Finite Group Databasebeta

← #10596200⋊C3 #13153325

Subgroups — #10596200⋊C3 #13153325

Order 1728.

789 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64155⋊C3C212yesmaximal
3#64153⋊C3C212yesmaximal, complemented
4#64105⋊C3C212yesmaximal
5#64101⋊C3C211yesmaximal, characteristic
6C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
7#64099⋊C3C211yesmaximal, characteristic, complemented
8#64098⋊C3C212yesmaximal, complemented
9#64097⋊C3C211yesmaximal, characteristic, complemented
10#64081⋊C3C211yesmaximal, characteristic
11#64079⋊C3C211yesmaximal, characteristic, complemented
12#10596200C311yesmaximal, characteristic, complemented
13C3×#5729—31nomaximal
14C2×Q8×H(3,𝔽3)C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2212yes—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C42×H(3,𝔽3)C2211yescharacteristic
22#3752⋊C3C2211yescharacteristic
23#3752⋊C3C2211yescharacteristic
24#3752⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2211yescharacteristic
31#3750⋊C3C2211yescharacteristic
32#3750⋊C3C2212yes—
33#3750⋊C3C2212yes—
34#3750⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×C4⋊Q8—21no—
38C32×C4⋊Q8D311yescharacteristic, complemented
39C32×C4⋊Q8C2×C311yescharacteristic, complemented
40#64155C2×C312yes—
41#64153C2×C312yescomplemented
42#64105C2×C312yes—
43#64101C2×C311yescharacteristic
44C4⋊C4×C3⋊D3C2×C311yescharacteristic
45#64099C2×C311yescharacteristic, complemented
46#64098C2×C312yescomplemented
47#64097C2×C311yescharacteristic, complemented
48#64081C2×C311yescharacteristic
49#64079C2×C311yescharacteristic, complemented
50C3×#946—32no—
51C3×#944—32no—
52C3×#896—32no—
53C3×#892—31no—
54C3×D3×C4⋊C4—31no—
55C3×#890—31no—
56C3×#889—32no—
57C3×#888—31no—
58C3×#872—31no—
59C3×#870—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61Q8×H(3,𝔽3)—42no—
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4×(C3⋊D3)⋊C3—42no—
73C4×(C3⋊D3)⋊C3Q814yescomplemented
74#634⋊C3—42no—
75C3×#544—91no—
76#5729—31no—
77#5729—91no—
78C2×C32×Q8—22no—
79C2×C32×Q8C2×D312yes—
80C2×C32×Q8C22×C312yes—
81C2×C4×C3⋊D3C22×C312yes—
82C2×C4×C3⋊D3C22×C311yescharacteristic
83C2×#634C22×C312yes—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—32no—
86C2×C3×Dic6—32no—
87C32×C4⋊C4—21no—
88C32×C4⋊C4—22no—
89C32×C4⋊C4—21no—
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C2×D312yes—
92C32×C4⋊C4C2×D311yescharacteristic
93C32×C4⋊C4C22×C311yescharacteristic
94C32×C4⋊C4C22×C312yes—
95C32×C4⋊C4C22×C311yescharacteristic
96C32×C42—21no—
97C32×C42C2×D311yescharacteristic
98C32×C42C22×C311yescharacteristic
99#3752C22×C311yescharacteristic
100#3752C22×C311yescharacteristic
101#3752C22×C312yes—
102C4⋊(C3⋊Dic3)C22×C311yescharacteristic
103C4⋊(C3⋊Dic3)C22×C311yescharacteristic
104C4⋊(C3⋊Dic3)C22×C312yes—
105#3750C22×C312yes—
106#3750C22×C312yes—
107#3750C22×C311yescharacteristic
108#3750C22×C311yescharacteristic
109#3750C22×C312yes—
110#3750C22×C312yes—
111#3750C22×C312yes—
112C4×C3⋊Dic3C22×C311yescharacteristic
113C4×C3⋊Dic3C22×C312yes—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—31no—
116C3×(C2×D3)⋊C4—31no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—31no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—31no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4—32no—
127C3×C4×Dic3—31no—
128C3×C4×Dic3—32no—
129C22×H(3,𝔽3)C2411yescharacteristic
130C2×(C3⋊D3)⋊C3—21no—
131C2×(C3⋊D3)⋊C3—21no—
132C2×(C3⋊D3)⋊C3C2×Q812yes—
133C4×H(3,𝔽3)—21no—
134C4×H(3,𝔽3)—22no—
135C4×H(3,𝔽3)—22no—
136C4×H(3,𝔽3)—21no—
137C4×H(3,𝔽3)C2×Q812yes—
138(C3⋊Dic3)⋊C3—21no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3C2×Q812yes—
143C3×C4⋊Q8—31no—
144C3×C4⋊Q8—31no—
145C3×C4⋊Q8—61no—
146C3×C4⋊Q8—91no—
147C3×C4⋊Q8—92no—
148C3×C4⋊Q8C3×D311yescharacteristic
149C3×#125—91no—
150C3×#125—92no—
151C3×#125—91no—
152C3×C22⋊Q8—92no—
153C3×C22⋊Q8—92no—
154C3×#117—91no—
155C3×#117—91no—
156C2×C3×C4⋊C4—91no—
157#946—32no—
158#946—92no—
159#944—32no—
160#944—92no—
161#896—32no—
162#896—92no—
163#892—31no—
164#892—91no—
165D3×C4⋊C4—31no—
166D3×C4⋊C4—91no—
167#890—31no—
168#890—91no—
169#889—32no—
170#889—92no—
171#888—31no—
172#888—91no—
173#872—31no—
174#872—91no—
175#870—31no—
176#870—91no—
177C22×C3⋊D3C23×C311yescharacteristic
178C22×C3×D3—31no—
179C32×Q8—42no—
180C32×Q8—42no—
181C32×Q8—44no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4—22no—
187C2×C32×C4C22×D312yes—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C23×C312yes—
193C2×C32×C4C23×C311yescharacteristic
194C2×C32×C4C23×C311yescharacteristic
195C2×C32×C4C23×C312yes—
196C2×C32×C4C23×C311yescharacteristic
197C2×C3⋊Dic3C23×C311yescharacteristic
198C2×C3⋊Dic3C23×C312yes—
199C2×C3⋊Dic3C23×C311yescharacteristic
200C2×C3⋊Dic3C23×C312yes—
201C2×C3⋊Dic3C23×C311yescharacteristic
202C4×C3⋊D3—42no—
203C4×C3⋊D3C3×Q814yescomplemented
204#634—42no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—32no—
210C3×C4×D3—34no—
211C3×C4×D3—122no—
212C3×Dic6—122no—
213#544—91noSylow
214C2×H(3,𝔽3)C22×Q811yescharacteristic
215C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
216C2×H(3,𝔽3)2−1+411yescharacteristic
217(C3⋊D3)⋊C3—22no—
218C2×C3×Q8—32no—
219C2×C3×Q8—32no—
220C2×C3×Q8—62no—
221C2×C3×Q8—92no—
222C2×C3×Q8C2×C3×D312yes—
223C22×C3×C4—91no—
224C22×C3×C4—92no—
225C2×C4×D3—31no—
226C2×C4×D3—32no—
227C2×C4×D3—91no—
228C2×C4×D3—92no—
229C2×Dic6—32no—
230C2×Dic6—92no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—32no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—62no—
240C3×C4⋊C4—92no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4—92no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4C2×C3×D311yescharacteristic
251C3×C4⋊C4C2×C3×D312yes—
252C3×C4⋊C4C2×C3×D311yescharacteristic
253C3×C22⋊C4—91no—
254C3×C22⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C42—31no—
257C3×C42—31no—
258C3×C42—61no—
259C3×C42—91no—
260C3×C42—92no—
261C3×C42C2×C3×D311yescharacteristic
262(C2×D3)⋊C4—31no—
263(C2×D3)⋊C4—31no—
264(C2×D3)⋊C4—32no—
265(C2×D3)⋊C4—91no—
266(C2×D3)⋊C4—91no—
267(C2×D3)⋊C4—92no—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—32no—
271C4⋊Dic3—91no—
272C4⋊Dic3—91no—
273C4⋊Dic3—92no—
274Dic3⋊C4—32no—
275Dic3⋊C4—32no—
276Dic3⋊C4—31no—
277Dic3⋊C4—31no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—32no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—91no—
284Dic3⋊C4—91no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288C4×Dic3—31no—
289C4×Dic3—32no—
290C4×Dic3—91no—
291C4×Dic3—92no—
292C22×C32—21no—
293C22×C32C23×D311yescharacteristic
294C22×C32C24×C311yescharacteristic
295C2×C3⋊D3—21no—
296C2×C3⋊D3—21no—
297C2×C3⋊D3C2×C3×Q812yes—
298C2×C3×D3—32no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C32×C4—22no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—22no—
308C32×C4—22no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—22no—
312C32×C4—42no—
313C32×C4—42no—
314C32×C4D3×Q812yes—
315C32×C4C2×C3×Q812yes—
316C3⋊Dic3—21no—
317C3⋊Dic3—22no—
318C3⋊Dic3—21no—
319C3⋊Dic3—22no—
320C3⋊Dic3C2×C3×Q812yes—
321C3×Dic3—32no—
322C3×Dic3—61no—
323C3×Dic3—62no—
324C3×Dic3—61no—
325C3×Dic3—62no—
326C4⋊Q8—92no—
327C4⋊Q8—91no—
328C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
329#125—92no—
330#125—91no—
331#125—91no—
332C22⋊Q8—92no—
333C22⋊Q8—92no—
334#117—91no—
335#117—91no—
336C2×C4⋊C4—91no—
337H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
338C23×C3—91no—
339C22×D3—31no—
340C22×D3—91no—
341C3×Q8—42no—
342C3×Q8—122no—
343C3×Q8—122no—
344C3×Q8—124no—
345C3×Q8—362no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—32no—
351C2×C3×C4—32no—
352C2×C3×C4—32no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—32no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—94no—
362C2×C3×C4—91no—
363C2×C3×C4—91no—
364C2×C3×C4—92no—
365C2×C3×C4—91no—
366C2×C3×C4—92no—
367C2×C3×C4—362no—
368C2×C3×C4C22×C3×D312yes—
369C2×C3×C4C22×C3×D311yescharacteristic
370C2×C3×C4C22×C3×D311yescharacteristic
371C2×C3×C4C22×C3×D312yes—
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—32no—
376C2×Dic3—31no—
377C2×Dic3—32no—
378C2×Dic3—91no—
379C2×Dic3—91no—
380C2×Dic3—92no—
381C2×Dic3—91no—
382C2×Dic3—92no—
383C4×D3—34no—
384C4×D3—94no—
385C4×D3—122no—
386C4×D3—362no—
387Dic6—122no—
388Dic6—362no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32#100411yescharacteristic
393C2×C32C2×D3×Q811yescharacteristic
394C2×C32#100711yescharacteristic
395C2×C32C22×C3×Q811yescharacteristic
396C2×C32C3×H(5,𝔽2)11yescharacteristic
397C2×C32C3×2−1+411yescharacteristic
398C3⋊D3—22no—
399C3×D3—62no—
400C2×Q8—92no—
401C2×Q8C2×(C3⋊D3)⋊C312yes—
402C22×C4—91no—
403C22×C4—92no—
404C4⋊C4—92no—
405C4⋊C4—91no—
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—91no—
409C4⋊C4—92no—
410C4⋊C4—92no—
411C4⋊C4—92no—
412C4⋊C4—92no—
413C4⋊C4—92no—
414C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
415C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
416C4⋊C4C2×(C3⋊D3)⋊C312yes—
417C22⋊C4—92no—
418C22⋊C4—91no—
419C22⋊C4—91no—
420C42—92no—
421C42—91no—
422C42C2×(C3⋊D3)⋊C311yescharacteristic
423C22×C3—31no—
424C22×C3—31no—
425C22×C3—61no—
426C22×C3—92no—
427C22×C3—181no—
428C22×C3—181no—
429C22×C3C23×C3×D311yescharacteristic
430C2×D3—32no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—92no—
434C2×D3—181no—
435C2×D3—181no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—32no—
441C3×C4—32no—
442C3×C4—62no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—62no—
448C3×C4—61no—
449C3×C4—61no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—62no—
453C3×C4—92no—
454C3×C4—122no—
455C3×C4—122no—
456C3×C4—182no—
457C3×C4—181no—
458C3×C4—181no—
459C3×C4—182no—
460C3×C4C3×D3×Q812yes—
461Dic3—32no—
462Dic3—61no—
463Dic3—62no—
464Dic3—61no—
465Dic3—62no—
466Dic3—92no—
467Dic3—181no—
468Dic3—182no—
469Dic3—181no—
470Dic3—182no—
471C32—21no—
472C32#572911yescharacteristic, complemented
473C32C3×#54411yescharacteristic, complemented
474C23—91no—
475Q8—42no—
476Q8—362no—
477C2×C4—91no—
478C2×C4—94no—
479C2×C4—91no—
480C2×C4—92no—
481C2×C4—91no—
482C2×C4—92no—
483C2×C4—362no—
484C2×C4C22×(C3⋊D3)⋊C312yes—
485C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
486C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
487C2×C4C22×(C3⋊D3)⋊C312yes—
488C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—31no—
494C2×C3—31no—
495C2×C3—61no—
496C2×C3—61no—
497C2×C3—61no—
498C2×C3—182no—
499C2×C3C3×#100411yescharacteristic
500C2×C3C2×C3×D3×Q811yescharacteristic
501C2×C3C3×#100711yescharacteristic
502D3—62no—
503D3—182no—
504C22—92no—
505C22—181no—
506C22—181no—
507C22C23×(C3⋊D3)⋊C311yescharacteristic
508C4—22no—
509C4—21no—
510C4—22no—
511C4—21no—
512C4—92no—
513C4—182no—
514C4—181no—
515C4—181no—
516C4—182no—
517C4Q8×(C3⋊D3)⋊C312yes—
518C3—31no—
519C3—31no—
520C3—61no—
521C3C3×#572911yescharacteristic
522C2—182no—
523C2#64361⋊C311yescharacteristic
524C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
525C2#1172262711yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #64153⋊C3C2semidirect
7. #64099⋊C3C2semidirect
8. #64098⋊C3C2semidirect
9. #64097⋊C3C2semidirect
11. #64079⋊C3C2semidirect
12. #10596200C3semidirect
38. C32×C4⋊Q8D3semidirect
39. C32×C4⋊Q8C2×C3semidirect
41. #64153C2×C3semidirect
45. #64099C2×C3semidirect
46. #64098C2×C3semidirect
47. #64097C2×C3semidirect
49. #64079C2×C3semidirect
73. C4×(C3⋊D3)⋊C3Q8semidirect
203. C4×C3⋊D3C3×Q8semidirect
328. C4⋊Q8(C3⋊D3)⋊C3semidirect
337. H(3,𝔽3)#544semidirect
472. C32#5729semidirect
473. C32C3×#544semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#64155⋊C32
#64153⋊C32
#64105⋊C32
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64099⋊C32
#64098⋊C32
#64097⋊C32
#64081⋊C32
#64079⋊C32
#105962003
C3×#57293