Finite Group Databasebeta

← #10596160⋊C3 #13153180

Subgroups — #10596160⋊C3 #13153180

Order 1728.

777 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#64106⋊C3C212yesmaximal
3#64105⋊C3C212yesmaximal
4#64101⋊C3C211yesmaximal, characteristic
5C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
6#64099⋊C3C211yesmaximal, characteristic, complemented
7#64098⋊C3C212yesmaximal, complemented
8#64097⋊C3C212yesmaximal, complemented
9#64097⋊C3C211yesmaximal, characteristic, complemented
10#64081⋊C3C211yesmaximal, characteristic
11#64078⋊C3C211yesmaximal, characteristic, complemented
12#10596160C311yesmaximal, characteristic, complemented
13C3×#5689—31nomaximal
14C2×C4×(C3⋊D3)⋊C3C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2212yes—
17C2×#634⋊C3C2212yes—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2212yes—
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C42×H(3,𝔽3)C2211yescharacteristic
23#3752⋊C3C2211yescharacteristic
24#3752⋊C3C2211yescharacteristic
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2212yes—
30#3750⋊C3C2211yescharacteristic
31#3750⋊C3C2211yescharacteristic
32#3750⋊C3C2212yes—
33#3750⋊C3C2212yes—
34#3750⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#125—21no—
38C32×#125D311yescharacteristic, complemented
39C32×#125C2×C311yescharacteristic, complemented
40#64106C2×C312yes—
41#64105C2×C312yes—
42#64101C2×C311yescharacteristic
43C4⋊C4×C3⋊D3C2×C311yescharacteristic
44#64099C2×C311yescharacteristic, complemented
45#64098C2×C312yescomplemented
46#64097C2×C311yescharacteristic, complemented
47#64097C2×C312yescomplemented
48#64081C2×C311yescharacteristic
49#64078C2×C311yescharacteristic, complemented
50C3×#897—32no—
51C3×#896—32no—
52C3×#892—31no—
53C3×D3×C4⋊C4—31no—
54C3×#890—31no—
55C3×#889—32no—
56C3×#888—32no—
57C3×#888—31no—
58C3×#872—31no—
59C3×#869—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2311yescharacteristic
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×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4×(C3⋊D3)⋊C3—42no—
72C4×(C3⋊D3)⋊C3Q814yescomplemented
73#634⋊C3—42no—
74#634⋊C3—42no—
75C3×#544—91no—
76#5689—31no—
77#5689—91no—
78C2×C4×C3⋊D3C22×C312yes—
79C2×C4×C3⋊D3C22×C311yescharacteristic
80C2×#634C22×C312yes—
81C2×#634C22×C312yes—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—32no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6—32no—
86C32×C4⋊C4—21no—
87C32×C4⋊C4—21no—
88C32×C4⋊C4—22no—
89C32×C4⋊C4—22no—
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C2×D312yes—
92C32×C4⋊C4C2×D312yes—
93C32×C4⋊C4C2×D311yescharacteristic
94C32×C4⋊C4C22×C311yescharacteristic
95C32×C4⋊C4C22×C312yes—
96C32×C4⋊C4C22×C312yes—
97C32×C4⋊C4C22×C311yescharacteristic
98C32×C42—21no—
99C32×C42C2×D311yescharacteristic
100C32×C42C22×C311yescharacteristic
101#3752C22×C311yescharacteristic
102#3752C22×C311yescharacteristic
103#3752C22×C312yes—
104C4⋊(C3⋊Dic3)C22×C311yescharacteristic
105C4⋊(C3⋊Dic3)C22×C312yes—
106C4⋊(C3⋊Dic3)C22×C311yescharacteristic
107C4⋊(C3⋊Dic3)C22×C312yes—
108#3750C22×C311yescharacteristic
109#3750C22×C311yescharacteristic
110#3750C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113C4×C3⋊Dic3C22×C311yescharacteristic
114C4×C3⋊Dic3C22×C312yes—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—31no—
117C3×(C2×D3)⋊C4—31no—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—32no—
121C3×C4⋊Dic3—31no—
122C3×Dic3⋊C4—31no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4—32no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—31no—
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—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3C2×Q812yes—
143C3×C4⋊Q8—91no—
144C3×C4⋊Q8—91no—
145C3×C4⋊Q8—92no—
146C3×#125—31no—
147C3×#125—31no—
148C3×#125—61no—
149C3×#125—92no—
150C3×#125—91no—
151C3×#125C3×D311yescharacteristic
152C3×C22⋊Q8—92no—
153C3×C22⋊Q8—92no—
154C3×#117—91no—
155C3×#117—91no—
156C2×C3×C4⋊C4—91no—
157#897—32no—
158#897—92no—
159#896—32no—
160#896—92no—
161#892—31no—
162#892—91no—
163D3×C4⋊C4—31no—
164D3×C4⋊C4—91no—
165#890—31no—
166#890—91no—
167#889—32no—
168#889—92no—
169#888—31no—
170#888—32no—
171#888—91no—
172#888—92no—
173#872—31no—
174#872—91no—
175#869—31no—
176#869—91no—
177C22×C3⋊D3C23×C311yescharacteristic
178C22×C3×D3—31no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—22no—
183C2×C32×C4—22no—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C23×C312yes—
190C2×C32×C4C23×C312yes—
191C2×C32×C4C23×C311yescharacteristic
192C2×C32×C4C23×C311yescharacteristic
193C2×C32×C4C23×C311yescharacteristic
194C2×C3⋊Dic3C23×C311yescharacteristic
195C2×C3⋊Dic3C23×C311yescharacteristic
196C2×C3⋊Dic3C23×C312yes—
197C2×C3⋊Dic3C23×C312yes—
198C2×C3⋊Dic3C23×C311yescharacteristic
199C4×C3⋊D3—42no—
200C4×C3⋊D3C3×Q814yescomplemented
201#634—42no—
202#634—42no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C3×C4×D3—34no—
209C3×C4×D3—122no—
210C3×Dic6—122no—
211C3×Dic6—122no—
212#544—91noSylow
213C2×H(3,𝔽3)C22×Q811yescharacteristic
214C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
215C2×H(3,𝔽3)2−1+411yescharacteristic
216(C3⋊D3)⋊C3—22no—
217C2×C3×Q8—92no—
218C2×C3×Q8—92no—
219C22×C3×C4—91no—
220C22×C3×C4—92no—
221C2×C4×D3—31no—
222C2×C4×D3—32no—
223C2×C4×D3—91no—
224C2×C4×D3—92no—
225C2×Dic6—32no—
226C2×Dic6—32no—
227C2×Dic6—92no—
228C2×Dic6—92no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—32no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—32no—
236C3×C4⋊C4—32no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—62no—
240C3×C4⋊C4—62no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4—92no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4C2×C3×D311yescharacteristic
251C3×C4⋊C4C2×C3×D312yes—
252C3×C4⋊C4C2×C3×D312yes—
253C3×C4⋊C4C2×C3×D311yescharacteristic
254C3×C22⋊C4—91no—
255C3×C22⋊C4—91no—
256C3×C22⋊C4—92no—
257C3×C42—31no—
258C3×C42—31no—
259C3×C42—61no—
260C3×C42—91no—
261C3×C42—92no—
262C3×C42C2×C3×D311yescharacteristic
263(C2×D3)⋊C4—31no—
264(C2×D3)⋊C4—31no—
265(C2×D3)⋊C4—32no—
266(C2×D3)⋊C4—91no—
267(C2×D3)⋊C4—91no—
268(C2×D3)⋊C4—92no—
269C4⋊Dic3—31no—
270C4⋊Dic3—32no—
271C4⋊Dic3—31no—
272C4⋊Dic3—32no—
273C4⋊Dic3—91no—
274C4⋊Dic3—92no—
275C4⋊Dic3—91no—
276C4⋊Dic3—92no—
277Dic3⋊C4—31no—
278Dic3⋊C4—31no—
279Dic3⋊C4—32no—
280Dic3⋊C4—32no—
281Dic3⋊C4—32no—
282Dic3⋊C4—91no—
283Dic3⋊C4—91no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3—31no—
288C4×Dic3—32no—
289C4×Dic3—91no—
290C4×Dic3—92no—
291C22×C32—21no—
292C22×C32C23×D311yescharacteristic
293C22×C32C24×C311yescharacteristic
294C2×C3⋊D3—21no—
295C2×C3⋊D3—21no—
296C2×C3⋊D3C2×C3×Q812yes—
297C2×C3×D3—32no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C32×C4—21no—
301C32×C4—22no—
302C32×C4—22no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—22no—
306C32×C4—22no—
307C32×C4—21no—
308C32×C4—22no—
309C32×C4—41no—
310C32×C4—42no—
311C32×C4—41no—
312C32×C4—42no—
313C32×C4D3×Q812yes—
314C32×C4C2×C3×Q812yes—
315C3⋊Dic3—21no—
316C3⋊Dic3—22no—
317C3⋊Dic3—22no—
318C3⋊Dic3—21no—
319C3⋊Dic3C2×C3×Q812yes—
320C3×Dic3—32no—
321C3×Dic3—61no—
322C3×Dic3—62no—
323C3×Dic3—62no—
324C3×Dic3—61no—
325C4⋊Q8—92no—
326C4⋊Q8—91no—
327C4⋊Q8—91no—
328#125—92no—
329#125—91no—
330#125(C3⋊D3)⋊C311yescharacteristic, complemented
331C22⋊Q8—92no—
332C22⋊Q8—92no—
333#117—91no—
334#117—91no—
335C2×C4⋊C4—91no—
336H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
337C23×C3—91no—
338C22×D3—31no—
339C22×D3—91no—
340C3×Q8—362no—
341C3×Q8—362no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—32no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—32no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—94no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—92no—
362C2×C3×C4—92no—
363C2×C3×C4—362no—
364C2×C3×C4C22×C3×D312yes—
365C2×C3×C4C22×C3×D312yes—
366C2×C3×C4C22×C3×D311yescharacteristic
367C2×C3×C4C22×C3×D311yescharacteristic
368C2×C3×C4C22×C3×D311yescharacteristic
369C2×Dic3—31no—
370C2×Dic3—31no—
371C2×Dic3—31no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—92no—
378C2×Dic3—92no—
379C4×D3—34no—
380C4×D3—94no—
381C4×D3—122no—
382C4×D3—362no—
383Dic6—122no—
384Dic6—122no—
385Dic6—362no—
386Dic6—362no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32C2×D3×Q811yescharacteristic
391C2×C32#100911yescharacteristic
392C2×C32#101011yescharacteristic
393C2×C32C22×C3×Q811yescharacteristic
394C2×C32C3×H(5,𝔽2)11yescharacteristic
395C2×C32C3×2−1+411yescharacteristic
396C3⋊D3—22no—
397C3×D3—62no—
398C2×Q8—92no—
399C2×Q8—92no—
400C22×C4—91no—
401C22×C4—92no—
402C4⋊C4—92no—
403C4⋊C4—91no—
404C4⋊C4—91no—
405C4⋊C4—91no—
406C4⋊C4—91no—
407C4⋊C4—92no—
408C4⋊C4—92no—
409C4⋊C4—92no—
410C4⋊C4—92no—
411C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
412C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
413C4⋊C4C2×(C3⋊D3)⋊C312yes—
414C4⋊C4C2×(C3⋊D3)⋊C312yes—
415C22⋊C4—91no—
416C22⋊C4—92no—
417C22⋊C4—91no—
418C42—92no—
419C42—91no—
420C42C2×(C3⋊D3)⋊C311yescharacteristic
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—61no—
424C22×C3—92no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3C23×C3×D311yescharacteristic
428C2×D3—32no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—92no—
432C2×D3—181no—
433C2×D3—181no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—32no—
439C3×C4—32no—
440C3×C4—61no—
441C3×C4—62no—
442C3×C4—62no—
443C3×C4—61no—
444C3×C4—62no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—62no—
448C3×C4—62no—
449C3×C4—92no—
450C3×C4—122no—
451C3×C4—121no—
452C3×C4—121no—
453C3×C4—122no—
454C3×C4—182no—
455C3×C4—181no—
456C3×C4—182no—
457C3×C4—181no—
458C3×C4C3×D3×Q812yes—
459Dic3—32no—
460Dic3—61no—
461Dic3—62no—
462Dic3—62no—
463Dic3—61no—
464Dic3—92no—
465Dic3—181no—
466Dic3—182no—
467Dic3—182no—
468Dic3—181no—
469C32—21no—
470C32#568911yescharacteristic, complemented
471C32C3×#54411yescharacteristic, complemented
472C23—91no—
473Q8—362no—
474Q8—362no—
475C2×C4—91no—
476C2×C4—94no—
477C2×C4—91no—
478C2×C4—91no—
479C2×C4—92no—
480C2×C4—92no—
481C2×C4—362no—
482C2×C4C22×(C3⋊D3)⋊C312yes—
483C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
484C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
485C2×C4C22×(C3⋊D3)⋊C312yes—
486C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—61no—
494C2×C3—61no—
495C2×C3—61no—
496C2×C3—182no—
497C2×C3C2×C3×D3×Q811yescharacteristic
498C2×C3C3×#100911yescharacteristic
499C2×C3C3×#101011yescharacteristic
500D3—62no—
501D3—182no—
502C22—92no—
503C22—181no—
504C22—181no—
505C22C23×(C3⋊D3)⋊C311yescharacteristic
506C4—22no—
507C4—21no—
508C4—22no—
509C4—21no—
510C4—92no—
511C4—182no—
512C4—181no—
513C4—182no—
514C4—181no—
515C4Q8×(C3⋊D3)⋊C312yes—
516C3—31no—
517C3—31no—
518C3—61no—
519C3C3×#568911yescharacteristic
520C2—182no—
521C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
522C2#1172263811yescharacteristic
523C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#125C2
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
6. #64099⋊C3C2semidirect
7. #64098⋊C3C2semidirect
8. #64097⋊C3C2semidirect
9. #64097⋊C3C2semidirect
11. #64078⋊C3C2semidirect
12. #10596160C3semidirect
38. C32×#125D3semidirect
39. C32×#125C2×C3semidirect
44. #64099C2×C3semidirect
45. #64098C2×C3semidirect
46. #64097C2×C3semidirect
47. #64097C2×C3semidirect
49. #64078C2×C3semidirect
72. C4×(C3⋊D3)⋊C3Q8semidirect
200. C4×C3⋊D3C3×Q8semidirect
330. #125(C3⋊D3)⋊C3semidirect
336. H(3,𝔽3)#544semidirect
470. C32#5689semidirect
471. 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)×#1252
#64106⋊C32
#64105⋊C32
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64099⋊C32
#64098⋊C32
#64097⋊C32
#64081⋊C32
#64078⋊C32
#105961603
C3×#56893