Finite Group Databasebeta

← #10596093⋊C3 #13152945

Subgroups — #10596093⋊C3 #13152945

Order 1728.

851 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3#64153⋊C3C211yesmaximal, characteristic, complemented
4#64133⋊C3C211yesmaximal, characteristic
5C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
6#64098⋊C3C212yesmaximal, complemented
7#64097⋊C3C211yesmaximal, characteristic, complemented
8#64096⋊C3C211yesmaximal, characteristic, complemented
9#64096⋊C3C212yesmaximal, complemented
10#64087⋊C3C212yesmaximal
11#64086⋊C3C212yesmaximal
12#10596093C311yesmaximal, characteristic, complemented
13C3×#5622—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×#634⋊C3C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2212yes—
22C22⋊C4×H(3,𝔽3)C2212yes—
23#3757⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2212yes—
26#3750⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×C22⋊Q8—21no—
38C32×C22⋊Q8D311yescharacteristic
39C32×C22⋊Q8C2×C311yescharacteristic
40Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
41#64153C2×C311yescharacteristic, complemented
42#64133C2×C311yescharacteristic
43C2×C4×C3⋊Dic3C2×C311yescharacteristic
44#64098C2×C312yescomplemented
45#64097C2×C311yescharacteristic, complemented
46#64096C2×C311yescharacteristic, complemented
47#64096C2×C312yescomplemented
48#64087C2×C312yes—
49#64086C2×C312yes—
50C3×Q8×Dic3—31no—
51C3×#944—31no—
52C3×#924—31no—
53C2×C3×C4×Dic3—31no—
54C3×#889—32no—
55C3×#888—31no—
56C3×#887—31no—
57C3×#887—32no—
58C3×#878—32no—
59C3×#877—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61Q8×H(3,𝔽3)—22no—
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2312yes—
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3—22no—
68C2×(C3⋊Dic3)⋊C3Q814yescomplemented
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75#634⋊C3—22no—
76#634⋊C3—24no—
77C3×#533—91no—
78#5622—31no—
79#5622—91no—
80C2×C32×Q8—21no—
81C2×C32×Q8C2×D311yescharacteristic
82C2×C32×Q8C22×C311yescharacteristic
83C22×C32×C4—21no—
84C22×C32×C4C2×D311yescharacteristic
85C22×C32×C4C22×C311yescharacteristic
86C22×C3⋊Dic3C22×C311yescharacteristic
87C22×C3⋊Dic3C22×C311yescharacteristic
88C2×#634C22×C312yes—
89C2×#634C22×C311yescharacteristic
90C22×C3×Dic3—31no—
91C22×C3×Dic3—31no—
92C2×C3×Dic6—32no—
93C2×C3×Dic6—31no—
94C32×C4⋊C4—21no—
95C32×C4⋊C4—22no—
96C32×C4⋊C4C2×D311yescharacteristic
97C32×C4⋊C4C2×D312yes—
98C32×C4⋊C4C22×C311yescharacteristic
99C32×C4⋊C4C22×C312yes—
100C32×C22⋊C4—22no—
101C32×C22⋊C4C2×D312yes—
102C32×C22⋊C4C22×C312yes—
103#3757C22×C312yes—
104C4⋊(C3⋊Dic3)C22×C312yes—
105C4⋊(C3⋊Dic3)C22×C311yescharacteristic
106#3750C22×C312yes—
107#3750C22×C312yes—
108#3750C22×C312yes—
109#3750C22×C312yes—
110#3750C22×C312yes—
111C4×C3⋊Dic3C22×C312yes—
112C4×C3⋊Dic3C22×C312yes—
113C4×C3⋊Dic3C22×C311yescharacteristic
114C4×C3⋊Dic3C22×C311yescharacteristic
115C4×C3⋊Dic3C22×C311yescharacteristic
116C4×C3⋊Dic3C22×C311yescharacteristic
117C3×#217—32no—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—31no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—31no—
128C3×C4×Dic3—31no—
129C3×C4×Dic3—31no—
130C3×C4×Dic3—31no—
131C22×H(3,𝔽3)—21no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)C2×Q812yes—
134C22×H(3,𝔽3)C2411yescharacteristic
135C4×H(3,𝔽3)—22no—
136C4×H(3,𝔽3)—21no—
137C4×H(3,𝔽3)—22no—
138C4×H(3,𝔽3)#4112yes—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3C2×Q812yes—
143(C3⋊Dic3)⋊C3C2×Q812yes—
144(C3⋊Dic3)⋊C3#4112yes—
145C3×C4⋊Q8—91no—
146C3×C4⋊Q8—91no—
147C3×#125—92no—
148C3×C22⋊Q8—31no—
149C3×C22⋊Q8—31no—
150C3×C22⋊Q8—61no—
151C3×C22⋊Q8—92no—
152C3×C22⋊Q8—91no—
153C3×C22⋊Q8C3×D311yescharacteristic
154C3×C4×Q8—92no—
155C3×C4×Q8—91no—
156C3×C4×Q8—91no—
157C3×#117—92no—
158C2×C3×C42—91no—
159Q8×Dic3—31no—
160Q8×Dic3—91no—
161#944—31no—
162#944—91no—
163#924—31no—
164#924—91no—
165C2×C4×Dic3—31no—
166C2×C4×Dic3—91no—
167#889—32no—
168#889—92no—
169#888—31no—
170#888—91no—
171#887—31no—
172#887—32no—
173#887—91no—
174#887—92no—
175#878—32no—
176#878—92no—
177#877—32no—
178#877—92no—
179C23×C32—21no—
180C23×C32C22×D311yescharacteristic
181C23×C32C23×C311yescharacteristic
182C32×Q8—22no—
183C32×Q8—22no—
184C32×Q8—42no—
185C2×C32×C4—22no—
186C2×C32×C4—22no—
187C2×C32×C4—21no—
188C2×C32×C4—22no—
189C2×C32×C4—22no—
190C2×C32×C4—21no—
191C2×C32×C4—42no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C23×C311yescharacteristic
197C2×C32×C4C23×C312yes—
198C2×C32×C4C23×C312yes—
199C2×C32×C4C23×C311yescharacteristic
200C2×C3⋊Dic3—22no—
201C2×C3⋊Dic3C3×Q814yescomplemented
202C2×C3⋊Dic3C23×C311yescharacteristic
203C2×C3⋊Dic3C23×C312yes—
204C2×C3⋊Dic3C23×C312yes—
205C2×C3⋊Dic3C23×C311yescharacteristic
206C2×C3⋊Dic3C23×C311yescharacteristic
207C2×C3⋊Dic3C23×C311yescharacteristic
208#634—22no—
209#634—24no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—62no—
218C3×Dic6—64no—
219C3×Dic6—62no—
220#533—91noSylow
221C2×H(3,𝔽3)—22no—
222C2×H(3,𝔽3)C22×Q811yescharacteristic
223C2×H(3,𝔽3)C2×#4111yescharacteristic
224C2×H(3,𝔽3)C2×#4111yescharacteristic
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—61no—
228C2×C3×Q8—91no—
229C2×C3×Q8—92no—
230C2×C3×Q8C2×C3×D311yescharacteristic
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—61no—
234C22×C3×C4—91no—
235C22×C3×C4—91no—
236C22×C3×C4C2×C3×D311yescharacteristic
237C22×Dic3—31no—
238C22×Dic3—31no—
239C22×Dic3—91no—
240C22×Dic3—91no—
241C2×Dic6—32no—
242C2×Dic6—31no—
243C2×Dic6—92no—
244C2×Dic6—91no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—32no—
247C3×C4⋊C4—32no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—62no—
250C3×C4⋊C4—61no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4—92no—
255C3×C4⋊C4—92no—
256C3×C4⋊C4—92no—
257C3×C4⋊C4—91no—
258C3×C4⋊C4C2×C3×D311yescharacteristic
259C3×C4⋊C4C2×C3×D312yes—
260C3×C22⋊C4—32no—
261C3×C22⋊C4—32no—
262C3×C22⋊C4—62no—
263C3×C22⋊C4—92no—
264C3×C22⋊C4C2×C3×D312yes—
265C3×C42—91no—
266C3×C42—91no—
267C3×C42—91no—
268C3×C42—91no—
269C3×C42—92no—
270C3×C42—92no—
271#217—32no—
272#217—92no—
273C4⋊Dic3—32no—
274C4⋊Dic3—31no—
275C4⋊Dic3—92no—
276C4⋊Dic3—91no—
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—32no—
281Dic3⋊C4—32no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3—32no—
288C4×Dic3—32no—
289C4×Dic3—31no—
290C4×Dic3—31no—
291C4×Dic3—31no—
292C4×Dic3—31no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C4×Dic3—91no—
296C4×Dic3—91no—
297C4×Dic3—91no—
298C4×Dic3—91no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—22no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32D3×Q812yes—
308C22×C32C2×C3×Q812yes—
309C22×C32C23×D311yescharacteristic
310C22×C32C24×C311yescharacteristic
311C32×C4—21no—
312C32×C4—22no—
313C32×C4—22no—
314C32×C4—22no—
315C32×C4—21no—
316C32×C4—22no—
317C32×C4—22no—
318C32×C4—42no—
319C32×C4—41no—
320C32×C4—42no—
321C32×C4#23712yes—
322C32×C4C3×#4112yes—
323C3⋊Dic3—22no—
324C3⋊Dic3—22no—
325C3⋊Dic3—21no—
326C3⋊Dic3C2×C3×Q812yes—
327C3⋊Dic3C2×C3×Q812yes—
328C3⋊Dic3C3×#4112yes—
329C3×Dic3—32no—
330C3×Dic3—32no—
331C3×Dic3—32no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—61no—
335C4⋊Q8—91no—
336C4⋊Q8—91no—
337#125—92no—
338C22⋊Q8—91no—
339C22⋊Q8—92no—
340C22⋊Q8(C3⋊D3)⋊C311yescharacteristic
341C4×Q8—91no—
342C4×Q8—91no—
343C4×Q8—92no—
344#117—92no—
345C2×C42—91no—
346H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
347C23×C3—31no—
348C23×C3—31no—
349C23×C3—61no—
350C23×C3C22×C3×D311yescharacteristic
351C3×Q8—22no—
352C3×Q8—62no—
353C3×Q8—62no—
354C3×Q8—122no—
355C3×Q8—182no—
356C3×Q8—184no—
357C2×C3×C4—22no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—32no—
362C2×C3×C4—32no—
363C2×C3×C4—31no—
364C2×C3×C4—32no—
365C2×C3×C4—32no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—61no—
369C2×C3×C4—62no—
370C2×C3×C4—62no—
371C2×C3×C4—61no—
372C2×C3×C4—91no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—94no—
376C2×C3×C4—91no—
377C2×C3×C4—92no—
378C2×C3×C4—92no—
379C2×C3×C4—122no—
380C2×C3×C4—182no—
381C2×C3×C4C22×C3×D311yescharacteristic
382C2×C3×C4C22×C3×D312yes—
383C2×C3×C4C22×C3×D312yes—
384C2×C3×C4C22×C3×D311yescharacteristic
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—34no—
389C2×Dic3—31no—
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—62no—
393C2×Dic3—94no—
394C2×Dic3—91no—
395C2×Dic3—91no—
396C2×Dic3—91no—
397C2×Dic3—91no—
398C2×Dic3—92no—
399C2×Dic3—92no—
400C2×Dic3—182no—
401Dic6—64no—
402Dic6—62no—
403Dic6—184no—
404Dic6—182no—
405C2×C32—22no—
406C2×C32—22no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—42no—
411C2×C32C2×#23711yescharacteristic
412C2×C32C2×D3×Q811yescharacteristic
413C2×C32D3×#4111yescharacteristic
414C2×C32C22×C3×Q811yescharacteristic
415C2×C32C2×C3×#4111yescharacteristic
416C2×C32C2×C3×#4111yescharacteristic
417C2×Q8—92no—
418C2×Q8—91no—
419C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
420C22×C4—91no—
421C22×C4—91no—
422C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
423C4⋊C4—92no—
424C4⋊C4—92no—
425C4⋊C4—92no—
426C4⋊C4—92no—
427C4⋊C4—92no—
428C4⋊C4—92no—
429C4⋊C4—91no—
430C4⋊C4C2×(C3⋊D3)⋊C312yes—
431C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
432C22⋊C4—92no—
433C22⋊C4C2×(C3⋊D3)⋊C312yes—
434C42—91no—
435C42—91no—
436C42—91no—
437C42—91no—
438C42—92no—
439C42—92no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—32no—
445C22×C3—32no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—62no—
452C22×C3—121no—
453C22×C3—121no—
454C22×C3C3×D3×Q812yes—
455C22×C3C23×C3×D311yescharacteristic
456C3×C4—21no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—32no—
460C3×C4—32no—
461C3×C4—62no—
462C3×C4—61no—
463C3×C4—61no—
464C3×C4—62no—
465C3×C4—62no—
466C3×C4—62no—
467C3×C4—62no—
468C3×C4—92no—
469C3×C4—92no—
470C3×C4—92no—
471C3×C4—121no—
472C3×C4—122no—
473C3×C4—122no—
474C3×C4—181no—
475C3×C4—182no—
476C3×C4—182no—
477C3×C4C3×#23712yes—
478Dic3—32no—
479Dic3—32no—
480Dic3—32no—
481Dic3—62no—
482Dic3—62no—
483Dic3—61no—
484Dic3—92no—
485Dic3—92no—
486Dic3—92no—
487Dic3—182no—
488Dic3—182no—
489Dic3—181no—
490C32—21no—
491C32#562211yescharacteristic, complemented
492C32C3×#53311yescharacteristic, complemented
493C23C22×(C3⋊D3)⋊C311yescharacteristic
494Q8—22no—
495Q8—184no—
496Q8—182no—
497C2×C4—22no—
498C2×C4—91no—
499C2×C4—91no—
500C2×C4—91no—
501C2×C4—94no—
502C2×C4—91no—
503C2×C4—92no—
504C2×C4—92no—
505C2×C4—182no—
506C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
507C2×C4C22×(C3⋊D3)⋊C312yes—
508C2×C4C22×(C3⋊D3)⋊C312yes—
509C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
510C2×C3—22no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—31no—
517C2×C3—62no—
518C2×C3—62no—
519C2×C3—61no—
520C2×C3—61no—
521C2×C3—61no—
522C2×C3—122no—
523C2×C3C2×C3×#23711yescharacteristic
524C2×C3C2×C3×D3×Q811yescharacteristic
525C2×C3C3×D3×#4111yescharacteristic
526C22—21no—
527C22—21no—
528C22Q8×(C3⋊D3)⋊C312yes—
529C22C23×(C3⋊D3)⋊C311yescharacteristic
530C4—21no—
531C4—22no—
532C4—22no—
533C4—92no—
534C4—92no—
535C4—92no—
536C4—182no—
537C4—181no—
538C4—182no—
539C4#3830⋊C312yes—
540C3—31no—
541C3—31no—
542C3—61no—
543C3C3×#562211yescharacteristic
544C2—22no—
545C2C2×#3830⋊C311yescharacteristic
546C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
547C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radicalC23×H(3,𝔽3)C23
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#533
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. Q8×(C3⋊Dic3)⋊C3C2semidirect
3. #64153⋊C3C2semidirect
6. #64098⋊C3C2semidirect
7. #64097⋊C3C2semidirect
8. #64096⋊C3C2semidirect
9. #64096⋊C3C2semidirect
12. #10596093C3semidirect
40. Q8×C3⋊Dic3C2×C3semidirect
41. #64153C2×C3semidirect
44. #64098C2×C3semidirect
45. #64097C2×C3semidirect
46. #64096C2×C3semidirect
47. #64096C2×C3semidirect
68. C2×(C3⋊Dic3)⋊C3Q8semidirect
201. C2×C3⋊Dic3C3×Q8semidirect
346. H(3,𝔽3)#533semidirect
491. C32#5622semidirect
492. C32C3×#533semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
Q8×(C3⋊Dic3)⋊C32
#64153⋊C32
#64133⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#64098⋊C32
#64097⋊C32
#64096⋊C32
#64087⋊C32
#64086⋊C32
#105960933
C3×#56223