Finite Group Databasebeta

← #13150864

Subgroups — #13150864

Order 1728.

1193 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88740C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4#11721322C214yesmaximal, complemented
5#11721159C212yesmaximal, complemented
6#11721156C212yesmaximal, complemented
7#11721107C212yesmaximal, complemented
8#11721094C212yesmaximal, complemented
9#10595044—31nomaximal
10#10595043—31nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235C2212yes—
13#88740—24no—
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×#6229C2211yescharacteristic
17C2×#638⋊C3C2212yes—
18C22×(C3⋊Dic3)⋊C3C2212yes—
19#3828⋊C3—24no—
20C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
21C2×C4×(C3⋊D3)⋊C3C2212yes—
22C2×#634⋊C3C2211yescharacteristic
23C2×#6200C2214yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2212yes—
26#88451C2214yes—
27#88450C2214yes—
28#88448C2214yes—
29C2×#3828D311yescharacteristic, complemented
30C2×C3×#235—31no—
31C2×C3×#235—31no—
32C2×C3×#235—31no—
33C2×C3×#235—61no—
34C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
35C2×C3×C4⋊Dic3—31no—
36#63980—34no—
37#63978—34no—
38#63916—32no—
39#63909—32no—
40#63908—32no—
41#63908—32no—
42#63871—32no—
43#63857—32no—
44#63856—32no—
45#63856—32no—
46C23×H(3,𝔽3)C2311yescharacteristic
47C22×#272C2312yes—
48C22×(C3⋊D3)⋊C3C2312yes—
49C2×C4×H(3,𝔽3)D414yescomplemented
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52#6235—24no—
53C2×#1084C2312yes—
54C4⋊#272—22no—
55C4×#272—24no—
56#6229—22no—
57#638⋊C3—24no—
58C2×(C3⋊Dic3)⋊C3—42no—
59C2×(C3⋊Dic3)⋊C3C2314yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C4⋊((C3⋊D3)⋊C3)—22no—
62C4×(C3⋊D3)⋊C3—24no—
63#634⋊C3—22no—
64#6200—44no—
65#5833—91no—
66#5833—91no—
67#5510—91no—
68C22×C32×C4—21no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C2×#638C2×D312yes—
72C22×C3⋊Dic3C2×D312yes—
73#3828—24no—
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C4×C3⋊D3C2×D312yes—
76C2×#634C2×D311yescharacteristic
77C2×C3×#67—32no—
78C2×C3×#67—32no—
79C2×C3×#67—32no—
80C2×C3×#67—62no—
81C22×C3×Dic3—32no—
82C3×#235—68no—
83C3×#235—64no—
84C3×#235—64no—
85C3×#235—64no—
86C2×C3×D12—31no—
87C2×C3×D12—31no—
88C2×C3×D12—31no—
89C2×C3×D12—61no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—62no—
94C2×C3×Dic6—31no—
95C2×C3×Dic6—31no—
96C2×C3×Dic6—31no—
97C2×C3×Dic6—61no—
98C2×#626—34no—
99C2×#625—34no—
100C4⋊(C3⋊Dic3)C2×D312yes—
101C4⋊(C3⋊Dic3)C2×D312yes—
102C3×C4⋊Dic3—32no—
103C3×C4⋊Dic3—32no—
104#3724—34no—
105Dic3⋊Dic3—34no—
106#3722—34no—
107(C2×D3)⋊Dic3—34no—
108(C2×D3)⋊Dic3—34no—
109(C2×D3)⋊Dic3—34no—
110C22×H(3,𝔽3)—21no—
111C22×H(3,𝔽3)—21no—
112C22×H(3,𝔽3)C2×D412yes—
113C22×H(3,𝔽3)C2411yescharacteristic
114C2×#272—22no—
115C2×#272—22no—
116C2×#272—22no—
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—22no—
120C4×H(3,𝔽3)C2×D412yes—
121C4×H(3,𝔽3)C2×D412yes—
122#1084—22no—
123(C3⋊Dic3)⋊C3—22no—
124(C3⋊Dic3)⋊C3—24no—
125C2×C3×#41—91no—
126C2×C3×#41—91no—
127C2×#235—31no—
128C2×#235—31no—
129C2×#235—31no—
130C2×#235—61no—
131C2×#235—91no—
132C2×C3×C4⋊C4—91no—
133#946—92no—
134#946—92no—
135#939—94no—
136#939—94no—
137#938—92no—
138#938—92no—
139#930—92no—
140#930—92no—
141#930—94no—
142C2×C4⋊Dic3—31no—
143C2×C4⋊Dic3—91no—
144#924—92no—
145#924—92no—
146#897—92no—
147#897—92no—
148#895—92no—
149#895—92no—
150C23×C32—21no—
151C23×C32C22×D311yescharacteristic
152C23×C32C22×D311yescharacteristic
153C22×C3⋊D3C22×D312yes—
154C22×C3×D3—32no—
155C22×C3×D3—32no—
156C22×C3×D3—32no—
157C22×C3×D3—62no—
158C2×C32×C4—24no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4D1214yescomplemented
162C2×C32×C4#6714yescomplemented
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167#638—24no—
168C2×C3⋊Dic3—42no—
169C2×C3⋊Dic3C22×D312yes—
170C2×C3⋊Dic3C22×D314yes—
171C4⋊(C3⋊D3)—22no—
172C4×C3⋊D3—24no—
173#634—22no—
174C3×#67—68no—
175C3×#67—64no—
176C3×#67—64no—
177C3×#67—64no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—34no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—122no—
184C3×D12—64no—
185C3×D12—62no—
186C3×D12—62no—
187C3×D12—62no—
188C3×C4×D3—68no—
189C3×C4×D3—64no—
190C3×C4×D3—64no—
191C3×C4×D3—64no—
192C3×Dic6—64no—
193C3×Dic6—62no—
194C3×Dic6—62no—
195C3×Dic6—62no—
196#626—124no—
197#625—124no—
198#537—271noSylow
199C2×H(3,𝔽3)—22no—
200C2×H(3,𝔽3)C22×D411yescharacteristic
201C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
202C2×H(3,𝔽3)2−1+411yescharacteristic
203#272—42no—
204(C3⋊D3)⋊C3—42no—
205C3×#41—184no—
206C3×#41—184no—
207C2×C3×Q8—91no—
208C2×C3×Q8—91no—
209C2×C3×D4—91no—
210C2×C3×D4—92no—
211C2×C3×D4—92no—
212C2×C3×D4—91no—
213C22×C3×C4—31no—
214C22×C3×C4—31no—
215C22×C3×C4—61no—
216C22×C3×C4—92no—
217C22×C3×C4—92no—
218C22×C3×C4—92no—
219C22×C3×C4D3211yescharacteristic
220C2×#67—32no—
221C2×#67—32no—
222C2×#67—32no—
223C2×#67—62no—
224C2×#67—92no—
225C2×#67—94no—
226C2×#67—94no—
227C22×Dic3—32no—
228C22×Dic3—92no—
229#235—64no—
230#235—64no—
231#235—68no—
232#235—64no—
233#235—184no—
234C2×D12—31no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×D12—91no—
239C2×D12—94no—
240C2×C4×D3—32no—
241C2×C4×D3—32no—
242C2×C4×D3—32no—
243C2×C4×D3—62no—
244C2×C4×D3—92no—
245C2×Dic6—31no—
246C2×Dic6—31no—
247C2×Dic6—31no—
248C2×Dic6—61no—
249C2×Dic6—91no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252#217—94no—
253#217—94no—
254(C2×D3)⋊C4—94no—
255(C2×D3)⋊C4—94no—
256(C2×D3)⋊C4—94no—
257(C2×D3)⋊C4—94no—
258C4⋊Dic3—32no—
259C4⋊Dic3—32no—
260C4⋊Dic3—92no—
261C4⋊Dic3—92no—
262C4⋊Dic3—94no—
263Dic3⋊C4—94no—
264Dic3⋊C4—94no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—22no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32C2×D1212yes—
274C22×C32C2×#6712yes—
275C22×C32C23×D311yescharacteristic
276C22×C32C23×D311yescharacteristic
277C2×C3⋊D3—22no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3—22no—
280C2×C3×D3—64no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—122no—
291C2×C3×D3—122no—
292C32×C4—22no—
293C32×C4—22no—
294C32×C4C2×D1212yes—
295C32×C4C2×D1212yes—
296C32×C4C2×#6712yes—
297C32×C4C2×#6712yes—
298C3⋊Dic3—24no—
299C3⋊Dic3—22no—
300C3×Dic3—64no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—64no—
304C3×Dic3—62no—
305C2×#41—91no—
306C2×#41—91no—
307C22⋊Q8—272no—
308C22⋊Q8—272no—
309#121—272no—
310#121—272no—
311#121—274no—
312C2×C4⋊C4—91no—
313H(3,𝔽3)#53711yescharacteristic, complemented, Sylow
314C23×C3—31no—
315C23×C3—31no—
316C23×C3—61no—
317C23×C3—92no—
318C23×C3—92no—
319C23×C3C2×D3211yescharacteristic
320C22×D3—32no—
321C22×D3—32no—
322C22×D3—32no—
323C22×D3—62no—
324C22×D3—92no—
325C3×Q8—182no—
326C3×Q8—182no—
327C3×D4—182no—
328C3×D4—184no—
329C3×D4—184no—
330C3×D4—182no—
331C2×C3×C4—34no—
332C2×C3×C4—34no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—64no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—92no—
341C2×C3×C4—94no—
342C2×C3×C4—92no—
343C2×C3×C4—184no—
344C2×C3×C4—184no—
345C2×C3×C4—362no—
346C2×C3×C4#62614yes—
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349#67—64no—
350#67—64no—
351#67—68no—
352#67—64no—
353#67—184no—
354#67—364no—
355#67—364no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—34no—
360C2×Dic3—62no—
361C2×Dic3—94no—
362C2×Dic3—92no—
363C2×Dic3—122no—
364C2×Dic3—362no—
365D12—62no—
366D12—62no—
367D12—64no—
368D12—62no—
369D12—182no—
370D12—364no—
371C4×D3—64no—
372C4×D3—64no—
373C4×D3—68no—
374C4×D3—64no—
375C4×D3—184no—
376Dic6—62no—
377Dic6—62no—
378Dic6—64no—
379Dic6—62no—
380Dic6—182no—
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—42no—
387C2×C32C22×D1211yescharacteristic
388C2×C32#100411yescharacteristic
389C2×C32#100711yescharacteristic
390C2×C32#100911yescharacteristic
391C2×C32#101011yescharacteristic
392C2×C32C22×#6711yescharacteristic
393C3⋊D3—42no—
394C3×D3—124no—
395C3×D3—122no—
396C3×D3—122no—
397C3×D3—122no—
398#41—184no—
399#41—184no—
400C2×Q8—91no—
401C2×Q8—91no—
402C2×D4—92no—
403C2×D4—92no—
404C2×D4—91no—
405C2×D4—91no—
406C2×D4—274no—
407C22×C4—92no—
408C22×C4—92no—
409C22×C4—92no—
410C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
411C4⋊C4—92no—
412C4⋊C4—92no—
413C4⋊C4—274no—
414C22⋊C4—274no—
415C22⋊C4—274no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—31no—
419C22×C3—31no—
420C22×C3—32no—
421C22×C3—32no—
422C22×C3—61no—
423C22×C3—61no—
424C22×C3—61no—
425C22×C3—61no—
426C22×C3—61no—
427C22×C3—62no—
428C22×C3—121no—
429C22×C3—121no—
430C22×C3—182no—
431C22×C3—182no—
432C22×C3—182no—
433C22×C3—182no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3C2×#62612yes—
437C22×C3C22×D3211yescharacteristic
438C2×D3—62no—
439C2×D3—62no—
440C2×D3—62no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—64no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—62no—
448C2×D3—122no—
449C2×D3—122no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—182no—
453C3×C4—32no—
454C3×C4—32no—
455C3×C4—32no—
456C3×C4—32no—
457C3×C4—62no—
458C3×C4—62no—
459C3×C4—184no—
460C3×C4—182no—
461C3×C4—182no—
462C3×C4C2×#62612yes—
463C3×C4C2×#62612yes—
464Dic3—62no—
465Dic3—62no—
466Dic3—64no—
467Dic3—64no—
468Dic3—62no—
469Dic3—182no—
470Dic3—184no—
471C32—21no—
472C32#551011yescharacteristic, complemented
473C32#583311yescharacteristic, complemented
474C23—92no—
475C23—92no—
476C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
477Q8—182no—
478Q8—182no—
479D4—184no—
480D4—184no—
481D4—182no—
482D4—182no—
483D4—1084no—
484C2×C4—94no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4—184no—
488C2×C4—184no—
489C2×C4—362no—
490C2×C4#620014yescomplemented
491C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
492C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
493C2×C3—22no—
494C2×C3—31no—
495C2×C3—31no—
496C2×C3—31no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—31no—
500C2×C3—62no—
501C2×C3—62no—
502C2×C3—61no—
503C2×C3—61no—
504C2×C3—61no—
505C2×C3—122no—
506C2×C3—362no—
507C2×C3—362no—
508C2×C3#6429711yescharacteristic
509C2×C3#6430711yescharacteristic
510C2×C3C22×#62611yescharacteristic
511D3—122no—
512D3—122no—
513D3—124no—
514D3—122no—
515D3—362no—
516C22—21no—
517C22—21no—
518C22—182no—
519C22—182no—
520C22—182no—
521C22—182no—
522C22—182no—
523C22—182no—
524C22C2×#620012yes—
525C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
526C4—182no—
527C4—182no—
528C4—184no—
529C4C2×#620012yes—
530C4C2×#620012yes—
531C3—31no—
532C3—31no—
533C3—61no—
534C3#1059504311yescharacteristic
535C2—22no—
536C2—362no—
537C2—362no—
538C2#1172248211yescharacteristic
539C2#1172249211yescharacteristic
540C2C22×#620011yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#537
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88740C2semidirect
2. C2×#3828⋊C3C2semidirect
3. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
4. #11721322C2semidirect
5. #11721159C2semidirect
6. #11721156C2semidirect
7. #11721107C2semidirect
8. #11721094C2semidirect
29. C2×#3828D3semidirect
34. C2×C4⋊(C3⋊Dic3)D3semidirect
49. C2×C4×H(3,𝔽3)D4semidirect
161. C2×C32×C4D12semidirect
162. C2×C32×C4#67semidirect
313. H(3,𝔽3)#537semidirect
472. C32#5510semidirect
473. C32#5833semidirect
490. C2×C4#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C2×#3828⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
#117213222
#117211592
#117211562
#117211072
#117210942
#105950443
#105950433