Finite Group Databasebeta

← #10596051⋊C3 #13152721

Subgroups — #10596051⋊C3 #13152721

Order 1728.

889 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic, complemented
3#64155⋊C3C211yesmaximal, characteristic, complemented
4#64107⋊C3C212yesmaximal, complemented
5#64104⋊C3C211yesmaximal, characteristic
6#64103⋊C3C212yesmaximal, complemented
7#64099⋊C3C211yesmaximal, characteristic, complemented
8C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
9C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
10#64081⋊C3C212yesmaximal, complemented
11C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
12#10596051C311yesmaximal, characteristic, complemented
13C3×#5580—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
20C2×C4×(C3⋊D3)⋊C3C2212yes—
21C4⋊C4×H(3,𝔽3)C2212yes—
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C42×H(3,𝔽3)C2211yescharacteristic
24C42×H(3,𝔽3)C2212yes—
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28#3752⋊C3C2212yes—
29#3752⋊C3C2212yes—
30C4⋊((C3⋊Dic3)⋊C3)C2212yes—
31C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
32#3750⋊C3C2212yes—
33#3750⋊C3C2212yes—
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
37C32×C4×Q8—21no—
38C32×C4×Q8D311yescharacteristic, complemented
39C32×C4×Q8C2×C311yescharacteristic, complemented
40#64156C2×C311yescharacteristic, complemented
41#64155C2×C311yescharacteristic, complemented
42#64107C2×C312yescomplemented
43#64104C2×C311yescharacteristic
44#64103C2×C312yescomplemented
45#64099C2×C311yescharacteristic, complemented
46C4×C4⋊(C3⋊D3)C2×C312yescomplemented
47C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
48#64081C2×C312yescomplemented
49C42×C3⋊D3C2×C311yescharacteristic, complemented
50C3×#947—31no—
51C3×#946—31no—
52C3×#898—32no—
53C3×#895—31no—
54C3×#894—32no—
55C3×#890—31no—
56C3×C4×D12—32no—
57C3×C4×D12—31no—
58C3×#872—32no—
59C3×C42×D3—31no—
60C22×(C3⋊D3)⋊C3C2312yes—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62Q8×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2312yes—
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C4⋊((C3⋊D3)⋊C3)—24no—
73C4⋊((C3⋊D3)⋊C3)—22no—
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—22no—
76C4×(C3⋊D3)⋊C3—22no—
77C4×(C3⋊D3)⋊C3—24no—
78C3×#529—91no—
79#5580—31no—
80#5580—91no—
81C2×C32×Q8—21no—
82C2×C32×Q8C2×D311yescharacteristic
83C2×C32×Q8C22×C311yescharacteristic
84C2×C4⋊(C3⋊D3)C22×C312yes—
85C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
86C2×C4×C3⋊D3C22×C311yescharacteristic
87C2×C4×C3⋊D3C22×C311yescharacteristic
88C2×C4×C3⋊D3C22×C311yescharacteristic
89C2×C4×C3⋊D3C22×C312yes—
90C2×C3×D12—32no—
91C2×C3×D12—31no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3—32no—
96C32×C4⋊C4—21no—
97C32×C4⋊C4—22no—
98C32×C4⋊C4C2×D311yescharacteristic
99C32×C4⋊C4C2×D312yes—
100C32×C4⋊C4C22×C311yescharacteristic
101C32×C4⋊C4C22×C312yes—
102C32×C42—21no—
103C32×C42—22no—
104C32×C42C2×D311yescharacteristic
105C32×C42C2×D312yes—
106C32×C42C22×C311yescharacteristic
107C32×C42C22×C312yes—
108#3752C22×C312yes—
109#3752C22×C312yes—
110#3752C22×C312yes—
111#3752C22×C312yes—
112#3752C22×C312yes—
113C4⋊(C3⋊Dic3)C22×C312yes—
114C4⋊(C3⋊Dic3)C22×C311yescharacteristic
115#3750C22×C312yes—
116#3750C22×C312yes—
117C4×C3⋊Dic3C22×C311yescharacteristic
118C4×C3⋊Dic3C22×C311yescharacteristic
119C4×C3⋊Dic3C22×C311yescharacteristic
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×C4⋊Dic3—32no—
126C3×C4⋊Dic3—31no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×C4×Dic3—31no—
130C3×C4×Dic3—31no—
131C3×C4×Dic3—31no—
132C22×H(3,𝔽3)C2411yescharacteristic
133C2×(C3⋊D3)⋊C3—21no—
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—22no—
136C2×(C3⋊D3)⋊C3—22no—
137C2×(C3⋊D3)⋊C3—22no—
138C2×(C3⋊D3)⋊C3#4112yes—
139C4×H(3,𝔽3)—22no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)—22no—
142C4×H(3,𝔽3)#4112yes—
143C4×H(3,𝔽3)#4112yes—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—21no—
148C3×#126—92no—
149C3×#125—91no—
150C3×#124—91no—
151C3×#123—92no—
152C3×C22⋊Q8—91no—
153C3×#121—91no—
154C3×C4×Q8—31no—
155C3×C4×Q8—31no—
156C3×C4×Q8—61no—
157C3×C4×Q8C3×D311yescharacteristic
158C3×C4×D4—91no—
159C3×C4×D4—92no—
160C3×#117—92no—
161C2×C3×C42—91no—
162#947—31no—
163#947—91no—
164#946—31no—
165#946—91no—
166#898—32no—
167#898—92no—
168#895—31no—
169#895—91no—
170#894—32no—
171#894—92no—
172#890—31no—
173#890—91no—
174C4×D12—32no—
175C4×D12—31no—
176C4×D12—92no—
177C4×D12—91no—
178#872—32no—
179#872—92no—
180C42×D3—31no—
181C42×D3—91no—
182C22×C3⋊D3C23×C312yes—
183C22×C3⋊D3C23×C311yescharacteristic
184C22×C3×D3—31no—
185C22×C3×D3—32no—
186C32×Q8—24no—
187C32×Q8—22no—
188C32×Q8—22no—
189C2×C32×C4—21no—
190C2×C32×C4—22no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—22no—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D312yes—
196C2×C32×C4C22×D312yes—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C23×C311yescharacteristic
200C2×C32×C4C23×C312yes—
201C2×C32×C4C23×C312yes—
202C2×C32×C4C23×C311yescharacteristic
203C2×C32×C4C23×C311yescharacteristic
204C2×C3⋊Dic3C23×C311yescharacteristic
205C2×C3⋊Dic3C23×C311yescharacteristic
206C2×C3⋊Dic3C23×C311yescharacteristic
207C2×C3⋊Dic3C23×C312yes—
208C4⋊(C3⋊D3)—24no—
209C4⋊(C3⋊D3)—22no—
210C4×C3⋊D3—22no—
211C4×C3⋊D3—22no—
212C4×C3⋊D3—22no—
213C4×C3⋊D3—24no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—32no—
218C3×D12—64no—
219C3×D12—62no—
220C3×C4×D3—62no—
221C3×C4×D3—62no—
222C3×C4×D3—62no—
223C3×C4×D3—64no—
224#529—91noSylow
225C2×H(3,𝔽3)C2×#4111yescharacteristic
226C2×H(3,𝔽3)C2×#4111yescharacteristic
227C2×H(3,𝔽3)C2×#4111yescharacteristic
228(C3⋊D3)⋊C3—22no—
229(C3⋊D3)⋊C3—42no—
230C2×C3×Q8—31no—
231C2×C3×Q8—31no—
232C2×C3×Q8—61no—
233C2×C3×Q8C2×C3×D311yescharacteristic
234C2×C3×D4—92no—
235C2×C3×D4—91no—
236C22×C3×C4—91no—
237C22×C3×C4—91no—
238C22×C3×C4—91no—
239C22×C3×C4—92no—
240C2×D12—32no—
241C2×D12—31no—
242C2×D12—92no—
243C2×D12—91no—
244C2×C4×D3—31no—
245C2×C4×D3—31no—
246C2×C4×D3—31no—
247C2×C4×D3—32no—
248C2×C4×D3—91no—
249C2×C4×D3—91no—
250C2×C4×D3—91no—
251C2×C4×D3—92no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—32no—
254C3×C4⋊C4—31no—
255C3×C4⋊C4—32no—
256C3×C4⋊C4—61no—
257C3×C4⋊C4—62no—
258C3×C4⋊C4—92no—
259C3×C4⋊C4—92no—
260C3×C4⋊C4—91no—
261C3×C4⋊C4—92no—
262C3×C4⋊C4C2×C3×D311yescharacteristic
263C3×C4⋊C4C2×C3×D312yes—
264C3×C22⋊C4—92no—
265C3×C22⋊C4—92no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C42—31no—
270C3×C42—32no—
271C3×C42—31no—
272C3×C42—32no—
273C3×C42—61no—
274C3×C42—62no—
275C3×C42—91no—
276C3×C42—91no—
277C3×C42—91no—
278C3×C42C2×C3×D311yescharacteristic
279C3×C42C2×C3×D312yes—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—32no—
282(C2×D3)⋊C4—32no—
283(C2×D3)⋊C4—32no—
284(C2×D3)⋊C4—32no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290C4⋊Dic3—32no—
291C4⋊Dic3—31no—
292C4⋊Dic3—92no—
293C4⋊Dic3—91no—
294Dic3⋊C4—32no—
295Dic3⋊C4—32no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—31no—
301C4×Dic3—91no—
302C4×Dic3—91no—
303C4×Dic3—91no—
304C22×C32—21no—
305C22×C32C23×D311yescharacteristic
306C22×C32C24×C311yescharacteristic
307C2×C3⋊D3—21no—
308C2×C3⋊D3—21no—
309C2×C3⋊D3—22no—
310C2×C3⋊D3—22no—
311C2×C3⋊D3—22no—
312C2×C3⋊D3C3×#4112yes—
313C2×C3×D3—32no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C32×C4—24no—
320C32×C4—22no—
321C32×C4—21no—
322C32×C4—22no—
323C32×C4—21no—
324C32×C4—22no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—22no—
328C32×C4—41no—
329C32×C4—42no—
330C32×C4#23512yes—
331C32×C4#23912yes—
332C32×C4C3×#4112yes—
333C32×C4C3×#4112yes—
334C3⋊Dic3—21no—
335C3⋊Dic3—21no—
336C3⋊Dic3—22no—
337C3⋊Dic3—21no—
338C3×Dic3—61no—
339C3×Dic3—61no—
340C3×Dic3—62no—
341C3×Dic3—61no—
342#126—92no—
343#125—91no—
344#124—91no—
345#123—92no—
346C22⋊Q8—91no—
347#121—91no—
348C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
349C4×D4—91no—
350C4×D4—92no—
351#117—92no—
352C2×C42—91no—
353H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
354C23×C3—91no—
355C23×C3—92no—
356C22×D3—31no—
357C22×D3—32no—
358C22×D3—91no—
359C22×D3—92no—
360C3×Q8—22no—
361C3×Q8—62no—
362C3×Q8—62no—
363C3×Q8—64no—
364C3×D4—182no—
365C3×D4—184no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—32no—
370C2×C3×C4—32no—
371C2×C3×C4—31no—
372C2×C3×C4—32no—
373C2×C3×C4—32no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—62no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—62no—
380C2×C3×C4—61no—
381C2×C3×C4—91no—
382C2×C3×C4—91no—
383C2×C3×C4—91no—
384C2×C3×C4—92no—
385C2×C3×C4—182no—
386C2×C3×C4—182no—
387C2×C3×C4—184no—
388C2×C3×C4—182no—
389C2×C3×C4C22×C3×D311yescharacteristic
390C2×C3×C4C22×C3×D312yes—
391C2×C3×C4C22×C3×D312yes—
392C2×C3×C4C22×C3×D311yescharacteristic
393C2×C3×C4C22×C3×D311yescharacteristic
394C2×Dic3—31no—
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—32no—
398C2×Dic3—91no—
399C2×Dic3—91no—
400C2×Dic3—91no—
401C2×Dic3—92no—
402D12—64no—
403D12—62no—
404D12—182no—
405D12—184no—
406C4×D3—62no—
407C4×D3—62no—
408C4×D3—62no—
409C4×D3—64no—
410C4×D3—182no—
411C4×D3—182no—
412C4×D3—184no—
413C4×D3—182no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32C2×#23511yescharacteristic
418C2×C32C2×#23911yescharacteristic
419C2×C32D3×#4111yescharacteristic
420C2×C32C2×C3×#4111yescharacteristic
421C2×C32C2×C3×#4111yescharacteristic
422C2×C32C2×C3×#4111yescharacteristic
423C3⋊D3—22no—
424C3⋊D3—42no—
425C3×D3—62no—
426C3×D3—122no—
427C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
428C2×D4—92no—
429C2×D4—91no—
430C22×C4—91no—
431C22×C4—91no—
432C22×C4—91no—
433C22×C4—92no—
434C4⋊C4—92no—
435C4⋊C4—92no—
436C4⋊C4—91no—
437C4⋊C4—92no—
438C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
439C4⋊C4C2×(C3⋊D3)⋊C312yes—
440C22⋊C4—92no—
441C22⋊C4—92no—
442C22⋊C4—92no—
443C22⋊C4—92no—
444C22⋊C4—92no—
445C42—91no—
446C42—91no—
447C42—91no—
448C42C2×(C3⋊D3)⋊C311yescharacteristic
449C42C2×(C3⋊D3)⋊C312yes—
450C22×C3—31no—
451C22×C3—31no—
452C22×C3—61no—
453C22×C3—92no—
454C22×C3—181no—
455C22×C3—181no—
456C22×C3—182no—
457C22×C3—182no—
458C22×C3—182no—
459C22×C3C23×C3×D311yescharacteristic
460C2×D3—32no—
461C2×D3—61no—
462C2×D3—61no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—92no—
467C2×D3—181no—
468C2×D3—181no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—182no—
472C3×C4—22no—
473C3×C4—21no—
474C3×C4—22no—
475C3×C4—32no—
476C3×C4—32no—
477C3×C4—32no—
478C3×C4—32no—
479C3×C4—61no—
480C3×C4—62no—
481C3×C4—61no—
482C3×C4—62no—
483C3×C4—62no—
484C3×C4—62no—
485C3×C4—64no—
486C3×C4—62no—
487C3×C4—62no—
488C3×C4—121no—
489C3×C4—122no—
490C3×C4—181no—
491C3×C4—181no—
492C3×C4—182no—
493C3×C4—181no—
494C3×C4C3×#23512yes—
495C3×C4C3×#23912yes—
496Dic3—61no—
497Dic3—61no—
498Dic3—62no—
499Dic3—61no—
500Dic3—181no—
501Dic3—181no—
502Dic3—182no—
503Dic3—181no—
504C32—21no—
505C32#558011yescharacteristic, complemented
506C32C3×#52911yescharacteristic, complemented
507C23—91no—
508C23—92no—
509Q8—22no—
510D4—184no—
511D4—182no—
512C2×C4—91no—
513C2×C4—91no—
514C2×C4—91no—
515C2×C4—92no—
516C2×C4—182no—
517C2×C4—182no—
518C2×C4—182no—
519C2×C4—184no—
520C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
521C2×C4C22×(C3⋊D3)⋊C312yes—
522C2×C4C22×(C3⋊D3)⋊C312yes—
523C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
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—61no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—182no—
535C2×C3—362no—
536C2×C3C2×C3×#23511yescharacteristic
537C2×C3C2×C3×#23911yescharacteristic
538C2×C3C3×D3×#4111yescharacteristic
539D3—62no—
540D3—122no—
541D3—182no—
542D3—362no—
543C22—92no—
544C22—181no—
545C22—181no—
546C22—182no—
547C22—182no—
548C22—182no—
549C22C23×(C3⋊D3)⋊C311yescharacteristic
550C4—22no—
551C4—21no—
552C4—22no—
553C4—181no—
554C4—181no—
555C4—182no—
556C4—181no—
557C4#3828⋊C312yes—
558C4#8872212yes—
559C3—31no—
560C3—31no—
561C3—61no—
562C3C3×#558011yescharacteristic
563C2—182no—
564C2—362no—
565C2C2×#3828⋊C311yescharacteristic
566C2C2×#8872211yescharacteristic
567C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
prime radical#64104⋊C3C2
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×H(3,𝔽3)C2semidirect
2. #64156⋊C3C2semidirect
3. #64155⋊C3C2semidirect
4. #64107⋊C3C2semidirect
6. #64103⋊C3C2semidirect
7. #64099⋊C3C2semidirect
8. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
9. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
10. #64081⋊C3C2semidirect
11. C42×(C3⋊D3)⋊C3C2semidirect
12. #10596051C3semidirect
38. C32×C4×Q8D3semidirect
39. C32×C4×Q8C2×C3semidirect
40. #64156C2×C3semidirect
41. #64155C2×C3semidirect
42. #64107C2×C3semidirect
44. #64103C2×C3semidirect
45. #64099C2×C3semidirect
46. C4×C4⋊(C3⋊D3)C2×C3semidirect
47. C4×C4⋊(C3⋊D3)C2×C3semidirect
48. #64081C2×C3semidirect
49. C42×C3⋊D3C2×C3semidirect
348. C4×Q8(C3⋊D3)⋊C3semidirect
353. H(3,𝔽3)#529semidirect
505. C32#5580semidirect
506. C32C3×#529semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
#64156⋊C32
#64155⋊C32
#64107⋊C32
#64104⋊C32
#64103⋊C32
#64099⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#64081⋊C32
C42×(C3⋊D3)⋊C32
#105960513
C3×#55803