Finite Group Databasebeta

← #13150848

Subgroups — #13150848

Order 1728.

1249 conjugacy classes of subgroups.

The subgroup lattice has 536 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)⋊C3C211yesmaximal, characteristic, complemented
4#11721322C214yesmaximal, complemented
5#11721158C211yesmaximal, characteristic, complemented
6C4×#6200C214yesmaximal, complemented
7#11721114C211yesmaximal, characteristic, complemented
8#11721110C211yesmaximal, characteristic, complemented
9#11721090C211yesmaximal, characteristic, complemented
10#10595032—31nomaximal
11#10595029—31nomaximal
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#6235C2212yes—
14#88740—24no—
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272C2212yes—
17C2×#6229C2211yescharacteristic
18C2×#638⋊C3C2212yes—
19C22×(C3⋊Dic3)⋊C3C2212yes—
20#3828⋊C3—24no—
21C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
22C2×C4×(C3⋊D3)⋊C3C2212yes—
23C2×#634⋊C3C2211yescharacteristic
24C2×#6200C2214yes—
25C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
26C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2212yes—
28#88451C2214yes—
29#88450C2214yes—
30#88448C2214yes—
31C2×#3828D311yescharacteristic, complemented
32C2×C3×#235—31no—
33C2×C3×#235—31no—
34C2×C3×#235—31no—
35C2×C3×#235—61no—
36C2×C4×C3⋊Dic3D311yescharacteristic, complemented
37C2×C3×C4×Dic3—31no—
38#63980—34no—
39#63978—34no—
40#63914—31no—
41#63911—31no—
42C4×#626—34no—
43C4×#625—34no—
44#63873—31no—
45#63866—31no—
46#63864—31no—
47#63863—31no—
48#63863—31no—
49#63860—31no—
50C23×H(3,𝔽3)C2311yescharacteristic
51C22×#272C2312yes—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×C4×H(3,𝔽3)D414yescomplemented
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56#6235—24no—
57C2×#1084C2312yes—
58C4⋊#272—22no—
59C4×#272—24no—
60#6229—22no—
61#638⋊C3—24no—
62C2×(C3⋊Dic3)⋊C3—24no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C4⋊((C3⋊D3)⋊C3)—22no—
67C4×(C3⋊D3)⋊C3—24no—
68#634⋊C3—22no—
69#6200—28no—
70#5832—91no—
71#5832—91no—
72#5477—91no—
73C22×C32×C4—21no—
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C2×#638C2×D312yes—
77C22×C3⋊Dic3C2×D312yes—
78#3828—24no—
79C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
80C2×C4×C3⋊D3C2×D312yes—
81C2×#634C2×D311yescharacteristic
82C2×C3×#67—32no—
83C2×C3×#67—32no—
84C2×C3×#67—32no—
85C2×C3×#67—62no—
86C22×C3×Dic3—32no—
87C3×#235—68no—
88C3×#235—64no—
89C3×#235—64no—
90C3×#235—64no—
91C2×C3×D12—31no—
92C2×C3×D12—31no—
93C2×C3×D12—31no—
94C2×C3×D12—61no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—32no—
98C2×C3×C4×D3—62no—
99C2×C3×Dic6—31no—
100C2×C3×Dic6—31no—
101C2×C3×Dic6—31no—
102C2×C3×Dic6—61no—
103C2×#626—34no—
104C2×#625—34no—
105C4×C3⋊Dic3C2×D311yescharacteristic
106C4×C3⋊Dic3C2×D311yescharacteristic
107C4×C3⋊Dic3C2×D312yes—
108C3×C4×Dic3—31no—
109C3×C4×Dic3—31no—
110C3×C4×Dic3—32no—
111#3724—34no—
112Dic3⋊Dic3—34no—
113#3722—34no—
114(C2×D3)⋊Dic3—34no—
115(C2×D3)⋊Dic3—34no—
116(C2×D3)⋊Dic3—34no—
117C22×H(3,𝔽3)—22no—
118C22×H(3,𝔽3)C2×D412yes—
119C22×H(3,𝔽3)C2411yescharacteristic
120C2×#272—22no—
121C2×#272—24no—
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—24no—
124C4×H(3,𝔽3)C2×D412yes—
125C4×H(3,𝔽3)C2×D412yes—
126#1084—22no—
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3—22no—
129(C3⋊Dic3)⋊C3#4114yescomplemented
130C2×C3×#41—91no—
131C2×C3×#41—91no—
132C2×#235—31no—
133C2×#235—31no—
134C2×#235—31no—
135C2×#235—61no—
136C2×#235—91no—
137C2×C3×C42—91no—
138#947—91no—
139#947—91no—
140#944—91no—
141#944—91no—
142#940—91no—
143#940—91no—
144#939—94no—
145#939—94no—
146#936—91no—
147#936—91no—
148#930—94no—
149C4×#67—94no—
150C4×#67—94no—
151C2×C4×Dic3—31no—
152C2×C4×Dic3—91no—
153#875—91no—
154#875—91no—
155#874—91no—
156C4×D12—94no—
157#869—91no—
158C23×C32—21no—
159C23×C32C22×D311yescharacteristic
160C23×C32C22×D311yescharacteristic
161C22×C3⋊D3C22×D312yes—
162C22×C3×D3—32no—
163C22×C3×D3—32no—
164C22×C3×D3—32no—
165C22×C3×D3—62no—
166C2×C32×C4—24no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4D1214yescomplemented
170C2×C32×C4#6714yescomplemented
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175#638—24no—
176C2×C3⋊Dic3—24no—
177C2×C3⋊Dic3C22×D312yes—
178C2×C3⋊Dic3C22×D312yes—
179C2×C3⋊Dic3C22×D312yes—
180C4⋊(C3⋊D3)—22no—
181C4×C3⋊D3—24no—
182#634—22no—
183C3×#67—68no—
184C3×#67—64no—
185C3×#67—64no—
186C3×#67—64no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—64no—
194C3×D12—64no—
195C3×D12—62no—
196C3×D12—62no—
197C3×D12—62no—
198C3×C4×D3—68no—
199C3×C4×D3—64no—
200C3×C4×D3—64no—
201C3×C4×D3—64no—
202C3×Dic6—64no—
203C3×Dic6—62no—
204C3×Dic6—62no—
205C3×Dic6—62no—
206#626—68no—
207#625—68no—
208#532—271noSylow
209C2×H(3,𝔽3)—22no—
210C2×H(3,𝔽3)C22×D411yescharacteristic
211C2×H(3,𝔽3)C2×#4112yes—
212#272—42no—
213(C3⋊D3)⋊C3—42no—
214C3×#41—184no—
215C3×#41—184no—
216C2×C3×Q8—91no—
217C2×C3×Q8—91no—
218C2×C3×D4—91no—
219C2×C3×D4—92no—
220C2×C3×D4—92no—
221C2×C3×D4—91no—
222C22×C3×C4—31no—
223C22×C3×C4—31no—
224C22×C3×C4—61no—
225C22×C3×C4—92no—
226C22×C3×C4—92no—
227C22×C3×C4—92no—
228C22×C3×C4D3211yescharacteristic
229C2×#67—32no—
230C2×#67—32no—
231C2×#67—32no—
232C2×#67—62no—
233C2×#67—92no—
234C2×#67—94no—
235C2×#67—94no—
236C22×Dic3—32no—
237C22×Dic3—92no—
238#235—64no—
239#235—64no—
240#235—68no—
241#235—64no—
242#235—184no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—61no—
247C2×D12—91no—
248C2×D12—94no—
249C2×C4×D3—32no—
250C2×C4×D3—32no—
251C2×C4×D3—32no—
252C2×C4×D3—62no—
253C2×C4×D3—92no—
254C2×Dic6—31no—
255C2×Dic6—31no—
256C2×Dic6—31no—
257C2×Dic6—61no—
258C2×Dic6—91no—
259C3×C42—91no—
260C3×C42—92no—
261C3×C42—91no—
262#217—94no—
263#217—94no—
264(C2×D3)⋊C4—94no—
265(C2×D3)⋊C4—94no—
266(C2×D3)⋊C4—94no—
267(C2×D3)⋊C4—94no—
268C4⋊Dic3—94no—
269Dic3⋊C4—94no—
270Dic3⋊C4—94no—
271C4×Dic3—32no—
272C4×Dic3—31no—
273C4×Dic3—31no—
274C4×Dic3—92no—
275C4×Dic3—91no—
276C4×Dic3—91no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—21no—
280C22×C32—22no—
281C22×C32—42no—
282C22×C32C2×D1212yes—
283C22×C32C2×#6712yes—
284C22×C32C23×D311yescharacteristic
285C22×C32C23×D311yescharacteristic
286C2×C3⋊D3—24no—
287C2×C3⋊D3—22no—
288C2×C3×D3—64no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—64no—
292C2×C3×D3—64no—
293C2×C3×D3—64no—
294C2×C3×D3—62no—
295C2×C3×D3—124no—
296C32×C4—22no—
297C32×C4—22no—
298C32×C4C2×D1212yes—
299C32×C4C2×D1212yes—
300C32×C4C2×#6712yes—
301C32×C4C2×#6712yes—
302C3⋊Dic3—22no—
303C3⋊Dic3—22no—
304C3⋊Dic3#23514yescomplemented
305C3×Dic3—34no—
306C3×Dic3—64no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C2×#41—91no—
312C2×#41—91no—
313C4⋊Q8—271no—
314#127—271no—
315#124—271no—
316#124—271no—
317#121—274no—
318C4×D4—274no—
319C2×C42—91no—
320H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
321C23×C3—31no—
322C23×C3—31no—
323C23×C3—61no—
324C23×C3—92no—
325C23×C3—92no—
326C23×C3C2×D3211yescharacteristic
327C22×D3—32no—
328C22×D3—32no—
329C22×D3—32no—
330C22×D3—62no—
331C22×D3—92no—
332C3×Q8—182no—
333C3×Q8—182no—
334C3×D4—182no—
335C3×D4—184no—
336C3×D4—184no—
337C3×D4—182no—
338C2×C3×C4—34no—
339C2×C3×C4—34no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—64no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—92no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4—184no—
352C2×C3×C4—184no—
353C2×C3×C4—184no—
354C2×C3×C4#62614yes—
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357#67—64no—
358#67—64no—
359#67—68no—
360#67—64no—
361#67—184no—
362#67—188no—
363#67—188no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—62no—
370C2×Dic3—64no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—92no—
374C2×Dic3—184no—
375D12—62no—
376D12—62no—
377D12—64no—
378D12—62no—
379D12—182no—
380D12—188no—
381C4×D3—64no—
382C4×D3—64no—
383C4×D3—68no—
384C4×D3—64no—
385C4×D3—184no—
386Dic6—62no—
387Dic6—62no—
388Dic6—64no—
389Dic6—62no—
390Dic6—182no—
391C2×C32—22no—
392C2×C32—22no—
393C2×C32—22no—
394C2×C32—21no—
395C2×C32—42no—
396C2×C32C22×D1211yescharacteristic
397C2×C32C2×#23512yes—
398C2×C32D3×#4112yes—
399C2×C32C22×#6711yescharacteristic
400C3⋊D3—42no—
401C3×D3—124no—
402C3×D3—122no—
403C3×D3—122no—
404C3×D3—122no—
405#41—184no—
406#41—184no—
407C2×Q8—91no—
408C2×Q8—91no—
409C2×D4—92no—
410C2×D4—92no—
411C2×D4—91no—
412C2×D4—91no—
413C2×D4—274no—
414C22×C4—92no—
415C22×C4—92no—
416C22×C4—92no—
417C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
418C4⋊C4—274no—
419C22⋊C4—274no—
420C22⋊C4—274no—
421C42—91no—
422C42—91no—
423C42—92no—
424C22×C3—22no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—32no—
428C22×C3—32no—
429C22×C3—62no—
430C22×C3—62no—
431C22×C3—61no—
432C22×C3—62no—
433C22×C3—122no—
434C22×C3—184no—
435C22×C3—182no—
436C22×C3—184no—
437C22×C3—182no—
438C22×C3C2×#62612yes—
439C22×C3C22×D3211yescharacteristic
440C2×D3—62no—
441C2×D3—62no—
442C2×D3—64no—
443C2×D3—64no—
444C2×D3—64no—
445C2×D3—64no—
446C2×D3—62no—
447C2×D3—124no—
448C2×D3—182no—
449C2×D3—184no—
450C3×C4—32no—
451C3×C4—32no—
452C3×C4—32no—
453C3×C4—32no—
454C3×C4—62no—
455C3×C4—62no—
456C3×C4—94no—
457C3×C4—182no—
458C3×C4—182no—
459C3×C4—182no—
460C3×C4C2×#62612yes—
461C3×C4C2×#62612yes—
462Dic3—34no—
463Dic3—62no—
464Dic3—62no—
465Dic3—64no—
466Dic3—62no—
467Dic3—62no—
468Dic3—94no—
469Dic3—182no—
470Dic3—182no—
471C32—21no—
472C32#547711yescharacteristic, complemented
473C32#583211yescharacteristic, complemented
474C23—92no—
475C23—92no—
476C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
477Q8—182no—
478Q8—182no—
479D4—184no—
480D4—184no—
481D4—182no—
482D4—182no—
483D4—548no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4—92no—
488C2×C4—184no—
489C2×C4—184no—
490C2×C4—184no—
491C2×C4#620014yescomplemented
492C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
493C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
494C2×C3—22no—
495C2×C3—31no—
496C2×C3—31no—
497C2×C3—32no—
498C2×C3—32no—
499C2×C3—62no—
500C2×C3—62no—
501C2×C3—61no—
502C2×C3—62no—
503C2×C3—122no—
504C2×C3—362no—
505C2×C3—362no—
506C2×C3D3×#23512yes—
507C2×C3C22×#62611yescharacteristic
508D3—122no—
509D3—122no—
510D3—124no—
511D3—122no—
512D3—362no—
513C22—22no—
514C22—182no—
515C22—182no—
516C22—184no—
517C22—184no—
518C22C2×#620012yes—
519C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
520C4—94no—
521C4—182no—
522C4—182no—
523C4—182no—
524C4C2×#620012yes—
525C4C2×#620012yes—
526C3—31no—
527C3—31no—
528C3—61no—
529C3#1059502911yescharacteristic
530C2—22no—
531C2—362no—
532C2—362no—
533C2#1172249012yes—
534C2C22×#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)#532
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. C2×#3828⋊C3C2semidirect
3. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
4. #11721322C2semidirect
5. #11721158C2semidirect
6. C4×#6200C2semidirect
7. #11721114C2semidirect
8. #11721110C2semidirect
9. #11721090C2semidirect
31. C2×#3828D3semidirect
36. C2×C4×C3⋊Dic3D3semidirect
53. C2×C4×H(3,𝔽3)D4semidirect
129. (C3⋊Dic3)⋊C3#41semidirect
169. C2×C32×C4D12semidirect
170. C2×C32×C4#67semidirect
304. C3⋊Dic3#235semidirect
320. H(3,𝔽3)#532semidirect
472. C32#5477semidirect
473. C32#5832semidirect
491. 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)⋊C32
#117213222
#117211582
C4×#62002
#117211142
#117211102
#117210902
#105950323
#105950293