Finite Group Databasebeta

← #10596049⋊C3 #13152719

Subgroups — #10596049⋊C3 #13152719

Order 1728.

847 conjugacy classes of subgroups.

The subgroup lattice has 541 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#64155⋊C3C211yesmaximal, characteristic
3#64153⋊C3C211yesmaximal, characteristic, complemented
4#64106⋊C3C211yesmaximal, characteristic
5#64105⋊C3C212yesmaximal
6#64098⋊C3C212yesmaximal, complemented
7#64097⋊C3C211yesmaximal, characteristic, complemented
8#64081⋊C3C212yesmaximal
9C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
10C4×#634⋊C3C211yesmaximal, characteristic, complemented
11C4×#634⋊C3C212yesmaximal, complemented
12#10596049C311yesmaximal, characteristic, complemented
13C3×#5578—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×#634⋊C3C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2212yes—
22C42×H(3,𝔽3)C2211yescharacteristic
23C42×H(3,𝔽3)C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32#3750⋊C3C2212yes—
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
37C32×C4×Q8—21no—
38C32×C4×Q8D311yescharacteristic, complemented
39C32×C4×Q8C2×C311yescharacteristic, complemented
40#64155C2×C311yescharacteristic
41#64153C2×C311yescharacteristic, complemented
42#64106C2×C311yescharacteristic
43#64105C2×C312yes—
44#64098C2×C312yescomplemented
45#64097C2×C311yescharacteristic, complemented
46#64081C2×C312yes—
47C42×C3⋊D3C2×C311yescharacteristic
48C4×#634C2×C312yescomplemented
49C4×#634C2×C311yescharacteristic, complemented
50C3×#946—31no—
51C3×#944—31no—
52C3×#897—31no—
53C3×#896—32no—
54C3×#889—32no—
55C3×#888—31no—
56C3×#872—32no—
57C3×C42×D3—31no—
58C3×C4×Dic6—32no—
59C3×C4×Dic6—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61Q8×H(3,𝔽3)—22no—
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4×(C3⋊D3)⋊C3—22no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3Q814yescomplemented
75#634⋊C3—22no—
76#634⋊C3—24no—
77C3×#533—91no—
78#5578—31no—
79#5578—91no—
80C2×C32×Q8—21no—
81C2×C32×Q8C2×D311yescharacteristic
82C2×C32×Q8C22×C311yescharacteristic
83C2×C4×C3⋊D3C22×C311yescharacteristic
84C2×C4×C3⋊D3C22×C311yescharacteristic
85C2×C4×C3⋊D3C22×C311yescharacteristic
86C2×#634C22×C312yes—
87C2×#634C22×C311yescharacteristic
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—31no—
91C2×C3×Dic6—32no—
92C2×C3×Dic6—31no—
93C32×C4⋊C4—21no—
94C32×C4⋊C4—22no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C4⋊C4C2×D312yes—
97C32×C4⋊C4C22×C311yescharacteristic
98C32×C4⋊C4C22×C312yes—
99C32×C42—21no—
100C32×C42—22no—
101C32×C42C2×D311yescharacteristic
102C32×C42C2×D312yes—
103C32×C42C22×C311yescharacteristic
104C32×C42C22×C312yes—
105#3752C22×C312yes—
106#3752C22×C312yes—
107C4⋊(C3⋊Dic3)C22×C311yescharacteristic
108C4⋊(C3⋊Dic3)C22×C312yes—
109#3750C22×C312yes—
110#3750C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113#3750C22×C312yes—
114C4×C3⋊Dic3C22×C311yescharacteristic
115C4×C3⋊Dic3C22×C312yes—
116C4×C3⋊Dic3C22×C311yescharacteristic
117C4×C3⋊Dic3C22×C311yescharacteristic
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—32no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×C4×Dic3—31no—
128C3×C4×Dic3—31no—
129C3×C4×Dic3—31no—
130C3×C4×Dic3—32no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C2×(C3⋊D3)⋊C3—21no—
133C2×(C3⋊D3)⋊C3—21no—
134C2×(C3⋊D3)⋊C3C2×Q812yes—
135C4×H(3,𝔽3)—22no—
136C4×H(3,𝔽3)—22no—
137C4×H(3,𝔽3)—21no—
138C4×H(3,𝔽3)C2×Q812yes—
139C4×H(3,𝔽3)#4112yes—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3C2×Q812yes—
144(C3⋊Dic3)⋊C3#4112yes—
145C3×C4⋊Q8—91no—
146C3×C4⋊Q8—91no—
147C3×#125—92no—
148C3×C22⋊Q8—92no—
149C3×C22⋊Q8—91no—
150C3×C22⋊Q8—91no—
151C3×C4×Q8—31no—
152C3×C4×Q8—31no—
153C3×C4×Q8—61no—
154C3×C4×Q8—91no—
155C3×C4×Q8—92no—
156C3×C4×Q8C3×D311yescharacteristic
157C3×#117—92no—
158C2×C3×C42—91no—
159#946—31no—
160#946—91no—
161#944—31no—
162#944—91no—
163#897—31no—
164#897—91no—
165#896—32no—
166#896—92no—
167#889—32no—
168#889—92no—
169#888—31no—
170#888—91no—
171#872—32no—
172#872—92no—
173C42×D3—31no—
174C42×D3—91no—
175C4×Dic6—32no—
176C4×Dic6—31no—
177C4×Dic6—92no—
178C4×Dic6—91no—
179C22×C3⋊D3C23×C311yescharacteristic
180C22×C3×D3—31no—
181C32×Q8—24no—
182C32×Q8—22no—
183C32×Q8—22no—
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—22no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D312yes—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C23×C311yescharacteristic
195C2×C32×C4C23×C312yes—
196C2×C32×C4C23×C312yes—
197C2×C32×C4C23×C311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199C2×C3⋊Dic3C23×C311yescharacteristic
200C2×C3⋊Dic3C23×C312yes—
201C2×C3⋊Dic3C23×C312yes—
202C2×C3⋊Dic3C23×C311yescharacteristic
203C2×C3⋊Dic3C23×C311yescharacteristic
204C4×C3⋊D3—22no—
205C4×C3⋊D3—22no—
206C4×C3⋊D3C3×Q814yescomplemented
207#634—22no—
208#634—24no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C3×C4×D3—34no—
215C3×C4×D3—62no—
216C3×C4×D3—62no—
217C3×Dic6—64no—
218C3×Dic6—62no—
219#533—91noSylow
220C2×H(3,𝔽3)C22×Q811yescharacteristic
221C2×H(3,𝔽3)C2×#4111yescharacteristic
222C2×H(3,𝔽3)C2×#4111yescharacteristic
223(C3⋊D3)⋊C3—22no—
224C2×C3×Q8—31no—
225C2×C3×Q8—31no—
226C2×C3×Q8—61no—
227C2×C3×Q8—91no—
228C2×C3×Q8—92no—
229C2×C3×Q8C2×C3×D311yescharacteristic
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C22×C3×C4—91no—
233C2×C4×D3—31no—
234C2×C4×D3—31no—
235C2×C4×D3—31no—
236C2×C4×D3—91no—
237C2×C4×D3—91no—
238C2×C4×D3—91no—
239C2×Dic6—32no—
240C2×Dic6—31no—
241C2×Dic6—92no—
242C2×Dic6—91no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—32no—
245C3×C4⋊C4—32no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—62no—
248C3×C4⋊C4—61no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4—92no—
255C3×C4⋊C4—91no—
256C3×C4⋊C4C2×C3×D311yescharacteristic
257C3×C4⋊C4C2×C3×D312yes—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—92no—
260C3×C42—31no—
261C3×C42—32no—
262C3×C42—31no—
263C3×C42—32no—
264C3×C42—61no—
265C3×C42—62no—
266C3×C42—91no—
267C3×C42—91no—
268C3×C42—91no—
269C3×C42—92no—
270C3×C42C2×C3×D311yescharacteristic
271C3×C42C2×C3×D312yes—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—32no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276C4⋊Dic3—31no—
277C4⋊Dic3—32no—
278C4⋊Dic3—91no—
279C4⋊Dic3—92no—
280Dic3⋊C4—32no—
281Dic3⋊C4—32no—
282Dic3⋊C4—32no—
283Dic3⋊C4—32no—
284Dic3⋊C4—32no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290C4×Dic3—31no—
291C4×Dic3—32no—
292C4×Dic3—31no—
293C4×Dic3—31no—
294C4×Dic3—91no—
295C4×Dic3—92no—
296C4×Dic3—91no—
297C4×Dic3—91no—
298C22×C32—21no—
299C22×C32C23×D311yescharacteristic
300C22×C32C24×C311yescharacteristic
301C2×C3⋊D3—21no—
302C2×C3⋊D3—21no—
303C2×C3⋊D3C2×C3×Q812yes—
304C2×C3×D3—32no—
305C2×C3×D3—61no—
306C2×C3×D3—61no—
307C32×C4—24no—
308C32×C4—22no—
309C32×C4—22no—
310C32×C4—21no—
311C32×C4—22no—
312C32×C4—21no—
313C32×C4—22no—
314C32×C4—22no—
315C32×C4—22no—
316C32×C4—42no—
317C32×C4—41no—
318C32×C4#23512yes—
319C32×C4D3×Q812yes—
320C32×C4C2×C3×Q812yes—
321C32×C4C3×#4112yes—
322C3⋊Dic3—22no—
323C3⋊Dic3—22no—
324C3⋊Dic3—21no—
325C3⋊Dic3C2×C3×Q812yes—
326C3⋊Dic3C3×#4112yes—
327C3×Dic3—32no—
328C3×Dic3—32no—
329C3×Dic3—62no—
330C3×Dic3—61no—
331C3×Dic3—62no—
332C4⋊Q8—91no—
333C4⋊Q8—91no—
334#125—92no—
335C22⋊Q8—91no—
336C22⋊Q8—92no—
337C22⋊Q8—91no—
338C4×Q8—92no—
339C4×Q8—91no—
340C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
341#117—92no—
342C2×C42—91no—
343H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
344C23×C3—91no—
345C22×D3—31no—
346C22×D3—91no—
347C3×Q8—22no—
348C3×Q8—62no—
349C3×Q8—62no—
350C3×Q8—64no—
351C3×Q8—182no—
352C3×Q8—184no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—32no—
357C2×C3×C4—32no—
358C2×C3×C4—31no—
359C2×C3×C4—32no—
360C2×C3×C4—32no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—62no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—62no—
367C2×C3×C4—61no—
368C2×C3×C4—94no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—182no—
375C2×C3×C4—182no—
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×C3×C4C22×C3×D312yes—
378C2×C3×C4C22×C3×D312yes—
379C2×C3×C4C22×C3×D311yescharacteristic
380C2×C3×C4C22×C3×D311yescharacteristic
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—32no—
385C2×Dic3—32no—
386C2×Dic3—91no—
387C2×Dic3—91no—
388C2×Dic3—91no—
389C2×Dic3—92no—
390C2×Dic3—92no—
391C4×D3—34no—
392C4×D3—62no—
393C4×D3—62no—
394C4×D3—94no—
395C4×D3—182no—
396C4×D3—182no—
397Dic6—64no—
398Dic6—62no—
399Dic6—184no—
400Dic6—182no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32C2×#23511yescharacteristic
405C2×C32C2×D3×Q811yescharacteristic
406C2×C32D3×#4111yescharacteristic
407C2×C32C22×C3×Q811yescharacteristic
408C2×C32C2×C3×#4111yescharacteristic
409C2×C32C2×C3×#4111yescharacteristic
410C3⋊D3—22no—
411C3×D3—62no—
412C2×Q8—92no—
413C2×Q8—91no—
414C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—91no—
418C4⋊C4—92no—
419C4⋊C4—92no—
420C4⋊C4—92no—
421C4⋊C4—92no—
422C4⋊C4—92no—
423C4⋊C4—92no—
424C4⋊C4—91no—
425C4⋊C4C2×(C3⋊D3)⋊C312yes—
426C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
427C22⋊C4—92no—
428C22⋊C4—92no—
429C42—91no—
430C42—91no—
431C42—91no—
432C42—92no—
433C42C2×(C3⋊D3)⋊C311yescharacteristic
434C42C2×(C3⋊D3)⋊C312yes—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—61no—
438C22×C3—92no—
439C22×C3—181no—
440C22×C3—181no—
441C22×C3C23×C3×D311yescharacteristic
442C2×D3—32no—
443C2×D3—61no—
444C2×D3—61no—
445C2×D3—92no—
446C2×D3—181no—
447C2×D3—181no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—21no—
451C3×C4—32no—
452C3×C4—32no—
453C3×C4—32no—
454C3×C4—32no—
455C3×C4—62no—
456C3×C4—61no—
457C3×C4—61no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—62no—
461C3×C4—64no—
462C3×C4—62no—
463C3×C4—62no—
464C3×C4—92no—
465C3×C4—92no—
466C3×C4—121no—
467C3×C4—122no—
468C3×C4—181no—
469C3×C4—182no—
470C3×C4—182no—
471C3×C4C3×#23512yes—
472C3×C4C3×D3×Q812yes—
473Dic3—32no—
474Dic3—32no—
475Dic3—62no—
476Dic3—61no—
477Dic3—62no—
478Dic3—92no—
479Dic3—92no—
480Dic3—182no—
481Dic3—182no—
482Dic3—181no—
483C32—21no—
484C32#557811yescharacteristic, complemented
485C32C3×#53311yescharacteristic, complemented
486C23—91no—
487Q8—22no—
488Q8—184no—
489Q8—182no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—94no—
493C2×C4—91no—
494C2×C4—92no—
495C2×C4—92no—
496C2×C4—182no—
497C2×C4—182no—
498C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
499C2×C4C22×(C3⋊D3)⋊C312yes—
500C2×C4C22×(C3⋊D3)⋊C312yes—
501C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
502C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—61no—
510C2×C3—61no—
511C2×C3—61no—
512C2×C3—182no—
513C2×C3C2×C3×#23511yescharacteristic
514C2×C3C2×C3×D3×Q811yescharacteristic
515C2×C3C3×D3×#4111yescharacteristic
516D3—62no—
517D3—182no—
518C22—92no—
519C22—181no—
520C22—181no—
521C22C23×(C3⋊D3)⋊C311yescharacteristic
522C4—22no—
523C4—21no—
524C4—22no—
525C4—92no—
526C4—92no—
527C4—181no—
528C4—182no—
529C4—182no—
530C4#3828⋊C312yes—
531C4Q8×(C3⋊D3)⋊C312yes—
532C3—31no—
533C3—31no—
534C3—61no—
535C3C3×#557811yescharacteristic
536C2—182no—
537C2C2×#3828⋊C311yescharacteristic
538C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
539C2#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 radicalC22×(C3⋊D3)⋊C3C23
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#533
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
3. #64153⋊C3C2semidirect
6. #64098⋊C3C2semidirect
7. #64097⋊C3C2semidirect
10. C4×#634⋊C3C2semidirect
11. C4×#634⋊C3C2semidirect
12. #10596049C3semidirect
38. C32×C4×Q8D3semidirect
39. C32×C4×Q8C2×C3semidirect
41. #64153C2×C3semidirect
44. #64098C2×C3semidirect
45. #64097C2×C3semidirect
48. C4×#634C2×C3semidirect
49. C4×#634C2×C3semidirect
74. C4×(C3⋊D3)⋊C3Q8semidirect
206. C4×C3⋊D3C3×Q8semidirect
340. C4×Q8(C3⋊D3)⋊C3semidirect
343. H(3,𝔽3)#533semidirect
484. C32#5578semidirect
485. C32C3×#533semidirect

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
#64155⋊C32
#64153⋊C32
#64106⋊C32
#64105⋊C32
#64098⋊C32
#64097⋊C32
#64081⋊C32
C42×(C3⋊D3)⋊C32
C4×#634⋊C32
#105960493
C3×#55783