Finite Group Databasebeta

← #10596114⋊C3 #13152964

Subgroups — #10596114⋊C3 #13152964

Order 1728.

830 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic, complemented
3Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#64139⋊C3C211yesmaximal, characteristic
5#64135⋊C3C211yesmaximal, characteristic, complemented
6#64107⋊C3C212yesmaximal, complemented
7#64106⋊C3C212yesmaximal, complemented
8#64102⋊C3C211yesmaximal, characteristic, complemented
9#64097⋊C3C211yesmaximal, characteristic, complemented
10#64094⋊C3C212yesmaximal, complemented
11#64092⋊C3C212yesmaximal, complemented
12#10596114C311yesmaximal, characteristic, complemented
13C3×#5643—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C2×#638⋊C3C2212yes—
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#634⋊C3C2212yes—
20C4⋊C4×H(3,𝔽3)C2212yes—
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C22⋊C4×H(3,𝔽3)C2212yes—
23#3757⋊C3C2211yescharacteristic
24#3757⋊C3C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28#3752⋊C3C2212yes—
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
32C4⋊((C3⋊Dic3)⋊C3)C2212yes—
33#3750⋊C3C2212yes—
34C4×(C3⋊Dic3)⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
37C32×C22⋊Q8—21no—
38C32×C22⋊Q8D311yescharacteristic, complemented
39C32×C22⋊Q8C2×C311yescharacteristic, complemented
40#64156C2×C311yescharacteristic, complemented
41Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
42#64139C2×C311yescharacteristic
43#64135C2×C311yescharacteristic, complemented
44#64107C2×C312yescomplemented
45#64106C2×C312yescomplemented
46#64102C2×C311yescharacteristic, complemented
47#64097C2×C311yescharacteristic, complemented
48#64094C2×C312yescomplemented
49#64092C2×C312yescomplemented
50C3×#947—31no—
51C3×Q8×Dic3—31no—
52C3×#930—31no—
53C3×#926—31no—
54C3×#898—32no—
55C3×#897—32no—
56C3×#893—31no—
57C3×#888—31no—
58C3×#885—32no—
59C3×#883—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62Q8×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)—22no—
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2312yes—
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68#638⋊C3—42no—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C4⋊((C3⋊D3)⋊C3)—22no—
75C4×(C3⋊D3)⋊C3—42no—
76#634⋊C3—42no—
77C3×#542—91no—
78#5643—31no—
79#5643—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
86C2×#638C22×C312yes—
87C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
88C2×C4×C3⋊D3C22×C312yes—
89C2×#634C22×C312yes—
90C2×C3×#67—32no—
91C2×C3×D12—31no—
92C2×C3×C4×D3—32no—
93C2×C3×Dic6—32no—
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×C311yescharacteristic
104#3757C22×C311yescharacteristic
105#3752C22×C312yes—
106#3752C22×C312yes—
107#3752C22×C312yes—
108#3752C22×C312yes—
109C4⋊(C3⋊Dic3)C22×C312yes—
110C4⋊(C3⋊Dic3)C22×C311yescharacteristic
111C4⋊(C3⋊Dic3)C22×C311yescharacteristic
112C4⋊(C3⋊Dic3)C22×C311yescharacteristic
113#3750C22×C312yes—
114C4×C3⋊Dic3C22×C312yes—
115C4×C3⋊Dic3C22×C311yescharacteristic
116C4×C3⋊Dic3C22×C311yescharacteristic
117C3×#217—31no—
118C3×#217—31no—
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×C4⋊Dic3—32no—
124C3×C4⋊Dic3—31no—
125C3×C4⋊Dic3—31no—
126C3×C4⋊Dic3—31no—
127C3×Dic3⋊C4—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—31no—
130C3×C4×Dic3—31no—
131C22×H(3,𝔽3)—21no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)C2411yescharacteristic
135C2×(C3⋊D3)⋊C3—22no—
136C2×(C3⋊D3)⋊C3—22no—
137C2×(C3⋊D3)⋊C3—22no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)—22no—
140C4×H(3,𝔽3)—22no—
141C4×H(3,𝔽3)#4112yes—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3—22no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—21no—
147C3×C4⋊Q8—91no—
148C3×#126—92no—
149C3×#124—92no—
150C3×#124—91no—
151C3×#123—92no—
152C3×C22⋊Q8—31no—
153C3×C22⋊Q8—31no—
154C3×C22⋊Q8—61no—
155C3×C22⋊Q8—92no—
156C3×C22⋊Q8C3×D311yescharacteristic
157C3×#121—91no—
158C3×C4×Q8—91no—
159C3×C4×D4—91no—
160C3×#117—91no—
161#947—31no—
162#947—91no—
163Q8×Dic3—31no—
164Q8×Dic3—91no—
165#930—31no—
166#930—91no—
167#926—31no—
168#926—91no—
169#898—32no—
170#898—92no—
171#897—32no—
172#897—92no—
173#893—31no—
174#893—91no—
175#888—31no—
176#888—91no—
177#885—32no—
178#885—92no—
179#883—32no—
180#883—92no—
181C23×C32—21no—
182C23×C32C22×D311yescharacteristic
183C23×C32C23×C311yescharacteristic
184C22×C3⋊D3C23×C312yes—
185C22×C3×D3—32no—
186C32×Q8—22no—
187C32×Q8—22no—
188C32×Q8—42no—
189C2×C32×C4—22no—
190C2×C32×C4—22no—
191C2×C32×C4—21no—
192C2×C32×C4—22no—
193C2×C32×C4—22no—
194C2×C32×C4—21no—
195C2×C32×C4—42no—
196C2×C32×C4C22×D312yes—
197C2×C32×C4C22×D312yes—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C23×C312yes—
201C2×C32×C4C23×C312yes—
202C2×C32×C4C23×C311yescharacteristic
203C2×C32×C4C23×C311yescharacteristic
204#638—42no—
205C2×C3⋊Dic3C23×C311yescharacteristic
206C2×C3⋊Dic3C23×C311yescharacteristic
207C2×C3⋊Dic3C23×C312yes—
208C2×C3⋊Dic3C23×C311yescharacteristic
209C2×C3⋊Dic3C23×C311yescharacteristic
210C4⋊(C3⋊D3)—22no—
211C4×C3⋊D3—42no—
212#634—42no—
213C3×#67—122no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—31no—
219C3×D12—62no—
220C3×C4×D3—122no—
221C3×Dic6—122no—
222#542—91noSylow
223C2×H(3,𝔽3)—41no—
224C2×H(3,𝔽3)C2×#4111yescharacteristic
225C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
226C2×H(3,𝔽3)2−1+411yescharacteristic
227(C3⋊D3)⋊C3—42no—
228C2×C3×Q8—31no—
229C2×C3×Q8—31no—
230C2×C3×Q8—61no—
231C2×C3×Q8—92no—
232C2×C3×Q8C2×C3×D311yescharacteristic
233C2×C3×D4—91no—
234C2×C3×D4—92no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—61no—
238C22×C3×C4—92no—
239C22×C3×C4C2×C3×D311yescharacteristic
240C2×#67—32no—
241C2×#67—92no—
242C2×D12—31no—
243C2×D12—91no—
244C2×C4×D3—32no—
245C2×C4×D3—92no—
246C2×Dic6—32no—
247C2×Dic6—92no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—32no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—32no—
252C3×C4⋊C4—61no—
253C3×C4⋊C4—62no—
254C3×C4⋊C4—92no—
255C3×C4⋊C4—91no—
256C3×C4⋊C4—91no—
257C3×C4⋊C4—91no—
258C3×C4⋊C4—92no—
259C3×C4⋊C4C2×C3×D311yescharacteristic
260C3×C4⋊C4C2×C3×D312yes—
261C3×C22⋊C4—32no—
262C3×C22⋊C4—32no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—92no—
265C3×C22⋊C4—92no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—91no—
269C3×C22⋊C4—91no—
270C3×C22⋊C4C2×C3×D312yes—
271C3×C42—91no—
272C3×C42—91no—
273C3×C42—92no—
274#217—31no—
275#217—31no—
276#217—91no—
277#217—91no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—32no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286C4⋊Dic3—32no—
287C4⋊Dic3—31no—
288C4⋊Dic3—31no—
289C4⋊Dic3—31no—
290C4⋊Dic3—92no—
291C4⋊Dic3—91no—
292C4⋊Dic3—91no—
293C4⋊Dic3—91no—
294Dic3⋊C4—32no—
295Dic3⋊C4—92no—
296C4×Dic3—32no—
297C4×Dic3—31no—
298C4×Dic3—31no—
299C4×Dic3—92no—
300C4×Dic3—91no—
301C4×Dic3—91no—
302C22×C32—22no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—21no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—41no—
311C22×C32—41no—
312C22×C32C23×D311yescharacteristic
313C22×C32C24×C311yescharacteristic
314C2×C3⋊D3—22no—
315C2×C3⋊D3—22no—
316C2×C3⋊D3—22no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C32×C4—21no—
321C32×C4—22no—
322C32×C4—22no—
323C32×C4—21no—
324C32×C4—22no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—42no—
328C32×C4—41no—
329C32×C4—42no—
330C32×C4#23712yes—
331C32×C4C3×#4112yes—
332C3⋊Dic3—21no—
333C3⋊Dic3—22no—
334C3⋊Dic3—21no—
335C3⋊Dic3—21no—
336C3⋊Dic3—21no—
337C3×Dic3—61no—
338C3×Dic3—61no—
339C3×Dic3—61no—
340C3×Dic3—62no—
341C3×Dic3—61no—
342C4⋊Q8—91no—
343#126—92no—
344#124—91no—
345#124—92no—
346#123—92no—
347C22⋊Q8—92no—
348C22⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
349#121—91no—
350C4×Q8—91no—
351C4×D4—91no—
352#117—91no—
353H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
354C23×C3—31no—
355C23×C3—31no—
356C23×C3—61no—
357C23×C3—92no—
358C23×C3C22×C3×D311yescharacteristic
359C22×D3—32no—
360C22×D3—92no—
361C3×Q8—22no—
362C3×Q8—62no—
363C3×Q8—62no—
364C3×Q8—122no—
365C3×Q8—362no—
366C3×D4—182no—
367C3×D4—362no—
368C2×C3×C4—22no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—32no—
372C2×C3×C4—32no—
373C2×C3×C4—31no—
374C2×C3×C4—32no—
375C2×C3×C4—32no—
376C2×C3×C4—31no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—61no—
380C2×C3×C4—62no—
381C2×C3×C4—62no—
382C2×C3×C4—61no—
383C2×C3×C4—91no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—92no—
388C2×C3×C4—122no—
389C2×C3×C4—362no—
390C2×C3×C4C22×C3×D312yes—
391C2×C3×C4C22×C3×D312yes—
392C2×C3×C4C22×C3×D311yescharacteristic
393C2×C3×C4C22×C3×D311yescharacteristic
394#67—122no—
395#67—362no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—32no—
399C2×Dic3—31no—
400C2×Dic3—31no—
401C2×Dic3—91no—
402C2×Dic3—91no—
403C2×Dic3—92no—
404C2×Dic3—91no—
405C2×Dic3—91no—
406D12—62no—
407D12—182no—
408C4×D3—122no—
409C4×D3—362no—
410Dic6—122no—
411Dic6—362no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—42no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32C2×#23711yescharacteristic
419C2×C32#100711yescharacteristic
420C2×C32#100911yescharacteristic
421C2×C32C2×C3×#4111yescharacteristic
422C2×C32C3×H(5,𝔽2)11yescharacteristic
423C2×C32C3×2−1+411yescharacteristic
424C3⋊D3—42no—
425C3×D3—122no—
426C2×Q8—92no—
427C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
428C2×D4—92no—
429C2×D4—91no—
430C22×C4—92no—
431C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
432C4⋊C4—91no—
433C4⋊C4—91no—
434C4⋊C4—91no—
435C4⋊C4—92no—
436C4⋊C4—92no—
437C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
438C4⋊C4C2×(C3⋊D3)⋊C312yes—
439C22⋊C4—91no—
440C22⋊C4—91no—
441C22⋊C4—92no—
442C22⋊C4—92no—
443C22⋊C4—92no—
444C22⋊C4—92no—
445C22⋊C4C2×(C3⋊D3)⋊C312yes—
446C42—91no—
447C42—91no—
448C42—92no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—31no—
453C22×C3—31no—
454C22×C3—61no—
455C22×C3—61no—
456C22×C3—61no—
457C22×C3—61no—
458C22×C3—61no—
459C22×C3—61no—
460C22×C3—62no—
461C22×C3—61no—
462C22×C3—121no—
463C22×C3—121no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3C23×C3×D311yescharacteristic
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—182no—
472C2×D3—182no—
473C2×D3—182no—
474C3×C4—21no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—32no—
478C3×C4—32no—
479C3×C4—62no—
480C3×C4—61no—
481C3×C4—61no—
482C3×C4—62no—
483C3×C4—62no—
484C3×C4—62no—
485C3×C4—62no—
486C3×C4—121no—
487C3×C4—122no—
488C3×C4—122no—
489C3×C4—181no—
490C3×C4—182no—
491C3×C4—181no—
492C3×C4—181no—
493C3×C4—181no—
494C3×C4C3×#23712yes—
495Dic3—61no—
496Dic3—61no—
497Dic3—61no—
498Dic3—62no—
499Dic3—61no—
500Dic3—181no—
501Dic3—182no—
502Dic3—181no—
503Dic3—181no—
504Dic3—181no—
505C32—21no—
506C32#564311yescharacteristic, complemented
507C32C3×#54211yescharacteristic, complemented
508C23—92no—
509C23C22×(C3⋊D3)⋊C311yescharacteristic
510Q8—22no—
511Q8—362no—
512D4—182no—
513D4—362no—
514C2×C4—22no—
515C2×C4—91no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—91no—
519C2×C4—92no—
520C2×C4—362no—
521C2×C4C22×(C3⋊D3)⋊C312yes—
522C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
523C2×C4C22×(C3⋊D3)⋊C312yes—
524C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—41no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—61no—
535C2×C3—121no—
536C2×C3—121no—
537C2×C3—122no—
538C2×C3—362no—
539C2×C3C2×C3×#23711yescharacteristic
540C2×C3C3×#100711yescharacteristic
541C2×C3C3×#100911yescharacteristic
542D3—122no—
543D3—362no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—182no—
548C22—182no—
549C22—182no—
550C22C23×(C3⋊D3)⋊C311yescharacteristic
551C4—21no—
552C4—22no—
553C4—22no—
554C4—181no—
555C4—181no—
556C4—181no—
557C4—182no—
558C4—181no—
559C4#3830⋊C312yes—
560C3—31no—
561C3—31no—
562C3—61no—
563C3C3×#564311yescharacteristic
564C2—41no—
565C2—362no—
566C2C2×#3830⋊C311yescharacteristic
567C2#1172262711yescharacteristic
568C2#1172263811yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radical#64139⋊C3C2
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C22⋊Q8C2semidirect
2. #64156⋊C3C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
5. #64135⋊C3C2semidirect
6. #64107⋊C3C2semidirect
7. #64106⋊C3C2semidirect
8. #64102⋊C3C2semidirect
9. #64097⋊C3C2semidirect
10. #64094⋊C3C2semidirect
11. #64092⋊C3C2semidirect
12. #10596114C3semidirect
38. C32×C22⋊Q8D3semidirect
39. C32×C22⋊Q8C2×C3semidirect
40. #64156C2×C3semidirect
41. Q8×C3⋊Dic3C2×C3semidirect
43. #64135C2×C3semidirect
44. #64107C2×C3semidirect
45. #64106C2×C3semidirect
46. #64102C2×C3semidirect
47. #64097C2×C3semidirect
48. #64094C2×C3semidirect
49. #64092C2×C3semidirect
348. C22⋊Q8(C3⋊D3)⋊C3semidirect
353. H(3,𝔽3)#542semidirect
506. C32#5643semidirect
507. C32C3×#542semidirect

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
#64156⋊C32
Q8×(C3⋊Dic3)⋊C32
#64139⋊C32
#64135⋊C32
#64107⋊C32
#64106⋊C32
#64102⋊C32
#64097⋊C32
#64094⋊C32
#64092⋊C32
#105961143
C3×#56433