Finite Group Databasebeta

← #10592520⋊C3 #13141981

Subgroups — #10592520⋊C3 #13141981

Order 1728.

583 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2M16×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#64081⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
5#63664⋊C3C211yesmaximal, characteristic, complemented
6#63652⋊C3C211yesmaximal, characteristic, complemented
7#63632⋊C3C211yesmaximal, characteristic, complemented
8#10592520C311yesmaximal, characteristic, complemented
9C3×#4821—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#88722C411yescharacteristic, complemented
12Q8×(C3⋊D3)⋊C3C411yescharacteristic, complemented
13#3830⋊C3—21no—
14#3830⋊C3C411yescharacteristic, complemented
15D4×(C3⋊D3)⋊C3—21no—
16D4×(C3⋊D3)⋊C3C411yescharacteristic, complemented
17#3828⋊C3—21no—
18#3828⋊C3C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3—21no—
20C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
21M16×H(3,𝔽3)C2211yescharacteristic, complemented
22C42×H(3,𝔽3)C2211yescharacteristic, complemented
23#3752⋊C3—21no—
24#3750⋊C3—21no—
25C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
26#3748⋊C3C2211yescharacteristic, complemented
27#3743⋊C3—21no—
28C8×(C3⋊D3)⋊C3—21no—
29#41×C3⋊D3C2×C311yescharacteristic
30C3×D3×#41—31no—
31M16×C3⋊D3C2×C311yescharacteristic, complemented
32#64081C2×C311yescharacteristic, complemented
33C3×D3×M16—31no—
34C3×#872—31no—
35C32×#104—21no—
36C32×#104D311yescharacteristic, complemented
37C32×#104C2×C311yescharacteristic, complemented
38#63664C2×C311yescharacteristic, complemented
39#63652C2×C311yescharacteristic, complemented
40#63632C2×C311yescharacteristic, complemented
41C3×#837—31no—
42C3×#825—31no—
43C3×#805—31no—
44C22×(C3⋊D3)⋊C3—21no—
45C22×(C3⋊D3)⋊C3D411yescharacteristic
46Q8×H(3,𝔽3)C2×C411yescharacteristic
47D4×H(3,𝔽3)—21no—
48D4×H(3,𝔽3)C2×C411yescharacteristic
49C2×C4×H(3,𝔽3)—21no—
50C2×C4×H(3,𝔽3)—21no—
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52#638⋊C3—21no—
53#638⋊C3—21no—
54#638⋊C3—21no—
55C2×(C3⋊Dic3)⋊C3—21no—
56C2×(C3⋊Dic3)⋊C3—21no—
57C2×(C3⋊Dic3)⋊C3D411yescharacteristic
58C4⋊((C3⋊D3)⋊C3)—21no—
59C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
60C4×(C3⋊D3)⋊C3—21no—
61C4×(C3⋊D3)⋊C3—21no—
62C4×(C3⋊D3)⋊C3—21no—
63C4×(C3⋊D3)⋊C3C2×C411yescharacteristic
64C4×(C3⋊D3)⋊C3C2×C411yescharacteristic
65C4×(C3⋊D3)⋊C3D411yescharacteristic
66C4×(C3⋊D3)⋊C3D411yescharacteristic
67#634⋊C3—21no—
68#634⋊C3C2×C411yescharacteristic
69C8×H(3,𝔽3)—21no—
70(C3⋊(C3⋊C8))⋊C3—21no—
71C3×#421—91no—
72#4821—31no—
73#4821—91no—
74C32×#41—21no—
75C32×#41C2×D311yescharacteristic
76C32×#41C22×C311yescharacteristic
77#3832C3×C411yescharacteristic, complemented
78Q8×C3⋊D3C3×C411yescharacteristic, complemented
79#3830—21no—
80#3830C3×C411yescharacteristic, complemented
81D4×C3⋊D3—21no—
82D4×C3⋊D3C3×C411yescharacteristic, complemented
83#3828—21no—
84#3828C22×C311yescharacteristic
85C2×C4×C3⋊D3—21no—
86C2×C4×C3⋊D3C22×C311yescharacteristic
87C3×#239—31no—
88C3×D3×Q8—31no—
89C3×#237—31no—
90C3×#237—61no—
91C3×D3×D4—31no—
92C3×D3×D4—61no—
93C3×#235—31no—
94C3×#235—61no—
95C2×C3×C4×D3—31no—
96C2×C3×C4×D3—61no—
97C32×M16—21no—
98C32×M16C2×D311yescharacteristic, complemented
99C32×M16C22×C311yescharacteristic, complemented
100C32×C42—21no—
101C32×C42C2×D311yescharacteristic, complemented
102C32×C42C22×C311yescharacteristic, complemented
103#3752—21no—
104#3750—21no—
105C4×C3⋊Dic3C22×C311yescharacteristic, complemented
106#3748C22×C311yescharacteristic, complemented
107#3743—21no—
108C8×C3⋊D3—21no—
109C3×(C2×D3)⋊C4—61no—
110C3×Dic3⋊C4—61no—
111C3×C4×Dic3—31no—
112C3×#208—31no—
113C3×ΓL(1,𝔽25)—61no—
114C3×D3×C8—61no—
115C22×H(3,𝔽3)—21no—
116C22×H(3,𝔽3)C2×D411yescharacteristic
117C2×(C3⋊D3)⋊C3—21no—
118C2×(C3⋊D3)⋊C3—41no—
119C2×(C3⋊D3)⋊C3—42no—
120C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
121C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
122C4×H(3,𝔽3)—21no—
123C4×H(3,𝔽3)—41no—
124C4×H(3,𝔽3)C22×C411yescharacteristic
125C4×H(3,𝔽3)C2×D411yescharacteristic
126(C3⋊Dic3)⋊C3—21no—
127(C3⋊Dic3)⋊C3—41no—
128(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
129(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
130C2×C3×#41—91no—
131D3×#41—31no—
132D3×#41—91no—
133C2×C3×M16—91no—
134C3×#117—91no—
135D3×M16—31no—
136D3×M16—91no—
137#872—31no—
138#872—91no—
139C3×#104—31no—
140C3×#104—31no—
141C3×#104—61no—
142C3×#104—91no—
143C3×#104—91no—
144C3×#104—91no—
145C3×#104C3×D311yescharacteristic
146#837—31no—
147#837—91no—
148#825—31no—
149#825—91no—
150#805—31no—
151#805—91no—
152C22×C3⋊D3—21no—
153C22×C3⋊D3C3×D411yescharacteristic
154C22×C3×D3—31no—
155C22×C3×D3—61no—
156C32×Q8—21no—
157C32×Q8C4×D311yescharacteristic
158C32×Q8C2×C3×C411yescharacteristic
159C32×D4—21no—
160C32×D4—21no—
161C32×D4—21no—
162C32×D4—41no—
163C32×D4C4×D311yescharacteristic
164C32×D4C2×C3×C411yescharacteristic
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C23×C311yescharacteristic
174#638—21no—
175#638—21no—
176#638—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3C3×D411yescharacteristic
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)C2×C3×C411yescharacteristic
182C4×C3⋊D3—21no—
183C4×C3⋊D3—21no—
184C4×C3⋊D3—21no—
185C4×C3⋊D3C2×C3×C411yescharacteristic
186C4×C3⋊D3C2×C3×C411yescharacteristic
187C4×C3⋊D3C3×D411yescharacteristic
188C4×C3⋊D3C3×D411yescharacteristic
189#634—21no—
190#634C2×C3×C411yescharacteristic
191C3×#67—61no—
192C3×#67—61no—
193C3×#67—61no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C3×D12—31no—
198C3×D12—61no—
199C3×C4×D3—31no—
200C3×C4×D3—31no—
201C3×C4×D3—31no—
202C3×C4×D3—31no—
203C3×C4×D3—61no—
204C3×C4×D3—61no—
205C3×C4×D3—61no—
206C3×Dic6—31no—
207C3×Dic6—61no—
208C32×C8—21no—
209C32×C8—21no—
210C32×C8—41no—
211C3⋊(C3⋊C8)—21no—
212C3×C3⋊C8—61no—
213#421—91noSylow
214C2×H(3,𝔽3)—21no—
215C2×H(3,𝔽3)—41no—
216C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
217(C3⋊D3)⋊C3—21no—
218(C3⋊D3)⋊C3—21no—
219(C3⋊D3)⋊C3—41no—
220C3×#41—31no—
221C3×#41—31no—
222C3×#41—61no—
223C3×#41—91no—
224C3×#41—91no—
225C3×#41—91no—
226C3×#41—181no—
227C3×#41—181no—
228C3×#41C2×C3×D311yescharacteristic
229C2×C3×Q8—91no—
230C2×C3×D4—91no—
231C2×C3×D4—181no—
232C22×C3×C4—91no—
233C22×C3×C4—181no—
234#239—31no—
235#239—91no—
236D3×Q8—31no—
237D3×Q8—91no—
238#237—31no—
239#237—61no—
240#237—91no—
241#237—181no—
242D3×D4—31no—
243D3×D4—61no—
244D3×D4—91no—
245D3×D4—181no—
246#235—31no—
247#235—61no—
248#235—91no—
249#235—181no—
250C2×C4×D3—31no—
251C2×C4×D3—61no—
252C2×C4×D3—91no—
253C2×C4×D3—181no—
254C3×M16—31no—
255C3×M16—31no—
256C3×M16—61no—
257C3×M16—91no—
258C3×M16—181no—
259C3×M16C2×C3×D311yescharacteristic
260C2×C3×C8—181no—
261C3×C4⋊C4—181no—
262C3×C22⋊C4—181no—
263C3×C42—31no—
264C3×C42—31no—
265C3×C42—61no—
266C3×C42—91no—
267C3×C42C2×C3×D311yescharacteristic
268(C2×D3)⋊C4—61no—
269(C2×D3)⋊C4—181no—
270Dic3⋊C4—61no—
271Dic3⋊C4—181no—
272C4×Dic3—31no—
273C4×Dic3—91no—
274#208—31no—
275#208—91no—
276ΓL(1,𝔽25)—61no—
277ΓL(1,𝔽25)—181no—
278D3×C8—61no—
279D3×C8—181no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—41no—
284C22×C32D3×D411yescharacteristic
285C22×C32C2×C3×D411yescharacteristic
286C2×C3⋊D3—21no—
287C2×C3⋊D3—41no—
288C2×C3⋊D3—42no—
289C2×C3⋊D3C3×C22⋊C411yescharacteristic
290C2×C3⋊D3C3×C22⋊C411yescharacteristic
291C2×C3×D3—31no—
292C2×C3×D3—31no—
293C2×C3×D3—61no—
294C2×C3×D3—121no—
295C2×C3×D3—122no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—41no—
301C32×C4—41no—
302C32×C4—41no—
303C32×C4—42no—
304C32×C4C2×C4×D311yescharacteristic
305C32×C4D3×D411yescharacteristic
306C32×C4C22×C3×C411yescharacteristic
307C32×C4C2×C3×D411yescharacteristic
308C3⋊Dic3—21no—
309C3⋊Dic3—41no—
310C3⋊Dic3C3×C22⋊C411yescharacteristic
311C3⋊Dic3C3×C22⋊C411yescharacteristic
312C3×Dic3—31no—
313C3×Dic3—31no—
314C3×Dic3—61no—
315C3×Dic3—121no—
316C2×#41—91no—
317C2×M16—91no—
318#117—91no—
319#104—91no—
320#104—91no—
321#104—91no—
322#104(C3⋊D3)⋊C311yescharacteristic, complemented
323H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
324C23×C3—91no—
325C23×C3—181no—
326C22×D3—31no—
327C22×D3—61no—
328C22×D3—91no—
329C22×D3—181no—
330C3×Q8—31no—
331C3×Q8—31no—
332C3×Q8—61no—
333C3×Q8—91no—
334C3×Q8—181no—
335C3×Q8C3×C4×D311yescharacteristic
336C3×D4—21no—
337C3×D4—31no—
338C3×D4—31no—
339C3×D4—61no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—91no—
343C3×D4—121no—
344C3×D4—181no—
345C3×D4—181no—
346C3×D4—181no—
347C3×D4—181no—
348C3×D4C3×C4×D311yescharacteristic
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—121no—
364C2×C3×C4—121no—
365C2×C3×C4—181no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4—181no—
369C2×C3×C4—181no—
370C2×C3×C4C22×C3×D311yescharacteristic
371#67—61no—
372#67—61no—
373#67—61no—
374#67—181no—
375#67—181no—
376#67—181no—
377C2×Dic3—31no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—91no—
381C2×Dic3—181no—
382C2×Dic3—181no—
383D12—31no—
384D12—61no—
385D12—91no—
386D12—181no—
387C4×D3—31no—
388C4×D3—31no—
389C4×D3—31no—
390C4×D3—31no—
391C4×D3—61no—
392C4×D3—61no—
393C4×D3—61no—
394C4×D3—91no—
395C4×D3—91no—
396C4×D3—91no—
397C4×D3—91no—
398C4×D3—181no—
399C4×D3—181no—
400C4×D3—181no—
401Dic6—31no—
402Dic6—61no—
403Dic6—91no—
404Dic6—181no—
405C3×C8—21no—
406C3×C8—61no—
407C3×C8—61no—
408C3×C8—121no—
409C3×C8—181no—
410C3⋊C8—61no—
411C3⋊C8—181no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—81no—
419C2×C32D3×C22⋊C411yescharacteristic
420C2×C32C2×C3×C22⋊C411yescharacteristic
421C3⋊D3—21no—
422C3⋊D3—21no—
423C3⋊D3—41no—
424C3×D3—61no—
425C3×D3—61no—
426C3×D3—121no—
427#41—91no—
428#41—91no—
429#41—91no—
430#41—181no—
431#41—181no—
432#41C2×(C3⋊D3)⋊C311yescharacteristic
433C2×Q8—91no—
434C2×D4—91no—
435C2×D4—181no—
436C22×C4—91no—
437C22×C4—181no—
438M16—91no—
439M16—181no—
440M16C2×(C3⋊D3)⋊C311yescharacteristic, complemented
441C2×C8—181no—
442C4⋊C4—181no—
443C22⋊C4—181no—
444C42—91no—
445C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
446C22×C3—21no—
447C22×C3—31no—
448C22×C3—31no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—91no—
453C22×C3—91no—
454C22×C3—121no—
455C22×C3—181no—
456C22×C3—361no—
457C22×C3—362no—
458C22×C3C3×D3×D411yescharacteristic
459C2×D3—31no—
460C2×D3—31no—
461C2×D3—61no—
462C2×D3—91no—
463C2×D3—91no—
464C2×D3—121no—
465C2×D3—122no—
466C2×D3—181no—
467C2×D3—361no—
468C2×D3—362no—
469C3×C4—21no—
470C3×C4—31no—
471C3×C4—31no—
472C3×C4—31no—
473C3×C4—31no—
474C3×C4—41no—
475C3×C4—61no—
476C3×C4—61no—
477C3×C4—61no—
478C3×C4—61no—
479C3×C4—91no—
480C3×C4—91no—
481C3×C4—121no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—122no—
485C3×C4—181no—
486C3×C4—361no—
487C3×C4C2×C3×C4×D311yescharacteristic
488C3×C4C3×D3×D411yescharacteristic
489Dic3—31no—
490Dic3—31no—
491Dic3—61no—
492Dic3—91no—
493Dic3—91no—
494Dic3—121no—
495Dic3—181no—
496Dic3—361no—
497C32—21no—
498C32#482111yescharacteristic, complemented
499C32C3×#42111yescharacteristic, complemented
500C23—91no—
501C23—181no—
502Q8—91no—
503Q8—181no—
504Q8C4×(C3⋊D3)⋊C311yescharacteristic
505D4—21no—
506D4—91no—
507D4—181no—
508D4—181no—
509D4—181no—
510D4—181no—
511D4C4×(C3⋊D3)⋊C311yescharacteristic
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—91no—
515C2×C4—91no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—91no—
519C2×C4—181no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4—181no—
523C2×C4—181no—
524C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
525C8—21no—
526C8—181no—
527C2×C3—21no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—41no—
531C2×C3—61no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—121no—
535C2×C3—121no—
536C2×C3—121no—
537C2×C3—181no—
538C2×C3—181no—
539C2×C3—241no—
540C2×C3—361no—
541C2×C3C3×D3×C22⋊C411yescharacteristic
542D3—61no—
543D3—61no—
544D3—121no—
545D3—181no—
546D3—181no—
547D3—361no—
548C22—21no—
549C22—91no—
550C22—91no—
551C22—181no—
552C22—361no—
553C22—362no—
554C22D4×(C3⋊D3)⋊C311yescharacteristic
555C4—21no—
556C4—41no—
557C4—91no—
558C4—91no—
559C4—181no—
560C4—361no—
561C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
562C4D4×(C3⋊D3)⋊C311yescharacteristic
563C3—31no—
564C3—31no—
565C3—61no—
566C3C3×#482111yescharacteristic
567C2—21no—
568C2—41no—
569C2—181no—
570C2—181no—
571C2—361no—
572C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC4D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#104C2
socleC2×C3C3×D3×C22⋊C4
prime radical#41×(C3⋊D3)⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#104(C3⋊D3)⋊C3
hypocenterC32C3×#421
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. M16×(C3⋊D3)⋊C3C2semidirect
3. #64081⋊C3C2semidirect
4. H(3,𝔽3)×#104C2semidirect
5. #63664⋊C3C2semidirect
6. #63652⋊C3C2semidirect
7. #63632⋊C3C2semidirect
8. #10592520C3semidirect
11. #88722C4semidirect
12. Q8×(C3⋊D3)⋊C3C4semidirect
14. #3830⋊C3C4semidirect
16. D4×(C3⋊D3)⋊C3C4semidirect
21. M16×H(3,𝔽3)C22semidirect
22. C42×H(3,𝔽3)C22semidirect
25. C4×(C3⋊Dic3)⋊C3C22semidirect
26. #3748⋊C3C22semidirect
31. M16×C3⋊D3C2×C3semidirect
32. #64081C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63664C2×C3semidirect
39. #63652C2×C3semidirect
40. #63632C2×C3semidirect
77. #3832C3×C4semidirect
78. Q8×C3⋊D3C3×C4semidirect
80. #3830C3×C4semidirect
82. D4×C3⋊D3C3×C4semidirect
98. C32×M16C2×D3semidirect
99. C32×M16C22×C3semidirect
101. C32×C42C2×D3semidirect
102. C32×C42C22×C3semidirect
105. C4×C3⋊Dic3C22×C3semidirect
106. #3748C22×C3semidirect
322. #104(C3⋊D3)⋊C3semidirect
323. H(3,𝔽3)#421semidirect
440. M16C2×(C3⋊D3)⋊C3semidirect
445. C42C2×(C3⋊D3)⋊C3semidirect
498. C32#4821semidirect
499. C32C3×#421semidirect

Maximal subgroups

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

subgroupindex
#41×(C3⋊D3)⋊C32
M16×(C3⋊D3)⋊C32
#64081⋊C32
H(3,𝔽3)×#1042
#63664⋊C32
#63652⋊C32
#63632⋊C32
#105925203
C3×#48213