Finite Group Databasebeta

← #10596161⋊C3 #13153181

Subgroups — #10596161⋊C3 #13153181

Order 1728.

854 conjugacy classes of subgroups.

The subgroup lattice has 539 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, complemented
3#64105⋊C3C212yesmaximal, complemented
4#64105⋊C3C212yesmaximal, complemented
5#64102⋊C3C211yesmaximal, characteristic
6#64102⋊C3C211yesmaximal, characteristic
7C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
8#64099⋊C3C211yesmaximal, characteristic, complemented
9#64097⋊C3C211yesmaximal, characteristic, complemented
10C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
11C4×#634⋊C3C211yesmaximal, characteristic, complemented
12#10596161C311yesmaximal, characteristic, complemented
13C3×#5690—31nomaximal
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic
19C2×#634⋊C3C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C4⋊C4×H(3,𝔽3)C2212yes—
23C4⋊C4×H(3,𝔽3)C2212yes—
24C42×H(3,𝔽3)C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2212yes—
31#3750⋊C3C2211yescharacteristic
32#3750⋊C3C2211yescharacteristic
33#3750⋊C3C2212yes—
34#3750⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
37C32×#125—21no—
38C32×#125D311yescharacteristic, complemented
39C32×#125C2×C311yescharacteristic, complemented
40#64106C2×C312yescomplemented
41#64105C2×C312yescomplemented
42#64105C2×C312yescomplemented
43#64102C2×C311yescharacteristic
44#64102C2×C311yescharacteristic
45C4⋊C4×C3⋊D3C2×C312yescomplemented
46#64099C2×C311yescharacteristic, complemented
47#64097C2×C311yescharacteristic, complemented
48C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
49C4×#634C2×C311yescharacteristic, complemented
50C3×#897—32no—
51C3×#896—32no—
52C3×#896—32no—
53C3×#893—31no—
54C3×#893—31no—
55C3×D3×C4⋊C4—32no—
56C3×#890—31no—
57C3×#888—31no—
58C3×C4×D12—31no—
59C3×C4×Dic6—31no—
60C22×(C3⋊D3)⋊C3C2312yes—
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4⋊((C3⋊D3)⋊C3)Q814yescomplemented
72C4×(C3⋊D3)⋊C3—24no—
73C4×(C3⋊D3)⋊C3—42no—
74C4×(C3⋊D3)⋊C3—42no—
75#634⋊C3—22no—
76#634⋊C3—42no—
77C3×#554—91no—
78#5690—31no—
79#5690—91no—
80C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
81C2×C4×C3⋊D3C22×C312yes—
82C2×C4×C3⋊D3C22×C312yes—
83C2×C4×C3⋊D3C22×C312yes—
84C2×#634C22×C311yescharacteristic
85C2×#634C22×C312yes—
86C2×C3×D12—31no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—32no—
92C32×C4⋊C4—21no—
93C32×C4⋊C4—21no—
94C32×C4⋊C4—22no—
95C32×C4⋊C4—22no—
96C32×C4⋊C4C2×D311yescharacteristic
97C32×C4⋊C4C2×D311yescharacteristic
98C32×C4⋊C4C2×D312yes—
99C32×C4⋊C4C2×D312yes—
100C32×C4⋊C4C22×C311yescharacteristic
101C32×C4⋊C4C22×C311yescharacteristic
102C32×C4⋊C4C22×C312yes—
103C32×C4⋊C4C22×C312yes—
104C32×C42—21no—
105C32×C42C2×D311yescharacteristic
106C32×C42C22×C311yescharacteristic
107#3752C22×C312yes—
108#3752C22×C312yes—
109#3752C22×C312yes—
110C4⋊(C3⋊Dic3)C22×C311yescharacteristic
111C4⋊(C3⋊Dic3)C22×C311yescharacteristic
112C4⋊(C3⋊Dic3)C22×C312yes—
113#3750C22×C311yescharacteristic
114#3750C22×C311yescharacteristic
115#3750C22×C312yes—
116#3750C22×C312yes—
117C4×C3⋊Dic3C22×C311yescharacteristic
118C4×C3⋊Dic3C22×C311yescharacteristic
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—32no—
124C3×C4⋊Dic3—31no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—31no—
128C3×Dic3⋊C4—31no—
129C3×C4×Dic3—31no—
130C3×C4×Dic3—31no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C2×(C3⋊D3)⋊C3—22no—
133C2×(C3⋊D3)⋊C3—22no—
134C2×(C3⋊D3)⋊C3C2×Q814yes—
135C4×H(3,𝔽3)—21no—
136C4×H(3,𝔽3)—22no—
137C4×H(3,𝔽3)—22no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)C2×Q812yes—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3#4112yes—
145C3×C4⋊Q8—91no—
146C3×#125—31no—
147C3×#125—31no—
148C3×#125—61no—
149C3×#125—91no—
150C3×#125C3×D311yescharacteristic
151C3×C22⋊Q8—92no—
152C3×C22⋊Q8—92no—
153C3×C22⋊Q8—92no—
154C3×C4×Q8—91no—
155C3×C4×D4—91no—
156C3×C4×D4—91no—
157C3×C4×D4—91no—
158C2×C3×C4⋊C4—92no—
159#897—32no—
160#897—92no—
161#896—32no—
162#896—32no—
163#896—92no—
164#896—92no—
165#893—31no—
166#893—31no—
167#893—91no—
168#893—91no—
169D3×C4⋊C4—32no—
170D3×C4⋊C4—92no—
171#890—31no—
172#890—91no—
173#888—31no—
174#888—91no—
175C4×D12—31no—
176C4×D12—91no—
177C4×Dic6—31no—
178C4×Dic6—91no—
179C22×C3⋊D3C23×C312yes—
180C22×C3×D3—32no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—22no—
185C2×C32×C4—22no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D312yes—
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192C2×C32×C4C23×C312yes—
193C2×C32×C4C23×C312yes—
194C2×C32×C4C23×C311yescharacteristic
195C2×C32×C4C23×C311yescharacteristic
196C2×C3⋊Dic3C23×C311yescharacteristic
197C2×C3⋊Dic3C23×C311yescharacteristic
198C2×C3⋊Dic3C23×C311yescharacteristic
199C2×C3⋊Dic3C23×C312yes—
200C2×C3⋊Dic3C23×C311yescharacteristic
201C4⋊(C3⋊D3)C3×Q814yescomplemented
202C4×C3⋊D3—24no—
203C4×C3⋊D3—42no—
204C4×C3⋊D3—42no—
205#634—22no—
206#634—42no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—32no—
212C3×D12—34no—
213C3×C4×D3—64no—
214C3×C4×D3—122no—
215C3×C4×D3—122no—
216C3×Dic6—62no—
217C3×Dic6—122no—
218#554—91noSylow
219C2×H(3,𝔽3)C22×Q811yescharacteristic
220C2×H(3,𝔽3)C2×#4111yescharacteristic
221C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
222(C3⋊D3)⋊C3—24no—
223C2×C3×Q8—91no—
224C2×C3×Q8—92no—
225C2×C3×D4—91no—
226C22×C3×C4—92no—
227C22×C3×C4—92no—
228C22×C3×C4—92no—
229C2×D12—31no—
230C2×D12—91no—
231C2×C4×D3—32no—
232C2×C4×D3—32no—
233C2×C4×D3—32no—
234C2×C4×D3—92no—
235C2×C4×D3—92no—
236C2×C4×D3—92no—
237C2×Dic6—31no—
238C2×Dic6—32no—
239C2×Dic6—91no—
240C2×Dic6—92no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—32no—
244C3×C4⋊C4—32no—
245C3×C4⋊C4—32no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—32no—
249C3×C4⋊C4—62no—
250C3×C4⋊C4—61no—
251C3×C4⋊C4—61no—
252C3×C4⋊C4—62no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—92no—
255C3×C4⋊C4—91no—
256C3×C4⋊C4—92no—
257C3×C4⋊C4—92no—
258C3×C4⋊C4—91no—
259C3×C4⋊C4—91no—
260C3×C4⋊C4C2×C3×D311yescharacteristic
261C3×C4⋊C4C2×C3×D311yescharacteristic
262C3×C4⋊C4C2×C3×D312yes—
263C3×C4⋊C4C2×C3×D312yes—
264C3×C22⋊C4—92no—
265C3×C22⋊C4—92no—
266C3×C22⋊C4—92no—
267C3×C42—31no—
268C3×C42—31no—
269C3×C42—61no—
270C3×C42—91no—
271C3×C42—91no—
272C3×C42C2×C3×D311yescharacteristic
273(C2×D3)⋊C4—32no—
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—32no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279C4⋊Dic3—31no—
280C4⋊Dic3—31no—
281C4⋊Dic3—32no—
282C4⋊Dic3—91no—
283C4⋊Dic3—91no—
284C4⋊Dic3—92no—
285Dic3⋊C4—31no—
286Dic3⋊C4—31no—
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—91no—
290Dic3⋊C4—91no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293C4×Dic3—31no—
294C4×Dic3—31no—
295C4×Dic3—91no—
296C4×Dic3—91no—
297C22×C32—21no—
298C22×C32C23×D311yescharacteristic
299C22×C32C24×C311yescharacteristic
300C2×C3⋊D3—22no—
301C2×C3⋊D3—22no—
302C2×C3⋊D3C2×C3×Q814yes—
303C2×C3×D3—34no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306C32×C4—21no—
307C32×C4—22no—
308C32×C4—22no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—22no—
312C32×C4—22no—
313C32×C4—21no—
314C32×C4—22no—
315C32×C4—41no—
316C32×C4—42no—
317C32×C4—42no—
318C32×C4—41no—
319C32×C4D3×Q812yes—
320C32×C4C2×C3×Q812yes—
321C3⋊Dic3—21no—
322C3⋊Dic3—21no—
323C3⋊Dic3—22no—
324C3⋊Dic3—21no—
325C3⋊Dic3C3×#4112yes—
326C3×Dic3—32no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—62no—
330C3×Dic3—61no—
331C4⋊Q8—91no—
332#125—91no—
333#125(C3⋊D3)⋊C311yescharacteristic, complemented
334C22⋊Q8—92no—
335C22⋊Q8—92no—
336C22⋊Q8—92no—
337C4×Q8—91no—
338C4×D4—91no—
339C4×D4—91no—
340C4×D4—91no—
341C2×C4⋊C4—92no—
342H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
343C23×C3—92no—
344C22×D3—32no—
345C22×D3—92no—
346C3×Q8—182no—
347C3×Q8—362no—
348C3×D4—94no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—32no—
354C2×C3×C4—32no—
355C2×C3×C4—31no—
356C2×C3×C4—32no—
357C2×C3×C4—32no—
358C2×C3×C4—31no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—92no—
369C2×C3×C4—184no—
370C2×C3×C4—362no—
371C2×C3×C4—362no—
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×C3×C4C22×C3×D312yes—
374C2×C3×C4C22×C3×D312yes—
375C2×C3×C4C22×C3×D311yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—31no—
381C2×Dic3—32no—
382C2×Dic3—91no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—92no—
387D12—34no—
388D12—94no—
389C4×D3—64no—
390C4×D3—122no—
391C4×D3—122no—
392C4×D3—184no—
393C4×D3—362no—
394C4×D3—362no—
395Dic6—62no—
396Dic6—122no—
397Dic6—182no—
398Dic6—362no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32C2×D3×Q811yescharacteristic
403C2×C32D3×#4111yescharacteristic
404C2×C32#100911yescharacteristic
405C2×C32C22×C3×Q811yescharacteristic
406C2×C32C2×C3×#4111yescharacteristic
407C2×C32C3×H(5,𝔽2)11yescharacteristic
408C3⋊D3—24no—
409C3×D3—64no—
410C2×Q8—91no—
411C2×Q8—92no—
412C2×D4—91no—
413C22×C4—92no—
414C22×C4—92no—
415C22×C4—92no—
416C4⋊C4—91no—
417C4⋊C4—92no—
418C4⋊C4—92no—
419C4⋊C4—91no—
420C4⋊C4—91no—
421C4⋊C4—91no—
422C4⋊C4—92no—
423C4⋊C4C2×(C3⋊D3)⋊C312yes—
424C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
425C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
426C4⋊C4C2×(C3⋊D3)⋊C312yes—
427C22⋊C4—92no—
428C22⋊C4—92no—
429C22⋊C4—92no—
430C42—91no—
431C42—91no—
432C42C2×(C3⋊D3)⋊C311yescharacteristic
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—61no—
436C22×C3—94no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3C23×C3×D311yescharacteristic
440C2×D3—34no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—94no—
444C2×D3—182no—
445C2×D3—182no—
446C3×C4—21no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—21no—
450C3×C4—32no—
451C3×C4—32no—
452C3×C4—61no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4—61no—
456C3×C4—62no—
457C3×C4—61no—
458C3×C4—62no—
459C3×C4—61no—
460C3×C4—62no—
461C3×C4—92no—
462C3×C4—122no—
463C3×C4—121no—
464C3×C4—121no—
465C3×C4—122no—
466C3×C4—181no—
467C3×C4—182no—
468C3×C4—181no—
469C3×C4—181no—
470C3×C4C3×D3×Q812yes—
471Dic3—32no—
472Dic3—61no—
473Dic3—61no—
474Dic3—62no—
475Dic3—61no—
476Dic3—92no—
477Dic3—181no—
478Dic3—181no—
479Dic3—182no—
480Dic3—181no—
481C32—21no—
482C32#569011yescharacteristic, complemented
483C32C3×#55411yescharacteristic, complemented
484C23—92no—
485Q8—182no—
486Q8—362no—
487D4—94no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—92no—
493C2×C4—184no—
494C2×C4—362no—
495C2×C4—362no—
496C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
497C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
498C2×C4C22×(C3⋊D3)⋊C312yes—
499C2×C4C22×(C3⋊D3)⋊C312yes—
500C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
501C2×C3—31no—
502C2×C3—31no—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—61no—
508C2×C3—61no—
509C2×C3—61no—
510C2×C3—184no—
511C2×C3C2×C3×D3×Q811yescharacteristic
512C2×C3C3×D3×#4111yescharacteristic
513C2×C3C3×#100911yescharacteristic
514D3—64no—
515D3—184no—
516C22—94no—
517C22—182no—
518C22—182no—
519C22C23×(C3⋊D3)⋊C311yescharacteristic
520C4—22no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—92no—
525C4—181no—
526C4—181no—
527C4—182no—
528C4—181no—
529C4Q8×(C3⋊D3)⋊C312yes—
530C3—31no—
531C3—31no—
532C3—61no—
533C3C3×#569011yescharacteristic
534C2—184no—
535C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
536C2#41×(C3⋊D3)⋊C311yescharacteristic
537C2#1172263811yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#125C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#554
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
2. #64106⋊C3C2semidirect
3. #64105⋊C3C2semidirect
4. #64105⋊C3C2semidirect
7. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
8. #64099⋊C3C2semidirect
9. #64097⋊C3C2semidirect
11. C4×#634⋊C3C2semidirect
12. #10596161C3semidirect
38. C32×#125D3semidirect
39. C32×#125C2×C3semidirect
40. #64106C2×C3semidirect
41. #64105C2×C3semidirect
42. #64105C2×C3semidirect
45. C4⋊C4×C3⋊D3C2×C3semidirect
46. #64099C2×C3semidirect
47. #64097C2×C3semidirect
49. C4×#634C2×C3semidirect
71. C4⋊((C3⋊D3)⋊C3)Q8semidirect
201. C4⋊(C3⋊D3)C3×Q8semidirect
333. #125(C3⋊D3)⋊C3semidirect
342. H(3,𝔽3)#554semidirect
482. C32#5690semidirect
483. C32C3×#554semidirect

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
#64102⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64099⋊C32
#64097⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#105961613
C3×#56903