Finite Group Databasebeta

← #13138052

Subgroups — #13138052

Order 1728.

661 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88649C211yesmaximal, characteristic, complemented
2#11721763C211yesmaximal, characteristic, complemented
3C2×#88437C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#11720323C211yesmaximal, characteristic, complemented
8#10590615—31nomaximal
9#10590613—31nomaximal
10C2×C4⋊#272—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic
15C2×#6200—21no—
16C2×((C3⋊D3)⋊C3)⋊C4—21no—
17#88649—22no—
18#88649—22no—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#88542—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88437—22no—
24#88436—22no—
25C2×#3799—31no—
26C2×#3795—31no—
27C3×#895—31no—
28C3×#895—31no—
29C3×#895—61no—
30C2×#3716—31no—
31C2×#3715—31no—
32C2×#3714—31no—
33C2×#3713—31no—
34#63637D311yescharacteristic, complemented
35#63636D311yescharacteristic, complemented
36C3×#810—31no—
37C3×#809—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×#272—21no—
41C22×#272D411yescharacteristic
42C22×(C3⋊D3)⋊C3—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084D411yescharacteristic
46C4⋊#272—21no—
47C4⋊#272—21no—
48C4⋊#272—42no—
49C4×#272—21no—
50C4×#272—21no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)D412yescomplemented
54#634⋊C3—21no—
55#634⋊C3D412yescomplemented
56#6200—24no—
57((C3⋊D3)⋊C3)⋊C4—24no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60#5037—91no—
61#4810—91no—
62#4781—91no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×#634C2×D311yescharacteristic
65C2×C3×D12—31no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×D12—61no—
70C2×C3×D12—61no—
71C2×C3×D12—62no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—61no—
75C2×C3×Dic6—31no—
76C2×#626—61no—
77C2×#625—61no—
78C2×#624—61no—
79C2×D3×Dic3—61no—
80#3799—62no—
81#3799—62no—
82#3795—62no—
83#3795—62no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×(C2×D3)⋊C4—61no—
90C3×(C2×D3)⋊C4—61no—
91C3×(C2×D3)⋊C4—62no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94#3716—62no—
95#3715—62no—
96#3714—62no—
97#3713—62no—
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C2×#272—21no—
100C2×#272—21no—
101C2×#272—21no—
102C2×#272—41no—
103C2×#272—41no—
104C2×#272—41no—
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3—41no—
108C4×H(3,𝔽3)—41no—
109C4×H(3,𝔽3)C2×D411yescharacteristic
110C4×H(3,𝔽3)C2×D411yescharacteristic
111#1084—21no—
112(C3⋊Dic3)⋊C3—22no—
113C2×#239—91no—
114C2×#237—91no—
115C2×#235—91no—
116C3×#121—91no—
117C2×#215—91no—
118C2×#214—91no—
119C2×#213—91no—
120C2×#213—91no—
121C2×D24—91no—
122C2×#204—91no—
123#895—31no—
124#895—31no—
125#895—61no—
126C3×#103—91no—
127C3×#102—91no—
128#830—91no—
129(C2×D3)⋊C8—91no—
130(C2×D3)⋊C8—91no—
131#810—31no—
132#810—91no—
133#809—31no—
134#809—91no—
135C22×C3⋊D3—21no—
136C22×C3×D3—31no—
137C22×C3×D3—31no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C22×C3×D3—62no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C2×C32×C4—22no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—21no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)D1212yescomplemented
152#634—21no—
153#634D1212yescomplemented
154C2×C3×Dic3—31no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C3×D12—32no—
159C3×D12—61no—
160C3×D12—61no—
161C3×D12—61no—
162C3×D12—61no—
163C3×D12—61no—
164C3×D12—121no—
165C3×D12—121no—
166C3×D12—124no—
167C3×D12—122no—
168C3×D12—122no—
169C3×C4×D3—61no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×C4×D3—61no—
173C3×C4×D3—121no—
174C3×C4×D3—121no—
175C3×Dic6—32no—
176C3×Dic6—61no—
177#626—64no—
178#625—64no—
179#624—64no—
180D3×Dic3—64no—
181C3⋊(C3⋊C8)—21no—
182C3⋊(C3⋊C8)—21no—
183C3×C3⋊C8—61no—
184C3×C3⋊C8—61no—
185#449—271noSylow
186C2×H(3,𝔽3)#12011yescharacteristic
187C2×H(3,𝔽3)#13511yescharacteristic
188C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
189#272—41no—
190#272—81no—
191(C3⋊D3)⋊C3—42no—
192C2×C3×Q8—91no—
193C2×C3×D4—91no—
194C2×C3×D4—91no—
195C2×C3×D4—181no—
196C22×C3×C4—91no—
197C2×#67—181no—
198C2×#67—181no—
199C22×Dic3—181no—
200#239—182no—
201#239—182no—
202#237—182no—
203#237—182no—
204#235—182no—
205#235—182no—
206C2×D12—31no—
207C2×D12—31no—
208C2×D12—31no—
209C2×D12—61no—
210C2×D12—61no—
211C2×D12—61no—
212C2×D12—62no—
213C2×D12—91no—
214C2×D12—181no—
215C2×C4×D3—31no—
216C2×C4×D3—31no—
217C2×C4×D3—61no—
218C2×C4×D3—181no—
219C2×C4×D3—181no—
220C2×Dic6—31no—
221C2×Dic6—91no—
222C2×C3×C8—91no—
223C2×C3×C8—91no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4D3211yescharacteristic
228C3×C22⋊C4—181no—
229#215—182no—
230#214—182no—
231#213—182no—
232#213—182no—
233(C2×D3)⋊C4—61no—
234(C2×D3)⋊C4—61no—
235(C2×D3)⋊C4—62no—
236C2×C3⋊C8—31no—
237C2×C3⋊C8—31no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—91no—
240D24—182no—
241#204—182no—
242C22×C32—21no—
243C22×C32D3×D411yescharacteristic
244C22×C32D3×D411yescharacteristic
245C2×C3⋊D3—22no—
246C2×C3⋊D3—41no—
247C2×C3⋊D3—41no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—61no—
251C2×C3×D3—61no—
252C2×C3×D3—61no—
253C2×C3×D3—61no—
254C2×C3×D3—61no—
255C2×C3×D3—61no—
256C2×C3×D3—121no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—122no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—42no—
272C32×C4—41no—
273C32×C4—41no—
274C32×C4C2×D1211yescharacteristic
275C32×C4C2×D1211yescharacteristic
276C32×C4D3×D411yescharacteristic
277C32×C4D3×D411yescharacteristic
278C3⋊Dic3—22no—
279C3×Dic3—62no—
280C3×Dic3—61no—
281C3×Dic3—61no—
282C3×Dic3—121no—
283C2×#41—271no—
284C2×SD16—271no—
285C2×D8—271no—
286#121—91no—
287#103—91no—
288#102—91no—
289C22⋊C8—271no—
290H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
291C23×C3—91no—
292C23×C3—181no—
293C23×C3—181no—
294C22×D3—31no—
295C22×D3—31no—
296C22×D3—61no—
297C22×D3—61no—
298C22×D3—61no—
299C22×D3—61no—
300C22×D3—62no—
301C22×D3—181no—
302C3×Q8—92no—
303C3×Q8—181no—
304C3×D4—92no—
305C3×D4—181no—
306C3×D4—181no—
307C3×D4—181no—
308C3×D4—362no—
309C2×C3×C4—21no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—62no—
315C2×C3×C4—61no—
316C2×C3×C4—91no—
317C2×C3×C4—181no—
318C2×C3×C4—181no—
319C2×C3×C4—181no—
320C2×C3×C4C2×D3211yescharacteristic
321#67—184no—
322#67—184no—
323C2×Dic3—31no—
324C2×Dic3—31no—
325C2×Dic3—61no—
326C2×Dic3—61no—
327C2×Dic3—181no—
328C2×Dic3—184no—
329D12—32no—
330D12—61no—
331D12—61no—
332D12—61no—
333D12—61no—
334D12—61no—
335D12—92no—
336D12—121no—
337D12—121no—
338D12—122no—
339D12—122no—
340D12—124no—
341D12—181no—
342D12—184no—
343C4×D3—61no—
344C4×D3—61no—
345C4×D3—61no—
346C4×D3—61no—
347C4×D3—121no—
348C4×D3—121no—
349C4×D3—184no—
350C4×D3—184no—
351Dic6—32no—
352Dic6—61no—
353Dic6—92no—
354Dic6—181no—
355C3×C8—181no—
356C3×C8—181no—
357C3⋊C8—61no—
358C3⋊C8—61no—
359C3⋊C8—181no—
360C3⋊C8—181no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32#88211yescharacteristic
365C2×C32#88211yescharacteristic
366C2×C32#91111yescharacteristic
367C2×C32#91411yescharacteristic
368C2×C32#91611yescharacteristic
369C2×C32#91911yescharacteristic
370C3⋊D3—42no—
371C3×D3—122no—
372C3×D3—121no—
373C3×D3—122no—
374C3×D3—121no—
375C3×D3—242no—
376C3×D3—241no—
377C3×D3—241no—
378#41—542no—
379#41—542no—
380C2×Q8—91no—
381C2×D4—91no—
382C2×D4—91no—
383C2×D4—181no—
384C2×D4—541no—
385C22×C4—91no—
386C22×C4—541no—
387SD16—542no—
388D8—542no—
389C2×C8—91no—
390C2×C8—91no—
391C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C22⋊C4—181no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—61no—
396C22×C3—182no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3—181no—
400C22×C3—361no—
401C22×C3—361no—
402C22×C3—361no—
403C22×C3—361no—
404C22×C3—361no—
405C22×C3#381111yescharacteristic
406C2×D3—61no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C2×D3—61no—
411C2×D3—61no—
412C2×D3—62no—
413C2×D3—62no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—121no—
417C2×D3—121no—
418C2×D3—121no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—122no—
425C2×D3—122no—
426C2×D3—122no—
427C2×D3—182no—
428C2×D3—361no—
429C2×D3—361no—
430C3×C4—31no—
431C3×C4—31no—
432C3×C4—31no—
433C3×C4—31no—
434C3×C4—41no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—121no—
438C3×C4—121no—
439C3×C4—122no—
440C3×C4—182no—
441C3×C4—181no—
442C3×C4D3×D1211yescharacteristic
443C3×C4D3×D1211yescharacteristic
444Dic3—61no—
445Dic3—61no—
446Dic3—62no—
447Dic3—121no—
448Dic3—182no—
449C32—21no—
450C32#478111yescharacteristic, complemented
451C32#481011yescharacteristic, complemented
452C23—91no—
453C23—181no—
454C23—181no—
455Q8—92no—
456Q8—181no—
457D4—92no—
458D4—181no—
459D4—181no—
460D4—181no—
461D4—362no—
462D4—544no—
463C2×C4—21no—
464C2×C4—91no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4—181no—
468C2×C4—544no—
469C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
470C8—181no—
471C8—181no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—31no—
477C2×C3—31no—
478C2×C3—61no—
479C2×C3—61no—
480C2×C3—61no—
481C2×C3—362no—
482C2×C3—361no—
483C2×C3—721no—
484C2×C3#6392311yescharacteristic
485C2×C3#6393711yescharacteristic
486C2×C3#6395011yescharacteristic
487D3—121no—
488D3—121no—
489D3—122no—
490D3—122no—
491D3—241no—
492D3—241no—
493D3—242no—
494D3—362no—
495C22—181no—
496C22—181no—
497C22—182no—
498C22—181no—
499C22—361no—
500C22—361no—
501C22—361no—
502C22—361no—
503C22—361no—
504C22#8867711yescharacteristic
505C4—41no—
506C4—181no—
507C4—182no—
508C4#8865411yescharacteristic
509C4#8865411yescharacteristic
510C3—31no—
511C3—31no—
512C3—61no—
513C3#1059061711yescharacteristic
514C2—361no—
515C2—362no—
516C2—721no—
517C2#1172116311yescharacteristic
518C2#1172119311yescharacteristic
519C2#1172123011yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#449
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88649C2semidirect
2. #11721763C2semidirect
3. C2×#88437C2semidirect
4. C2×#88436C2semidirect
5. #63636⋊C3C2semidirect
6. #63637⋊C3C2semidirect
7. #11720323C2semidirect
34. #63637D3semidirect
35. #63636D3semidirect
53. C4⋊((C3⋊D3)⋊C3)D4semidirect
55. #634⋊C3D4semidirect
151. C4⋊(C3⋊D3)D12semidirect
153. #634D12semidirect
290. H(3,𝔽3)#449semidirect
450. C32#4781semidirect
451. C32#4810semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
#117217632
C2×#884372
C2×#884362
#63636⋊C32
#63637⋊C32
#117203232
#105906153
#105906133