Finite Group Databasebeta

← #10596119⋊C3 #13153036

Subgroups — #10596119⋊C3 #13153036

Order 1728.

1023 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3830⋊C3C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#123C211yesmaximal, characteristic, complemented
4#64144⋊C3C211yesmaximal, characteristic, complemented
5#64138⋊C3C211yesmaximal, characteristic, complemented
6#64105⋊C3C212yesmaximal, complemented
7#64097⋊C3C212yesmaximal, complemented
8#64095⋊C3C211yesmaximal, characteristic, complemented
9#64094⋊C3C212yesmaximal, complemented
10#64087⋊C3C212yesmaximal, complemented
11#64086⋊C3C211yesmaximal, characteristic, complemented
12#10596119C311yesmaximal, characteristic, complemented
13C3×#5648—31nomaximal
14C2×D4×H(3,𝔽3)C2211yescharacteristic
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C2×#638⋊C3C2211yescharacteristic
17C2×#638⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
21#3830⋊C3—24no—
22C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
23C2×#634⋊C3—24no—
24C2×#634⋊C3C2212yes—
25C2×#634⋊C3C2212yes—
26C2×#634⋊C3C2211yescharacteristic
27C4⋊C4×H(3,𝔽3)C2212yes—
28C22⋊C4×H(3,𝔽3)C2211yescharacteristic
29C22⋊C4×H(3,𝔽3)C2212yes—
30#3757⋊C3C2211yescharacteristic
31#3757⋊C3C2212yes—
32#3752⋊C3C2212yes—
33#3752⋊C3C2212yes—
34C4⋊((C3⋊Dic3)⋊C3)C2212yes—
35#3750⋊C3C2212yes—
36#3750⋊C3C2212yes—
37#3750⋊C3C2212yes—
38C4×(C3⋊Dic3)⋊C3C2212yes—
39C2×#3830C2×C311yescharacteristic
40C22×#634C2×C311yescharacteristic, complemented
41C2×C3×#237—31no—
42C22×C3×Dic6—31no—
43C32×#123—21no—
44C32×#123D311yescharacteristic, complemented
45C32×#123C2×C311yescharacteristic, complemented
46#64144C2×C311yescharacteristic, complemented
47#64138C2×C311yescharacteristic, complemented
48#64105C2×C312yescomplemented
49#64097C2×C312yescomplemented
50#64095C2×C311yescharacteristic, complemented
51#64094C2×C312yescomplemented
52#64087C2×C312yescomplemented
53#64086C2×C311yescharacteristic, complemented
54C3×#935—31no—
55C3×#929—31no—
56C3×#896—32no—
57C3×#888—32no—
58C3×#886—31no—
59C3×#885—32no—
60C3×#878—32no—
61C3×#877—31no—
62C23×H(3,𝔽3)C2311yescharacteristic
63C23×H(3,𝔽3)C2311yescharacteristic
64C22×(C3⋊D3)⋊C3C2311yescharacteristic
65D4×H(3,𝔽3)—22no—
66C2×C4×H(3,𝔽3)—22no—
67C2×C4×H(3,𝔽3)C2312yes—
68C2×C4×H(3,𝔽3)C2311yescharacteristic
69C2×C4×H(3,𝔽3)C2312yes—
70#638⋊C3—22no—
71#638⋊C3—22no—
72C2×(C3⋊Dic3)⋊C3—22no—
73C2×(C3⋊Dic3)⋊C3—22no—
74C2×(C3⋊Dic3)⋊C3D414yes—
75C2×(C3⋊Dic3)⋊C3C2312yes—
76C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
77C2×(C3⋊Dic3)⋊C3C2312yes—
78C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
79C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
80C4×(C3⋊D3)⋊C3—22no—
81#634⋊C3—24no—
82#634⋊C3—24no—
83#634⋊C3—22no—
84C3×#536—91no—
85#5648—31no—
86#5648—91no—
87C2×C32×D4—21no—
88C2×C32×D4C2×D311yescharacteristic
89C2×C32×D4C22×C311yescharacteristic
90C22×C32×C4—21no—
91C22×C32×C4C2×D311yescharacteristic
92C22×C32×C4C22×C311yescharacteristic
93C2×#638C22×C311yescharacteristic
94C2×#638C22×C311yescharacteristic
95C22×C3⋊Dic3C22×C311yescharacteristic
96C22×C3⋊Dic3C22×C311yescharacteristic
97C22×C3⋊Dic3C22×C311yescharacteristic
98#3830—24no—
99C2×C4×C3⋊D3C22×C311yescharacteristic
100C2×#634—24no—
101C2×#634C22×C312yes—
102C2×#634C22×C312yes—
103C2×#634C22×C311yescharacteristic
104C2×C3×#67—31no—
105C2×C3×#67—31no—
106C22×C3×Dic3—31no—
107C22×C3×Dic3—31no—
108C22×C3×Dic3—31no—
109C3×#237—64no—
110C2×C3×C4×D3—31no—
111C2×C3×Dic6—32no—
112C2×C3×Dic6—32no—
113C2×C3×Dic6—31no—
114C2×C3×Dic6—64no—
115C32×C4⋊C4—22no—
116C32×C4⋊C4C2×D312yes—
117C32×C4⋊C4C22×C312yes—
118C32×C22⋊C4—21no—
119C32×C22⋊C4—22no—
120C32×C22⋊C4C2×D311yescharacteristic
121C32×C22⋊C4C2×D312yes—
122C32×C22⋊C4C22×C311yescharacteristic
123C32×C22⋊C4C22×C312yes—
124#3757C22×C311yescharacteristic
125#3757C22×C312yes—
126#3752C22×C312yes—
127#3752C22×C312yes—
128C4⋊(C3⋊Dic3)C22×C312yes—
129#3750C22×C312yes—
130#3750C22×C312yes—
131#3750C22×C312yes—
132C4×C3⋊Dic3C22×C312yes—
133C3×#217—31no—
134C3×#217—32no—
135C3×(C2×D3)⋊C4—32no—
136C3×(C2×D3)⋊C4—32no—
137C3×C4⋊Dic3—32no—
138C3×Dic3⋊C4—32no—
139C3×Dic3⋊C4—32no—
140C3×Dic3⋊C4—32no—
141C3×C4×Dic3—32no—
142C22×H(3,𝔽3)—22no—
143C22×H(3,𝔽3)—21no—
144C22×H(3,𝔽3)—22no—
145C22×H(3,𝔽3)C2×D412yes—
146C22×H(3,𝔽3)C2411yescharacteristic
147C2×(C3⋊D3)⋊C3—21no—
148C2×(C3⋊D3)⋊C3—22no—
149C4×H(3,𝔽3)—22no—
150C4×H(3,𝔽3)—21no—
151C4×H(3,𝔽3)—22no—
152(C3⋊Dic3)⋊C3—22no—
153(C3⋊Dic3)⋊C3—21no—
154(C3⋊Dic3)⋊C3—22no—
155(C3⋊Dic3)⋊C3C2×D412yes—
156(C3⋊Dic3)⋊C3C2×D412yes—
157C2×C3×#41—91no—
158C22×C3×Q8—91no—
159C2×#237—31no—
160C2×#237—91no—
161C22×Dic6—31no—
162C22×Dic6—91no—
163C3×C4⋊Q8—92no—
164C3×#124—92no—
165C3×#123—31no—
166C3×#123—31no—
167C3×#123—61no—
168C3×#123—91no—
169C3×#123—91no—
170C3×#123—91no—
171C3×#123C3×D311yescharacteristic
172C3×C22⋊Q8—92no—
173C3×C22⋊Q8—92no—
174C3×#117—91no—
175#935—31no—
176#935—91no—
177#929—31no—
178#929—91no—
179#896—32no—
180#896—92no—
181#888—32no—
182#888—92no—
183#886—31no—
184#886—91no—
185#885—32no—
186#885—92no—
187#878—32no—
188#878—92no—
189#877—31no—
190#877—91no—
191C23×C32—21no—
192C23×C32—21no—
193C23×C32C22×D311yescharacteristic
194C23×C32C22×D311yescharacteristic
195C23×C32C23×C311yescharacteristic
196C23×C32C23×C311yescharacteristic
197C22×C3⋊D3C23×C311yescharacteristic
198C22×C3×D3—31no—
199C32×D4—22no—
200C32×D4—22no—
201C32×D4—42no—
202C2×C32×C4—22no—
203C2×C32×C4—22no—
204C2×C32×C4—22no—
205C2×C32×C4—22no—
206C2×C32×C4—21no—
207C2×C32×C4—42no—
208C2×C32×C4C22×D312yes—
209C2×C32×C4C22×D311yescharacteristic
210C2×C32×C4C22×D312yes—
211C2×C32×C4C23×C312yes—
212C2×C32×C4C23×C311yescharacteristic
213C2×C32×C4C23×C312yes—
214#638—22no—
215#638—22no—
216C2×C3⋊Dic3—22no—
217C2×C3⋊Dic3—22no—
218C2×C3⋊Dic3C3×D414yes—
219C2×C3⋊Dic3C23×C312yes—
220C2×C3⋊Dic3C23×C311yescharacteristic
221C2×C3⋊Dic3C23×C312yes—
222C2×C3⋊Dic3C23×C311yescharacteristic
223C2×C3⋊Dic3C23×C311yescharacteristic
224C4×C3⋊D3—22no—
225#634—24no—
226#634—24no—
227#634—22no—
228C3×#67—62no—
229C3×#67—62no—
230C2×C3×Dic3—34no—
231C2×C3×Dic3—31no—
232C2×C3×Dic3—31no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—31no—
235C2×C3×Dic3—32no—
236C2×C3×Dic3—62no—
237C2×C3×Dic3—62no—
238C3×C4×D3—62no—
239C3×Dic6—64no—
240C3×Dic6—64no—
241C3×Dic6—62no—
242#536—91noSylow
243C2×H(3,𝔽3)—22no—
244C2×H(3,𝔽3)—41no—
245C2×H(3,𝔽3)C22×D411yescharacteristic
246C2×H(3,𝔽3)2−1+412yes—
247(C3⋊D3)⋊C3—41no—
248C3×#41—184no—
249C2×C3×Q8—92no—
250C2×C3×Q8—92no—
251C2×C3×Q8—91no—
252C2×C3×Q8—184no—
253C2×C3×D4—31no—
254C2×C3×D4—31no—
255C2×C3×D4—61no—
256C2×C3×D4—91no—
257C2×C3×D4—91no—
258C2×C3×D4C2×C3×D311yescharacteristic
259C22×C3×C4—31no—
260C22×C3×C4—31no—
261C22×C3×C4—61no—
262C22×C3×C4—91no—
263C22×C3×C4—91no—
264C22×C3×C4—91no—
265C22×C3×C4—91no—
266C22×C3×C4C2×C3×D311yescharacteristic
267C2×#67—31no—
268C2×#67—31no—
269C2×#67—91no—
270C2×#67—91no—
271C22×Dic3—31no—
272C22×Dic3—31no—
273C22×Dic3—31no—
274C22×Dic3—91no—
275C22×Dic3—91no—
276C22×Dic3—91no—
277#237—64no—
278#237—184no—
279C2×C4×D3—31no—
280C2×C4×D3—91no—
281C2×Dic6—32no—
282C2×Dic6—32no—
283C2×Dic6—31no—
284C2×Dic6—64no—
285C2×Dic6—92no—
286C2×Dic6—92no—
287C2×Dic6—91no—
288C2×Dic6—184no—
289C3×C4⋊C4—32no—
290C3×C4⋊C4—32no—
291C3×C4⋊C4—62no—
292C3×C4⋊C4—92no—
293C3×C4⋊C4—92no—
294C3×C4⋊C4—92no—
295C3×C4⋊C4—92no—
296C3×C4⋊C4C2×C3×D312yes—
297C3×C22⋊C4—31no—
298C3×C22⋊C4—32no—
299C3×C22⋊C4—31no—
300C3×C22⋊C4—32no—
301C3×C22⋊C4—61no—
302C3×C22⋊C4—62no—
303C3×C22⋊C4—92no—
304C3×C22⋊C4—92no—
305C3×C22⋊C4—91no—
306C3×C22⋊C4—92no—
307C3×C22⋊C4C2×C3×D311yescharacteristic
308C3×C22⋊C4C2×C3×D312yes—
309C3×C42—92no—
310#217—31no—
311#217—32no—
312#217—91no—
313#217—92no—
314(C2×D3)⋊C4—32no—
315(C2×D3)⋊C4—32no—
316(C2×D3)⋊C4—92no—
317(C2×D3)⋊C4—92no—
318C4⋊Dic3—32no—
319C4⋊Dic3—92no—
320Dic3⋊C4—32no—
321Dic3⋊C4—32no—
322Dic3⋊C4—32no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326C4×Dic3—32no—
327C4×Dic3—92no—
328C22×C32—22no—
329C22×C32—22no—
330C22×C32—21no—
331C22×C32—22no—
332C22×C32—21no—
333C22×C32—22no—
334C22×C32—21no—
335C22×C32—22no—
336C22×C32—42no—
337C22×C32—41no—
338C22×C32—42no—
339C22×C32D3×D412yes—
340C22×C32C2×C3×D412yes—
341C22×C32C23×D311yescharacteristic
342C22×C32C24×C311yescharacteristic
343C2×C3⋊D3—21no—
344C2×C3⋊D3—22no—
345C2×C3×D3—61no—
346C2×C3×D3—62no—
347C32×C4—22no—
348C32×C4—21no—
349C32×C4—22no—
350C32×C4—22no—
351C32×C4—21no—
352C32×C4—22no—
353C32×C4—42no—
354C32×C4—42no—
355C32×C4—41no—
356C3⋊Dic3—22no—
357C3⋊Dic3—21no—
358C3⋊Dic3—22no—
359C3⋊Dic3C2×C3×D412yes—
360C3⋊Dic3C2×C3×D412yes—
361C3×Dic3—32no—
362C3×Dic3—32no—
363C3×Dic3—62no—
364C3×Dic3—61no—
365C3×Dic3—62no—
366C2×#41—91no—
367C22×Q8—91no—
368C4⋊Q8—92no—
369#124—92no—
370#123—91no—
371#123—91no—
372#123—91no—
373#123(C3⋊D3)⋊C311yescharacteristic, complemented
374C22⋊Q8—92no—
375C22⋊Q8—92no—
376#117—91no—
377H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
378C23×C3—31no—
379C23×C3—31no—
380C23×C3—31no—
381C23×C3—31no—
382C23×C3—61no—
383C23×C3—61no—
384C23×C3—91no—
385C23×C3C22×C3×D311yescharacteristic
386C23×C3C22×C3×D311yescharacteristic
387C22×D3—31no—
388C22×D3—91no—
389C3×Q8—184no—
390C3×Q8—184no—
391C3×Q8—182no—
392C3×D4—22no—
393C3×D4—62no—
394C3×D4—62no—
395C3×D4—122no—
396C3×D4—182no—
397C3×D4—182no—
398C2×C3×C4—22no—
399C2×C3×C4—32no—
400C2×C3×C4—32no—
401C2×C3×C4—31no—
402C2×C3×C4—32no—
403C2×C3×C4—31no—
404C2×C3×C4—32no—
405C2×C3×C4—62no—
406C2×C3×C4—62no—
407C2×C3×C4—62no—
408C2×C3×C4—62no—
409C2×C3×C4—61no—
410C2×C3×C4—91no—
411C2×C3×C4—91no—
412C2×C3×C4—94no—
413C2×C3×C4—92no—
414C2×C3×C4—92no—
415C2×C3×C4—91no—
416C2×C3×C4—122no—
417C2×C3×C4—182no—
418C2×C3×C4—182no—
419C2×C3×C4—182no—
420C2×C3×C4C22×C3×D312yes—
421C2×C3×C4C22×C3×D311yescharacteristic
422C2×C3×C4C22×C3×D312yes—
423#67—62no—
424#67—62no—
425#67—182no—
426#67—182no—
427C2×Dic3—31no—
428C2×Dic3—31no—
429C2×Dic3—34no—
430C2×Dic3—32no—
431C2×Dic3—31no—
432C2×Dic3—32no—
433C2×Dic3—62no—
434C2×Dic3—62no—
435C2×Dic3—94no—
436C2×Dic3—91no—
437C2×Dic3—91no—
438C2×Dic3—92no—
439C2×Dic3—91no—
440C2×Dic3—92no—
441C2×Dic3—182no—
442C2×Dic3—182no—
443C4×D3—62no—
444C4×D3—182no—
445Dic6—64no—
446Dic6—64no—
447Dic6—62no—
448Dic6—184no—
449Dic6—184no—
450Dic6—182no—
451C2×C32—22no—
452C2×C32—22no—
453C2×C32—21no—
454C2×C32—22no—
455C2×C32—42no—
456C2×C32—41no—
457C2×C32—41no—
458C2×C32—81no—
459C2×C32C2×D3×D411yescharacteristic
460C2×C32#101012yes—
461C2×C32C22×C3×D411yescharacteristic
462C2×C32C3×2−1+412yes—
463C3⋊D3—41no—
464C3×D3—121no—
465#41—184no—
466C2×Q8—92no—
467C2×Q8—92no—
468C2×Q8—91no—
469C2×Q8—184no—
470C2×D4—91no—
471C2×D4—91no—
472C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
473C22×C4—91no—
474C22×C4—91no—
475C22×C4—91no—
476C22×C4—91no—
477C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
478C4⋊C4—92no—
479C4⋊C4—92no—
480C4⋊C4—92no—
481C4⋊C4—92no—
482C4⋊C4C2×(C3⋊D3)⋊C312yes—
483C22⋊C4—91no—
484C22⋊C4—92no—
485C22⋊C4—92no—
486C22⋊C4—92no—
487C22⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
488C22⋊C4C2×(C3⋊D3)⋊C312yes—
489C42—92no—
490C22×C3—22no—
491C22×C3—21no—
492C22×C3—22no—
493C22×C3—31no—
494C22×C3—31no—
495C22×C3—32no—
496C22×C3—32no—
497C22×C3—62no—
498C22×C3—62no—
499C22×C3—61no—
500C22×C3—61no—
501C22×C3—62no—
502C22×C3—62no—
503C22×C3—61no—
504C22×C3—62no—
505C22×C3—122no—
506C22×C3—121no—
507C22×C3—122no—
508C22×C3—181no—
509C22×C3—182no—
510C22×C3C3×D3×D412yes—
511C22×C3C23×C3×D311yescharacteristic
512C2×D3—61no—
513C2×D3—62no—
514C2×D3—181no—
515C2×D3—182no—
516C3×C4—22no—
517C3×C4—21no—
518C3×C4—22no—
519C3×C4—62no—
520C3×C4—61no—
521C3×C4—62no—
522C3×C4—61no—
523C3×C4—62no—
524C3×C4—62no—
525C3×C4—92no—
526C3×C4—92no—
527C3×C4—122no—
528C3×C4—121no—
529C3×C4—122no—
530C3×C4—181no—
531C3×C4—182no—
532C3×C4—182no—
533Dic3—32no—
534Dic3—32no—
535Dic3—62no—
536Dic3—61no—
537Dic3—62no—
538Dic3—92no—
539Dic3—92no—
540Dic3—182no—
541Dic3—181no—
542Dic3—182no—
543C32—21no—
544C32#564811yescharacteristic, complemented
545C32C3×#53611yescharacteristic, complemented
546C23—91no—
547C23C22×(C3⋊D3)⋊C311yescharacteristic
548C23C22×(C3⋊D3)⋊C311yescharacteristic
549Q8—184no—
550Q8—184no—
551Q8—182no—
552D4—22no—
553D4—182no—
554D4—182no—
555C2×C4—22no—
556C2×C4—91no—
557C2×C4—91no—
558C2×C4—94no—
559C2×C4—92no—
560C2×C4—92no—
561C2×C4—91no—
562C2×C4—182no—
563C2×C4—182no—
564C2×C4—182no—
565C2×C4C22×(C3⋊D3)⋊C312yes—
566C2×C4C22×(C3⋊D3)⋊C312yes—
567C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
568C2×C3—22no—
569C2×C3—31no—
570C2×C3—31no—
571C2×C3—32no—
572C2×C3—32no—
573C2×C3—41no—
574C2×C3—62no—
575C2×C3—62no—
576C2×C3—61no—
577C2×C3—62no—
578C2×C3—122no—
579C2×C3—121no—
580C2×C3—121no—
581C2×C3—241no—
582C2×C3—361no—
583C2×C3C2×C3×D3×D411yescharacteristic
584C2×C3C3×#101012yes—
585D3—121no—
586D3—361no—
587C22—22no—
588C22—21no—
589C22—22no—
590C22—181no—
591C22—182no—
592C22D4×(C3⋊D3)⋊C312yes—
593C22C23×(C3⋊D3)⋊C311yescharacteristic
594C4—22no—
595C4—21no—
596C4—22no—
597C4—92no—
598C4—92no—
599C4—181no—
600C4—182no—
601C4—182no—
602C3—31no—
603C3—31no—
604C3—61no—
605C3C3×#564811yescharacteristic
606C2—22no—
607C2—41no—
608C2—361no—
609C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
610C2#1172263912yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#123C2
prime radicalC2×#3830⋊C3C2
hypercenter#123(C3⋊D3)⋊C3
hypocenterC32C3×#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
3. H(3,𝔽3)×#123C2semidirect
4. #64144⋊C3C2semidirect
5. #64138⋊C3C2semidirect
6. #64105⋊C3C2semidirect
7. #64097⋊C3C2semidirect
8. #64095⋊C3C2semidirect
9. #64094⋊C3C2semidirect
10. #64087⋊C3C2semidirect
11. #64086⋊C3C2semidirect
12. #10596119C3semidirect
40. C22×#634C2×C3semidirect
44. C32×#123D3semidirect
45. C32×#123C2×C3semidirect
46. #64144C2×C3semidirect
47. #64138C2×C3semidirect
48. #64105C2×C3semidirect
49. #64097C2×C3semidirect
50. #64095C2×C3semidirect
51. #64094C2×C3semidirect
52. #64087C2×C3semidirect
53. #64086C2×C3semidirect
373. #123(C3⋊D3)⋊C3semidirect
377. H(3,𝔽3)#536semidirect
544. C32#5648semidirect
545. C32C3×#536semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
C22×#634⋊C32
H(3,𝔽3)×#1232
#64144⋊C32
#64138⋊C32
#64105⋊C32
#64097⋊C32
#64095⋊C32
#64094⋊C32
#64087⋊C32
#64086⋊C32
#105961193
C3×#56483