C33C2×D411yescharacteristic
124C3⋊(C3⋊D3)—22no—
125C3×C3⋊D3—22no—
126C32×D3—22no—
127C32×D3—41no—
128C23×D3—91no—
129C23×D3—91no—
130C23×D3—91no—
131C23×D3—181no—
132C2×C3×D4—91no—
133C2×C3×D4—91no—
134C2×#67—31no—
135C2×#67—31no—
136C2×#67—31no—
137C2×#67—91no—
138C2×#67—91no—
139C2×#67—181no—
140C2×D12—31no—
141C2×D12—91no—
142C3×C22⋊C4—91no—
143#217—31no—
144#217—91no—
145(C2×D3)⋊C4—31no—
146(C2×D3)⋊C4—91no—
147(C2×D3)⋊C4—91no—
148(C2×D3)⋊C4—91no—
149(C2×D3)⋊C4—91no—
150(C2×D3)⋊C4—91no—
151(C2×D3)⋊C4—181no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—32no—
156C22×C32—41no—
157C22×C32—61no—
158C22×C32—61no—
159C22×C32—61no—
160C22×C32—61no—
161C22×C32—61no—
162C22×C32C22×D311yescharacteristic
163C22×C32C22×D311yescharacteristic
164C22×C32C22×D311yescharacteristic
165C2×C3⋊D3—21no—
166C2×C3⋊D3—21no—
167C2×C3⋊D3—32no—
168C2×C3⋊D3—32no—
169C2×C3⋊D3—32no—
170C2×C3⋊D3—62no—
171C2×C3⋊D3—62no—
172C2×C3⋊D3—62no—
173C2×C3⋊D3—68no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—62no—
177C2×C3⋊D3—62no—
178C2×C3⋊D3—61no—
179C2×C3⋊D3—61no—
180C2×C3⋊D3—61no—
181C2×C3⋊D3—61no—
182C2×C3⋊D3—122no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—121no—
185C2×C3⋊D3—121no—
186C2×C3⋊D3—121no—
187C2×C3⋊D3—122no—
188C2×C3⋊D3—122no—
189C2×C3⋊D3#6712yescomplemented
190C2×C3×D3—22no—
191C2×C3×D3—22no—
192C2×C3×D3—21no—
193C2×C3×D3—21no—
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C2×C3×D3—22no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—21no—
202C2×C3×D3—21no—
203C2×C3×D3—22no—
204C2×C3×D3—32no—
205C2×C3×D3—32no—
206C2×C3×D3—41no—
207C2×C3×D3—41no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—62no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3D1212yescomplemented
216C2×C3×D3#6712yescomplemented
217D32—616no—
218D32—68no—
219D32—68no—
220C32×C4—61no—
221C3⋊Dic3—21no—
222C3⋊Dic3—21no—
223C3×Dic3—21no—
224C3×Dic3—21no—
225C3×Dic3—41no—
226C3×Dic3—62no—
227C3×Dic3—61no—
228C3×Dic3—61no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—121no—
232#120—271noSylow
233C33#12011yescharacteristic, complemented, Sylow
234C23×C3—31no—
235C23×C3—31no—
236C23×C3—31no—
237C23×C3—31no—
238C23×C3—61no—
239C23×C3—61no—
240C23×C3—91no—
241C22×D3—31no—
242C22×D3—31no—
243C22×D3—61no—
244C22×D3—91no—
245C22×D3—91no—
246C22×D3—91no—
247C22×D3—181no—
248C22×D3—181no—
249C22×D3—181no—
250C22×D3—182no—
251C22×D3—182no—
252C22×D3—182no—
253C22×D3—182no—
254C22×D3—182no—
255C22×D3—182no—
256C22×D3—182no—
257C22×D3—182no—
258C22×D3—184no—
259C22×D3—184no—
260C22×D3—184no—
261C22×D3—182no—
262C22×D3—361no—
263C22×D3D3211yescharacteristic
264C22×D3D3211yescharacteristic, complemented
265C3×D4—182no—
266C3×D4—182no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272#67—62no—
273#67—62no—
274#67—62no—
275#67—182no—
276#67—182no—
277#67—184no—
278C2×Dic3—31no—
279C2×Dic3—31no—
280C2×Dic3—31no—
281C2×Dic3—31no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3D3211yescharacteristic, complemented
285D12—62no—
286D12—182no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32—62no—
300C2×C32—121no—
301C2×C32C2×D1211yescharacteristic
302C2×C32D3×D411yescharacteristic
303C2×C32D3×D411yescharacteristic
304C2×C32D3×D411yescharacteristic
305C2×C32D3×D411yescharacteristic
306C2×C32D3×D411yescharacteristic
307C2×C32D3×D411yescharacteristic
308C2×C32C2×#6711yescharacteristic
309C2×C32C2×#6711yescharacteristic
310C3⋊D3—22no—
311C3⋊D3—62no—
312C3⋊D3—64no—
313C3⋊D3—62no—
314C3⋊D3—62no—
315C3⋊D3—122no—
316C3⋊D3—122no—
317C3⋊D3—124no—
318C3×D3—24no—
319C3×D3—22no—
320C3×D3—22no—
321C3×D3—42no—
322C3×D3—41no—
323C3×D3—41no—
324C3×D3—64no—
325C3×D3—62no—
326C3×D3—62no—
327C24—271no—
328C2×D4—91no—
329C2×D4—91no—
330C2×D4—271no—
331C22⋊C4—91no—
332C22⋊C4—271no—
333C22⋊C4—271no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—32no—
338C22×C3—32no—
339C22×C3—41no—
340C22×C3—62no—
341C22×C3—62no—
342C22×C3—62no—
343C22×C3—62no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—92no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—181no—
358C22×C3—181no—
359C22×C3C2×D3211yescharacteristic
360C22×C3C2×D3211yescharacteristic
361C22×C3C2×D3211yescharacteristic
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—21no—
367C2×D3—32no—
368C2×D3—32no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—62no—
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—61no—
376C2×D3—92no—
377C2×D3—92no—
378C2×D3—92no—
379C2×D3—182no—
380C2×D3—182no—
381C2×D3—182no—
382C2×D3—188no—
383C2×D3—1816no—
384C2×D3—188no—
385C2×D3—188no—
386C2×D3—182no—
387C2×D3—182no—
388C2×D3—181no—
389C2×D3—181no—
390C2×D3—181no—
391C2×D3—181no—
392C2×D3—181no—
393C2×D3—181no—
394C2×D3—362no—
395C2×D3—361no—
396C2×D3—361no—
397C2×D3—361no—
398C2×D3—361no—
399C2×D3—362no—
400C2×D3—362no—
401C2×D3#62612yescomplemented
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—121no—
405C3×C4—181no—
406C3×C4—181no—
407Dic3—21no—
408Dic3—62no—
409Dic3—61no—
410Dic3—61no—
411Dic3—61no—
412Dic3—61no—
413Dic3—121no—
414C32—21no—
415C32—21no—
416C32—21no—
417C32—41no—
418C32#88211yescharacteristic, complemented
419C32#93711yescharacteristic, complemented
420C32#93711yescharacteristic, complemented
421C23—31no—
422C23—31no—
423C23—91no—
424C23—271no—
425C23—542no—
426C23—542no—
427C23—542no—
428D4—182no—
429D4—182no—
430D4—542no—
431C2×C4—31no—
432C2×C4—91no—
433C2×C4—91no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—41no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—64no—
447C2×C3—62no—
448C2×C3—62no—
449C2×C3—122no—
450C2×C3—121no—
451C2×C3—121no—
452C2×C3—182no—
453C2×C3D3×D1211yescharacteristic
454C2×C3#380211yescharacteristic
455C2×C3C2×#62611yescharacteristic
456C2×C3D3×#6711yescharacteristic
457C2×C3D3×#6711yescharacteristic
458C2×C3D3×#6711yescharacteristic
459C2×C3#381111yescharacteristic
460C2×C3#381111yescharacteristic
461C2×C3#381111yescharacteristic
462D3—22no—
463D3—41no—
464D3—64no—
465D3—62no—
466D3—62no—
467D3—184no—
468D3—182no—
469D3—182no—
470D3—182no—
471D3—362no—
472D3—362no—
473D3—364no—
474C22—32no—
475C22—61no—
476C22—61no—
477C22—61no—
478C22—61no—
479C22—61no—
480C22—92no—
481C22—181no—
482C22—181no—
483C22—272no—
484C22—548no—
485C22—541no—
486C22—541no—
487C22D3311yescharacteristic
488C4—61no—
489C4—181no—
490C4—181no—
491C3—21no—
492C3—21no—
493C3—21no—
494C3—41no—
495C3#6392211yescharacteristic, complemented
496C3#6398111yescharacteristic, complemented
497C3#6398311yescharacteristic, complemented
498C2—62no—
499C2—121no—
500C2—182no—
501C2—542no—
502C2D3×#62611yescharacteristic
503C2#8895211yescharacteristic
504C2#8895411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
hypocenterC33#120
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×D3×C3⋊D3C2semidirect
2. C2×#6291C2semidirect
3. C2×C3×#626C2semidirect
4. C2×C3×#625C2semidirect
5. #88814C2semidirect
6. #88804C2semidirect
7. C3×(C2×D3)⋊Dic3C2semidirect
14. C22×C32×D3C22semidirect
18. C2×C3×C3⋊Dic3C22semidirect
19. C2×C3×C3⋊Dic3C22semidirect
20. C2×C32×Dic3C22semidirect
36. C2×#626D3semidirect
37. C2×#625D3semidirect
47. (C2×D3)⋊Dic3D3semidirect
51. C2×C3⋊(C3⋊D3)D4semidirect
54. C2×C3×C3⋊D3D4semidirect
60. C2×C32×D3D4semidirect
88. C22×C3×D3C2×D3semidirect
89. C22×C3×D3C2×D3semidirect
101. C2×C3⋊Dic3C2×D3semidirect
102. C2×C3⋊Dic3C2×D3semidirect
113. C2×C3×Dic3C2×D3semidirect
114. C2×C3×Dic3C2×D3semidirect
189. C2×C3⋊D3#67semidirect
215. C2×C3×D3D12semidirect
216. C2×C3×D3#67semidirect
233. C33#120semidirect
264. C22×D3D32semidirect
284. C2×Dic3D32semidirect
401. C2×D3#626semidirect
418. C32#882semidirect
419. C32#937semidirect
420. C32#937semidirect
495. C3#63922semidirect
496. C3#63981semidirect
497. C3#63983semidirect

Maximal subgroups

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

subgroupindex
C22×D3×C3⋊D32
C2×#62912
C2×C3×#6262
C2×C3×#6252
#888142
#888042
C3×(C2×D3)⋊Dic32
#639833
#639813
#639223