Finite Group Databasebeta

← #13150846

Subgroups — #13150846

Order 1728.

1319 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88740C211yesmaximal, characteristic, complemented
2C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#11721321C212yesmaximal, complemented
5#11721301C212yesmaximal, complemented
6#11721156C212yesmaximal, complemented
7C4×#6200C214yesmaximal, complemented
8#11721107C212yesmaximal, complemented
9#10595034—31nomaximal
10#10595022—31nomaximal
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#6235C2212yes—
14#88740—24no—
15C2×C4⋊#272C2211yescharacteristic, complemented
16C2×C4×#272C2212yescomplemented
17C2×#6229C2211yescharacteristic, complemented
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3—24no—
20C2×C4×(C3⋊D3)⋊C3C2212yes—
21C2×C4×(C3⋊D3)⋊C3C2212yes—
22C2×#6200C2214yes—
23#3757⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
25C4×(C3⋊Dic3)⋊C3C2212yescomplemented
26#88451C2214yescomplemented
27#88450C2214yes—
28#88448C2214yescomplemented
29C22×C4×C3⋊D3D311yescharacteristic, complemented
30C2×C3×#235—31no—
31C2×C3×#235—31no—
32C2×C3×#235—61no—
33C22×C3×C4×D3—31no—
34#64135D311yescharacteristic, complemented
35C3×#926—31no—
36#63981—32no—
37#63976—32no—
38#63962—32no—
39#63961—32no—
40#63916—32no—
41#63908—32no—
42C4×#626—34no—
43C4×#625—34no—
44#63871—32no—
45#63856—32no—
46C23×H(3,𝔽3)C2311yescharacteristic
47C22×#272C2312yes—
48C22×(C3⋊D3)⋊C3—22no—
49C22×(C3⋊D3)⋊C3—24no—
50C22×(C3⋊D3)⋊C3C2312yes—
51C2×C4×H(3,𝔽3)D414yescomplemented
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54#6235—24no—
55C2×#1084C2312yes—
56C4⋊#272—22no—
57C4×#272—24no—
58#6229—22no—
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3C2314yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C4×(C3⋊D3)⋊C3—24no—
63C4×(C3⋊D3)⋊C3—24no—
64#6200—28no—
65#5828—91no—
66#5791—91no—
67#5524—91no—
68C23×C3⋊D3C2×D311yescharacteristic
69C23×C3×D3—31no—
70C22×C32×C4—21no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C2×C4×C3⋊D3—24no—
75C2×C4×C3⋊D3C2×D312yes—
76C2×C4×C3⋊D3C2×D312yes—
77C2×C3×#67—32no—
78C2×C3×#67—32no—
79C2×C3×#67—62no—
80C22×C3×Dic3—31no—
81C3×#235—68no—
82C3×#235—64no—
83C3×#235—64no—
84C2×C3×D12—31no—
85C2×C3×D12—31no—
86C2×C3×D12—61no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—62no—
92C2×C3×C4×D3—64no—
93C2×C3×Dic6—31no—
94C2×C3×Dic6—31no—
95C2×C3×Dic6—61no—
96C2×#626—34no—
97C2×#625—34no—
98#3757C2×D312yes—
99C4⋊(C3⋊Dic3)C2×D312yescomplemented
100C4×C3⋊Dic3C2×D312yescomplemented
101C3×#217—32no—
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)—22no—
111C22×H(3,𝔽3)C2×D412yes—
112C22×H(3,𝔽3)C2411yescharacteristic
113C2×#272—22no—
114C2×#272—24no—
115C2×(C3⋊D3)⋊C3—28no—
116C2×(C3⋊D3)⋊C3—22no—
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3#4114yescomplemented
119C4×H(3,𝔽3)C2×D412yes—
120C4×H(3,𝔽3)C2×D412yes—
121#1084—22no—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—24no—
124C2×C3×#41—91no—
125C23×C3×C4—91no—
126C2×#235—31no—
127C2×#235—31no—
128C2×#235—61no—
129C22×C4×D3—31no—
130C22×C4×D3—91no—
131C3×#117—91no—
132#953—92no—
133#946—92no—
134#938—92no—
135#937—92no—
136#935—92no—
137#930—92no—
138#929—92no—
139C4×#67—94no—
140C4×#67—94no—
141#926—31no—
142#926—91no—
143#924—92no—
144#897—92no—
145#895—92no—
146#886—92no—
147#882—92no—
148C4×D12—94no—
149C23×C32—21no—
150C23×C32C22×D311yescharacteristic
151C23×C32C22×D311yescharacteristic
152C22×C3⋊D3—22no—
153C22×C3⋊D3—24no—
154C22×C3⋊D3C22×D312yes—
155C22×C3×D3—32no—
156C22×C3×D3—32no—
157C22×C3×D3—32no—
158C22×C3×D3—62no—
159C22×C3×D3—64no—
160C22×C3×D3—62no—
161C2×C32×C4—24no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4D1214yescomplemented
165C2×C32×C4#6714yescomplemented
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3C22×D312yes—
172C2×C3⋊Dic3C22×D314yes—
173C4×C3⋊D3—24no—
174C4×C3⋊D3—24no—
175C3×#67—68no—
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—62no—
184C3×D12—64no—
185C3×D12—62no—
186C3×D12—62no—
187C3×C4×D3—68no—
188C3×C4×D3—64no—
189C3×C4×D3—64no—
190C3×C4×D3—64no—
191C3×C4×D3—64no—
192C3×Dic6—64no—
193C3×Dic6—62no—
194C3×Dic6—62no—
195#626—68no—
196#625—68no—
197#525—271noSylow
198C2×H(3,𝔽3)—22no—
199C2×H(3,𝔽3)C22×D411yescharacteristic
200C2×H(3,𝔽3)C2×#4112yes—
201#272—42no—
202(C3⋊D3)⋊C3—24no—
203C24×C3—91no—
204C23×D3—31no—
205C23×D3—91no—
206C3×#41—184no—
207C2×C3×Q8—91no—
208C2×C3×D4—92no—
209C2×C3×D4—91no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—61no—
213C22×C3×C4—92no—
214C22×C3×C4—92no—
215C22×C3×C4—91no—
216C22×C3×C4—92no—
217C22×C3×C4—184no—
218C22×C3×C4D3211yescharacteristic
219C2×#67—32no—
220C2×#67—32no—
221C2×#67—62no—
222C2×#67—94no—
223C2×#67—94no—
224C22×Dic3—31no—
225C22×Dic3—91no—
226#235—64no—
227#235—64no—
228#235—68no—
229C2×D12—31no—
230C2×D12—31no—
231C2×D12—61no—
232C2×D12—94no—
233C2×C4×D3—32no—
234C2×C4×D3—32no—
235C2×C4×D3—32no—
236C2×C4×D3—32no—
237C2×C4×D3—62no—
238C2×C4×D3—64no—
239C2×C4×D3—92no—
240C2×C4×D3—92no—
241C2×C4×D3—184no—
242C2×Dic6—31no—
243C2×Dic6—31no—
244C2×Dic6—61no—
245C3×C4⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C42—92no—
248#217—32no—
249#217—92no—
250#217—94no—
251#217—94no—
252(C2×D3)⋊C4—94no—
253(C2×D3)⋊C4—94no—
254(C2×D3)⋊C4—94no—
255(C2×D3)⋊C4—94no—
256C4⋊Dic3—32no—
257C4⋊Dic3—92no—
258C4⋊Dic3—94no—
259Dic3⋊C4—94no—
260Dic3⋊C4—94no—
261C4×Dic3—32no—
262C4×Dic3—92no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—42no—
268C22×C32C2×D1212yes—
269C22×C32C2×#6712yes—
270C22×C32C23×D311yescharacteristic
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—28no—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3#23514yescomplemented
276C2×C3×D3—34no—
277C2×C3×D3—68no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—64no—
281C2×C3×D3—62no—
282C2×C3×D3—64no—
283C2×C3×D3—64no—
284C2×C3×D3—62no—
285C2×C3×D3—124no—
286C32×C4—22no—
287C32×C4—22no—
288C32×C4C2×D1212yes—
289C32×C4C2×D1212yes—
290C32×C4C2×#6712yes—
291C32×C4C2×#6712yes—
292C3⋊Dic3—24no—
293C3⋊Dic3—22no—
294C3×Dic3—64no—
295C3×Dic3—62no—
296C3×Dic3—62no—
297C3×Dic3—64no—
298C3×Dic3—62no—
299C2×#41—91no—
300C23×C4—91no—
301#123—272no—
302C22⋊Q8—272no—
303#121—272no—
304#120—272no—
305C4×D4—274no—
306#117—91no—
307H(3,𝔽3)#52511yescharacteristic, complemented, Sylow
308C23×C3—31no—
309C23×C3—31no—
310C23×C3—61no—
311C23×C3—92no—
312C23×C3—92no—
313C23×C3—182no—
314C23×C3—184no—
315C23×C3C2×D3211yescharacteristic
316C22×D3—32no—
317C22×D3—32no—
318C22×D3—32no—
319C22×D3—62no—
320C22×D3—64no—
321C22×D3—62no—
322C22×D3—92no—
323C22×D3—184no—
324C22×D3—182no—
325C3×Q8—182no—
326C3×D4—184no—
327C3×D4—182no—
328C2×C3×C4—34no—
329C2×C3×C4—34no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—64no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—92no—
338C2×C3×C4—94no—
339C2×C3×C4—92no—
340C2×C3×C4—184no—
341C2×C3×C4—184no—
342C2×C3×C4—182no—
343C2×C3×C4—184no—
344C2×C3×C4#62614yes—
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347#67—64no—
348#67—64no—
349#67—68no—
350#67—188no—
351#67—188no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—34no—
355C2×Dic3—32no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—94no—
359C2×Dic3—92no—
360C2×Dic3—182no—
361D12—62no—
362D12—62no—
363D12—64no—
364D12—188no—
365C4×D3—64no—
366C4×D3—64no—
367C4×D3—68no—
368C4×D3—64no—
369C4×D3—64no—
370C4×D3—184no—
371C4×D3—184no—
372Dic6—62no—
373Dic6—62no—
374Dic6—64no—
375C2×C32—22no—
376C2×C32—22no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—42no—
380C2×C32C22×D1211yescharacteristic
381C2×C32C2×#23512yes—
382C2×C32D3×#4112yes—
383C2×C32C22×#6711yescharacteristic
384C3⋊D3—24no—
385C3×D3—64no—
386C3×D3—124no—
387C3×D3—122no—
388C3×D3—122no—
389C24—91no—
390#41—184no—
391C2×Q8—91no—
392C2×D4—92no—
393C2×D4—91no—
394C2×D4—274no—
395C22×C4—91no—
396C22×C4—92no—
397C22×C4—92no—
398C22×C4—92no—
399C22×C4—184no—
400C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
401C4⋊C4—92no—
402C4⋊C4—274no—
403C22⋊C4—92no—
404C22⋊C4—274no—
405C22⋊C4—274no—
406C42—92no—
407C22×C3—22no—
408C22×C3—31no—
409C22×C3—31no—
410C22×C3—32no—
411C22×C3—32no—
412C22×C3—62no—
413C22×C3—62no—
414C22×C3—61no—
415C22×C3—62no—
416C22×C3—94no—
417C22×C3—122no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3—188no—
421C22×C3—184no—
422C22×C3—182no—
423C22×C3C2×#62612yes—
424C22×C3C22×D3211yescharacteristic
425C2×D3—34no—
426C2×D3—68no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—64no—
432C2×D3—64no—
433C2×D3—64no—
434C2×D3—94no—
435C2×D3—124no—
436C2×D3—182no—
437C2×D3—182no—
438C2×D3—188no—
439C3×C4—32no—
440C3×C4—32no—
441C3×C4—32no—
442C3×C4—32no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—184no—
446C3×C4—182no—
447C3×C4—182no—
448C3×C4C2×#62612yes—
449C3×C4C2×#62612yes—
450Dic3—62no—
451Dic3—62no—
452Dic3—64no—
453Dic3—64no—
454Dic3—62no—
455Dic3—182no—
456Dic3—184no—
457C32—21no—
458C32#552411yescharacteristic, complemented
459C32#579111yescharacteristic, complemented
460C23—92no—
461C23—92no—
462C23—184no—
463C23—182no—
464C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
465Q8—182no—
466D4—184no—
467D4—182no—
468D4—548no—
469C2×C4—94no—
470C2×C4—92no—
471C2×C4—92no—
472C2×C4—182no—
473C2×C4—184no—
474C2×C4—184no—
475C2×C4—184no—
476C2×C4#620014yescomplemented
477C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
478C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
479C2×C3—22no—
480C2×C3—31no—
481C2×C3—31no—
482C2×C3—32no—
483C2×C3—32no—
484C2×C3—62no—
485C2×C3—62no—
486C2×C3—61no—
487C2×C3—62no—
488C2×C3—122no—
489C2×C3—184no—
490C2×C3—362no—
491C2×C3D3×#23512yes—
492C2×C3C22×#62611yescharacteristic
493D3—64no—
494D3—122no—
495D3—122no—
496D3—124no—
497D3—184no—
498C22—22no—
499C22—94no—
500C22—188no—
501C22—182no—
502C22—182no—
503C22—182no—
504C22—184no—
505C22C2×#620012yes—
506C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
507C4—182no—
508C4—182no—
509C4—184no—
510C4C2×#620012yes—
511C4C2×#620012yes—
512C3—31no—
513C3—31no—
514C3—61no—
515C3#1059502311yescharacteristic
516C2—22no—
517C2—184no—
518C2—362no—
519C2#1172249012yes—
520C2C22×#620011yescharacteristic
subgroupG/H
centerC2×C4C2×((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)#525
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88740C2semidirect
2. C22×C4×(C3⋊D3)⋊C3C2semidirect
3. #64135⋊C3C2semidirect
4. #11721321C2semidirect
5. #11721301C2semidirect
6. #11721156C2semidirect
7. C4×#6200C2semidirect
8. #11721107C2semidirect
15. C2×C4⋊#272C22semidirect
16. C2×C4×#272C22semidirect
17. C2×#6229C22semidirect
24. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
25. C4×(C3⋊Dic3)⋊C3C22semidirect
26. #88451C22semidirect
28. #88448C22semidirect
29. C22×C4×C3⋊D3D3semidirect
34. #64135D3semidirect
51. C2×C4×H(3,𝔽3)D4semidirect
99. C4⋊(C3⋊Dic3)C2×D3semidirect
100. C4×C3⋊Dic3C2×D3semidirect
118. C2×(C3⋊D3)⋊C3#41semidirect
164. C2×C32×C4D12semidirect
165. C2×C32×C4#67semidirect
275. C2×C3⋊D3#235semidirect
307. H(3,𝔽3)#525semidirect
458. C32#5524semidirect
459. C32#5791semidirect
476. 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
C22×C4×(C3⋊D3)⋊C32
#64135⋊C32
#117213212
#117213012
#117211562
C4×#62002
#117211072
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