Finite Group Databasebeta

← #10596052⋊C3 #13152722

Subgroups — #10596052⋊C3 #13152722

Order 1728.

819 conjugacy classes of subgroups.

The subgroup lattice has 555 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, complemented
3#64155⋊C3C211yesmaximal, characteristic, complemented
4#64103⋊C3C212yesmaximal
5#64101⋊C3C211yesmaximal, characteristic, complemented
6C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
7#64098⋊C3C212yesmaximal, complemented
8#11721387C212yesmaximal, complemented
9C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
10#64081⋊C3C212yesmaximal, complemented
11#64079⋊C3C211yesmaximal, characteristic, complemented
12#10596052C311yesmaximal, characteristic, complemented
13C3×#5581—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C42×H(3,𝔽3)C2211yescharacteristic
22C42×H(3,𝔽3)C2212yes—
23#3752⋊C3C2211yescharacteristic
24#3752⋊C3C2212yes—
25#3752⋊C3C2211yescharacteristic
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31#3750⋊C3C2212yes—
32#3750⋊C3C2212yes—
33#3750⋊C3C2212yes—
34#3750⋊C3C2212yes—
35#3750⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×C4×Q8—21no—
38C32×C4×Q8D311yescharacteristic, complemented
39C32×C4×Q8C2×C311yescharacteristic, complemented
40#64155C2×C311yescharacteristic, complemented
41#64155C2×C311yescharacteristic, complemented
42#64103C2×C312yes—
43#64101C2×C311yescharacteristic, complemented
44C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
45#64098C2×C312yescomplemented
46#64085C2×C312yescomplemented
47C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
48#64081C2×C312yescomplemented
49#64079C2×C311yescharacteristic, complemented
50C3×#946—31no—
51C3×#946—31no—
52C3×#894—32no—
53C3×#892—31no—
54C3×D3×C4⋊C4—31no—
55C3×#889—32no—
56C3×#876—32no—
57C3×C4×D12—31no—
58C3×#872—32no—
59C3×#870—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62Q8×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
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)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C4⋊((C3⋊D3)⋊C3)—22no—
73C4×(C3⋊D3)⋊C3—24no—
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—22no—
76C3×#552—91no—
77#5581—31no—
78#5581—91no—
79C2×C32×Q8—21no—
80C2×C32×Q8C2×D311yescharacteristic
81C2×C32×Q8C22×C311yescharacteristic
82C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
83C2×C4×C3⋊D3C22×C312yes—
84C2×C4×C3⋊D3C22×C311yescharacteristic
85C2×C4×C3⋊D3C22×C311yescharacteristic
86C2×C3×D12—31no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C32×C4⋊C4—22no—
91C32×C4⋊C4—21no—
92C32×C4⋊C4C2×D312yes—
93C32×C4⋊C4C2×D311yescharacteristic
94C32×C4⋊C4C22×C312yes—
95C32×C4⋊C4C22×C311yescharacteristic
96C32×C42—21no—
97C32×C42—22no—
98C32×C42C2×D311yescharacteristic
99C32×C42C2×D312yes—
100C32×C42C22×C311yescharacteristic
101C32×C42C22×C312yes—
102#3752C22×C311yescharacteristic
103#3752C22×C312yes—
104#3752C22×C311yescharacteristic
105#3752C22×C312yes—
106#3752C22×C312yes—
107C4⋊(C3⋊Dic3)C22×C311yescharacteristic
108C4⋊(C3⋊Dic3)C22×C311yescharacteristic
109C4⋊(C3⋊Dic3)C22×C311yescharacteristic
110#3750C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113#3750C22×C312yes—
114#3750C22×C312yes—
115C4×C3⋊Dic3C22×C312yes—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—31no—
120C3×(C2×D3)⋊C4—31no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—31no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×C4×Dic3—32no—
130C22×H(3,𝔽3)C2411yescharacteristic
131C2×(C3⋊D3)⋊C3—21no—
132C2×(C3⋊D3)⋊C3—21no—
133C2×(C3⋊D3)⋊C3—21no—
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3#4112yes—
137C4×H(3,𝔽3)—22no—
138C4×H(3,𝔽3)—22no—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)#4112yes—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—22no—
146C3×#126—92no—
147C3×#125—91no—
148C3×#125—92no—
149C3×#123—92no—
150C3×C22⋊Q8—91no—
151C3×C22⋊Q8—91no—
152C3×C4×Q8—31no—
153C3×C4×Q8—31no—
154C3×C4×Q8—61no—
155C3×C4×Q8C3×D311yescharacteristic
156C3×C4×D4—91no—
157C3×#117—92no—
158C3×#117—91no—
159C2×C3×C4⋊C4—91no—
160#946—31no—
161#946—31no—
162#946—91no—
163#946—91no—
164#894—32no—
165#894—92no—
166#892—31no—
167#892—91no—
168D3×C4⋊C4—31no—
169D3×C4⋊C4—91no—
170#889—32no—
171#889—92no—
172#876—32no—
173#876—92no—
174C4×D12—31no—
175C4×D12—91no—
176#872—32no—
177#872—92no—
178#870—31no—
179#870—91no—
180C22×C3⋊D3C23×C311yescharacteristic
181C22×C3⋊D3C23×C311yescharacteristic
182C22×C3×D3—31no—
183C22×C3×D3—31no—
184C32×Q8—24no—
185C32×Q8—22no—
186C32×Q8—22no—
187C2×C32×C4—21no—
188C2×C32×C4—22no—
189C2×C32×C4—21no—
190C2×C32×C4—22no—
191C2×C32×C4—21no—
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D312yes—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C23×C312yes—
198C2×C32×C4C23×C311yescharacteristic
199C2×C32×C4C23×C311yescharacteristic
200C2×C32×C4C23×C312yes—
201C2×C32×C4C23×C311yescharacteristic
202C2×C3⋊Dic3C23×C311yescharacteristic
203C2×C3⋊Dic3C23×C312yes—
204C2×C3⋊Dic3C23×C312yes—
205C2×C3⋊Dic3C23×C311yescharacteristic
206C4⋊(C3⋊D3)—22no—
207C4×C3⋊D3—24no—
208C4×C3⋊D3—22no—
209C4×C3⋊D3—22no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—32no—
214C3×D12—62no—
215C3×C4×D3—64no—
216C3×C4×D3—62no—
217C3×C4×D3—62no—
218#552—91noSylow
219C2×H(3,𝔽3)C2×#4111yescharacteristic
220C2×H(3,𝔽3)C2×#4111yescharacteristic
221C2×H(3,𝔽3)2−1+411yescharacteristic
222(C3⋊D3)⋊C3—22no—
223(C3⋊D3)⋊C3—41no—
224C2×C3×Q8—31no—
225C2×C3×Q8—31no—
226C2×C3×Q8—61no—
227C2×C3×Q8C2×C3×D311yescharacteristic
228C2×C3×D4—91no—
229C22×C3×C4—92no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C2×D12—31no—
233C2×D12—91no—
234C2×C4×D3—32no—
235C2×C4×D3—31no—
236C2×C4×D3—31no—
237C2×C4×D3—92no—
238C2×C4×D3—91no—
239C2×C4×D3—91no—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—32no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4—62no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4C2×C3×D312yes—
255C3×C4⋊C4C2×C3×D311yescharacteristic
256C3×C22⋊C4—91no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—91no—
260C3×C22⋊C4—92no—
261C3×C42—31no—
262C3×C42—32no—
263C3×C42—32no—
264C3×C42—31no—
265C3×C42—62no—
266C3×C42—61no—
267C3×C42—92no—
268C3×C42C2×C3×D311yescharacteristic
269C3×C42C2×C3×D312yes—
270(C2×D3)⋊C4—31no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—31no—
273(C2×D3)⋊C4—32no—
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—91no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—91no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—31no—
283C4⋊Dic3—91no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296C4×Dic3—32no—
297C4×Dic3—92no—
298C22×C32—21no—
299C22×C32C23×D311yescharacteristic
300C22×C32C24×C311yescharacteristic
301C2×C3⋊D3—21no—
302C2×C3⋊D3—21no—
303C2×C3⋊D3—21no—
304C2×C3⋊D3—21no—
305C2×C3⋊D3—21no—
306C2×C3⋊D3C3×#4112yes—
307C2×C3×D3—32no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C2×C3×D3—61no—
313C32×C4—22no—
314C32×C4—24no—
315C32×C4—22no—
316C32×C4—21no—
317C32×C4—22no—
318C32×C4—21no—
319C32×C4—22no—
320C32×C4—21no—
321C32×C4—22no—
322C32×C4—21no—
323C32×C4—22no—
324C32×C4—42no—
325C32×C4—41no—
326C32×C4#23512yes—
327C32×C4C3×#4112yes—
328C3⋊Dic3—21no—
329C3⋊Dic3—21no—
330C3⋊Dic3—22no—
331C3⋊Dic3—22no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336#126—92no—
337#125—92no—
338#125—91no—
339#123—92no—
340C22⋊Q8—91no—
341C22⋊Q8—91no—
342C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
343C4×D4—91no—
344#117—91no—
345#117—92no—
346C2×C4⋊C4—91no—
347H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
348C23×C3—91no—
349C23×C3—91no—
350C22×D3—31no—
351C22×D3—31no—
352C22×D3—91no—
353C22×D3—91no—
354C3×Q8—22no—
355C3×Q8—62no—
356C3×Q8—62no—
357C3×Q8—64no—
358C3×D4—182no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—32no—
363C2×C3×C4—32no—
364C2×C3×C4—31no—
365C2×C3×C4—32no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—32no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—61no—
372C2×C3×C4—62no—
373C2×C3×C4—61no—
374C2×C3×C4—91no—
375C2×C3×C4—92no—
376C2×C3×C4—91no—
377C2×C3×C4—92no—
378C2×C3×C4—182no—
379C2×C3×C4—182no—
380C2×C3×C4—184no—
381C2×C3×C4C22×C3×D312yes—
382C2×C3×C4C22×C3×D311yescharacteristic
383C2×C3×C4C22×C3×D311yescharacteristic
384C2×C3×C4C22×C3×D312yes—
385C2×C3×C4C22×C3×D311yescharacteristic
386C2×Dic3—31no—
387C2×Dic3—32no—
388C2×Dic3—31no—
389C2×Dic3—32no—
390C2×Dic3—91no—
391C2×Dic3—92no—
392C2×Dic3—92no—
393C2×Dic3—91no—
394D12—62no—
395D12—182no—
396C4×D3—64no—
397C4×D3—62no—
398C4×D3—62no—
399C4×D3—182no—
400C4×D3—182no—
401C4×D3—184no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32C2×#23511yescharacteristic
406C2×C32#100711yescharacteristic
407C2×C32D3×#4111yescharacteristic
408C2×C32C2×C3×#4111yescharacteristic
409C2×C32C2×C3×#4111yescharacteristic
410C2×C32C3×2−1+411yescharacteristic
411C3⋊D3—22no—
412C3⋊D3—41no—
413C3×D3—62no—
414C3×D3—121no—
415C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
416C2×D4—91no—
417C22×C4—92no—
418C22×C4—91no—
419C22×C4—91no—
420C4⋊C4—91no—
421C4⋊C4—92no—
422C4⋊C4—92no—
423C4⋊C4—91no—
424C4⋊C4—92no—
425C4⋊C4—91no—
426C4⋊C4—92no—
427C4⋊C4—92no—
428C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
429C4⋊C4C2×(C3⋊D3)⋊C312yes—
430C22⋊C4—91no—
431C22⋊C4—92no—
432C22⋊C4—92no—
433C22⋊C4—91no—
434C22⋊C4—92no—
435C42—92no—
436C42C2×(C3⋊D3)⋊C312yes—
437C42C2×(C3⋊D3)⋊C311yescharacteristic
438C22×C3—31no—
439C22×C3—31no—
440C22×C3—61no—
441C22×C3—92no—
442C22×C3—181no—
443C22×C3—181no—
444C22×C3—181no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3C23×C3×D311yescharacteristic
448C2×D3—32no—
449C2×D3—61no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—61no—
453C2×D3—61no—
454C2×D3—92no—
455C2×D3—181no—
456C2×D3—181no—
457C2×D3—181no—
458C2×D3—181no—
459C2×D3—181no—
460C3×C4—22no—
461C3×C4—21no—
462C3×C4—22no—
463C3×C4—21no—
464C3×C4—32no—
465C3×C4—32no—
466C3×C4—62no—
467C3×C4—61no—
468C3×C4—62no—
469C3×C4—61no—
470C3×C4—62no—
471C3×C4—62no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—64no—
475C3×C4—62no—
476C3×C4—62no—
477C3×C4—122no—
478C3×C4—121no—
479C3×C4—181no—
480C3×C4—181no—
481C3×C4—182no—
482C3×C4—182no—
483C3×C4C3×#23512yes—
484Dic3—61no—
485Dic3—61no—
486Dic3—62no—
487Dic3—62no—
488Dic3—181no—
489Dic3—181no—
490Dic3—182no—
491Dic3—182no—
492C32—21no—
493C32#558111yescharacteristic, complemented
494C32C3×#55211yescharacteristic, complemented
495C23—91no—
496C23—91no—
497Q8—22no—
498D4—182no—
499C2×C4—91no—
500C2×C4—92no—
501C2×C4—91no—
502C2×C4—92no—
503C2×C4—184no—
504C2×C4—182no—
505C2×C4—182no—
506C2×C4C22×(C3⋊D3)⋊C312yes—
507C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
508C2×C4C22×(C3⋊D3)⋊C312yes—
509C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
510C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—31no—
517C2×C3—61no—
518C2×C3—61no—
519C2×C3—61no—
520C2×C3—182no—
521C2×C3—361no—
522C2×C3C2×C3×#23511yescharacteristic
523C2×C3C3×#100711yescharacteristic
524C2×C3C3×D3×#4111yescharacteristic
525D3—62no—
526D3—121no—
527D3—182no—
528D3—361no—
529C22—92no—
530C22—181no—
531C22—181no—
532C22—181no—
533C22—181no—
534C22—181no—
535C22C23×(C3⋊D3)⋊C311yescharacteristic
536C4—22no—
537C4—21no—
538C4—22no—
539C4—21no—
540C4—181no—
541C4—181no—
542C4—182no—
543C4—182no—
544C4#3828⋊C312yes—
545C3—31no—
546C3—31no—
547C3—61no—
548C3C3×#558111yescharacteristic
549C2—182no—
550C2—361no—
551C2C2×#3828⋊C311yescharacteristic
552C2#1172262711yescharacteristic
553C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×H(3,𝔽3)C2semidirect
2. #64155⋊C3C2semidirect
3. #64155⋊C3C2semidirect
5. #64101⋊C3C2semidirect
6. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
7. #64098⋊C3C2semidirect
8. #11721387C2semidirect
10. #64081⋊C3C2semidirect
11. #64079⋊C3C2semidirect
12. #10596052C3semidirect
38. C32×C4×Q8D3semidirect
39. C32×C4×Q8C2×C3semidirect
40. #64155C2×C3semidirect
41. #64155C2×C3semidirect
43. #64101C2×C3semidirect
44. C4⋊C4×C3⋊D3C2×C3semidirect
45. #64098C2×C3semidirect
46. #64085C2×C3semidirect
48. #64081C2×C3semidirect
49. #64079C2×C3semidirect
342. C4×Q8(C3⋊D3)⋊C3semidirect
347. H(3,𝔽3)#552semidirect
493. C32#5581semidirect
494. C32C3×#552semidirect

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
#64103⋊C32
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64098⋊C32
#117213872
C4×C4⋊((C3⋊D3)⋊C3)2
#64081⋊C32
#64079⋊C32
#105960523
C3×#55813