Finite Group Databasebeta

← #13147875

Subgroups — #13147875

Order 1728.

1116 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 157 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×#88652C212yesmaximal, complemented
2C4×C4⋊#272C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4#64078⋊C3C211yesmaximal, characteristic, complemented
5#11721159C212yesmaximal, complemented
6#11721152C212yesmaximal, complemented
7#11721106C212yesmaximal, complemented
8#11721102C214yesmaximal, complemented
9#10593896—31nomaximal
10#10593895—31nomaximal
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2212yes—
15C2×#634⋊C3C2212yes—
16C2×#6200C2214yes—
17C2×#6199C2214yes—
18#88652—28no—
19C42×H(3,𝔽3)C2211yescharacteristic
20#88542C2212yes—
21C4⋊#1084C2211yescharacteristic
22#3752⋊C3C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#88450C2214yes—
27#88449C2214yes—
28C2×#3796—32no—
29C2×#3795—32no—
30C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
31#64078D311yescharacteristic, complemented
32C3×C4×D12—31no—
33C3×C4×D12—31no—
34C3×C4×D12—31no—
35C3×C4×D12—61no—
36C3×#869—31no—
37#63909—32no—
38#63908—32no—
39#63891—34no—
40#63890—34no—
41D3×C4⋊Dic3—32no—
42D3×C4⋊Dic3—32no—
43#63871—32no—
44#63857—32no—
45C22×#272C2312yes—
46C22×(C3⋊D3)⋊C3C2312yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×#1084C2312yes—
51C4⋊#272D414yescomplemented
52C4×#272—24no—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C2×(C3⋊Dic3)⋊C3C2314yes—
55C4⋊((C3⋊D3)⋊C3)—22no—
56C4×(C3⋊D3)⋊C3—24no—
57#634⋊C3—24no—
58#6200—28no—
59#6199—28no—
60#5726—91no—
61#5555—91no—
62#5551—91no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4×C3⋊D3C2×D312yes—
65C2×#634C2×D312yes—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—61no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—32no—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—62no—
74C2×C3×Dic6—32no—
75C2×#626—34no—
76C2×#625—34no—
77C2×D3×Dic3—34no—
78C2×D3×Dic3—34no—
79#3796—68no—
80#3795—68no—
81C32×C42—21no—
82C32×C42C2×D311yescharacteristic
83C32×C42C2×D311yescharacteristic
84#3752C2×D312yes—
85C4⋊(C3⋊Dic3)C2×D312yes—
86C4⋊(C3⋊Dic3)C2×D312yes—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—62no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—32no—
96C3×C4⋊Dic3—32no—
97C3×C4⋊Dic3—61no—
98Dic3⋊Dic3—34no—
99Dic3⋊Dic3—34no—
100#3722—34no—
101(C2×D3)⋊Dic3—34no—
102C22×H(3,𝔽3)C2411yescharacteristic
103C2×#272—22no—
104C2×#272—22no—
105C2×#272C2×D414yes—
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3—22no—
108C2×(C3⋊D3)⋊C3—22no—
109C4×H(3,𝔽3)—21no—
110C4×H(3,𝔽3)C2×D412yes—
111C4×H(3,𝔽3)#4112yescomplemented
112#1084—22no—
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—24no—
115C2×#239—92no—
116C2×#237—92no—
117C2×#235—92no—
118C3×C4⋊Q8—91no—
119C3×C4×D4—91no—
120C3×C4×D4—91no—
121#946—92no—
122#938—92no—
123#930—92no—
124C2×C4⋊Dic3—92no—
125#924—92no—
126#897—92no—
127#895—92no—
128#894—94no—
129D3×C4⋊C4—92no—
130D3×C4⋊C4—92no—
131#886—94no—
132#883—94no—
133C4×D12—31no—
134C4×D12—31no—
135C4×D12—31no—
136C4×D12—61no—
137C4×D12—91no—
138#869—31no—
139#869—91no—
140C22×C3⋊D3C22×D312yes—
141C22×C3×D3—32no—
142C22×C3×D3—32no—
143C22×C3×D3—32no—
144C22×C3×D3—62no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3C22×D314yes—
155C2×C3⋊Dic3C22×D312yes—
156C4⋊(C3⋊D3)—22no—
157C4×C3⋊D3—24no—
158#634—24no—
159C2×C3×Dic3—34no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—62no—
164C3×D12—34no—
165C3×D12—34no—
166C3×D12—64no—
167C3×D12—62no—
168C3×C4×D3—64no—
169C3×C4×D3—64no—
170C3×C4×D3—64no—
171C3×C4×D3—124no—
172C3×Dic6—64no—
173#626—68no—
174#625—68no—
175D3×Dic3—68no—
176D3×Dic3—68no—
177#547—271noSylow
178C2×H(3,𝔽3)C22×D411yescharacteristic
179C2×H(3,𝔽3)C2×#4111yescharacteristic
180C2×H(3,𝔽3)2−1+411yescharacteristic
181#272—24no—
182(C3⋊D3)⋊C3—42no—
183C2×C3×Q8—92no—
184C2×C3×D4—91no—
185C2×C3×D4—91no—
186C22×C3×C4—92no—
187C22×C3×C4—92no—
188C2×#67—94no—
189C2×#67—94no—
190C22×Dic3—94no—
191#239—188no—
192#237—188no—
193#235—188no—
194C2×D12—31no—
195C2×D12—31no—
196C2×D12—31no—
197C2×D12—61no—
198C2×D12—91no—
199C2×D12—94no—
200C2×C4×D3—32no—
201C2×C4×D3—32no—
202C2×C4×D3—32no—
203C2×C4×D3—62no—
204C2×C4×D3—92no—
205C2×C4×D3—94no—
206C2×C4×D3—94no—
207C2×Dic6—32no—
208C2×Dic6—92no—
209C3×C4⋊C4—91no—
210C3×C4⋊C4—92no—
211C3×C4⋊C4—92no—
212C3×C4⋊C4—91no—
213C3×C22⋊C4—92no—
214C3×C22⋊C4—92no—
215C3×C42—31no—
216C3×C42—31no—
217C3×C42—61no—
218C3×C42D3211yescharacteristic
219#217—94no—
220(C2×D3)⋊C4—32no—
221(C2×D3)⋊C4—32no—
222(C2×D3)⋊C4—32no—
223(C2×D3)⋊C4—62no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4—94no—
226(C2×D3)⋊C4—94no—
227C4⋊Dic3—31no—
228C4⋊Dic3—31no—
229C4⋊Dic3—32no—
230C4⋊Dic3—32no—
231C4⋊Dic3—31no—
232C4⋊Dic3—61no—
233C4⋊Dic3—92no—
234C4⋊Dic3—92no—
235C4⋊Dic3—91no—
236C4⋊Dic3—94no—
237Dic3⋊C4—94no—
238Dic3⋊C4—94no—
239C22×C32—21no—
240C22×C32C23×D311yescharacteristic
241C22×C32C23×D311yescharacteristic
242C2×C3⋊D3—22no—
243C2×C3⋊D3—22no—
244C2×C3⋊D3—22no—
245C2×C3×D3—34no—
246C2×C3×D3—34no—
247C2×C3×D3—64no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3—122no—
256C2×C3×D3—122no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—22no—
260C32×C4—22no—
261C32×C4—41no—
262C32×C4#23512yescomplemented
263C32×C4D3×D412yes—
264C32×C4D3×D412yes—
265C32×C4#23912yescomplemented
266C3⋊Dic3—22no—
267C3⋊Dic3—24no—
268C3×Dic3—62no—
269C3×Dic3—62no—
270C3×Dic3—64no—
271C3×Dic3—62no—
272C3×Dic3—122no—
273C2×#41—272no—
274C4⋊Q8—91no—
275#123—274no—
276C22⋊Q8—272no—
277#121—272no—
278C4×D4—91no—
279C4×D4—91no—
280C2×C4⋊C4—272no—
281H(3,𝔽3)#54711yescharacteristic, complemented, Sylow
282C23×C3—92no—
283C23×C3—92no—
284C22×D3—32no—
285C22×D3—32no—
286C22×D3—32no—
287C22×D3—62no—
288C22×D3—92no—
289C3×Q8—184no—
290C3×D4—94no—
291C3×D4—182no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—92no—
302C2×C3×C4—94no—
303C2×C3×C4—92no—
304C2×C3×C4—184no—
305C2×C3×C4—184no—
306C2×C3×C4C2×D3211yescharacteristic
307C2×C3×C4C2×D3211yescharacteristic
308C2×C3×C4C2×D3211yescharacteristic
309#67—188no—
310#67—188no—
311C2×Dic3—32no—
312C2×Dic3—32no—
313C2×Dic3—34no—
314C2×Dic3—32no—
315C2×Dic3—62no—
316C2×Dic3—92no—
317C2×Dic3—94no—
318C2×Dic3—188no—
319D12—34no—
320D12—34no—
321D12—64no—
322D12—62no—
323D12—182no—
324D12—188no—
325C4×D3—64no—
326C4×D3—64no—
327C4×D3—64no—
328C4×D3—124no—
329C4×D3—184no—
330C4×D3—188no—
331C4×D3—188no—
332Dic6—64no—
333Dic6—184no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32C2×#23511yescharacteristic
338C2×C32C2×D3×D411yescharacteristic
339C2×C32C2×D3×D411yescharacteristic
340C2×C32C2×#23911yescharacteristic
341C2×C32#100711yescharacteristic
342C2×C32#101011yescharacteristic
343C3⋊D3—42no—
344C3×D3—64no—
345C3×D3—64no—
346C3×D3—124no—
347C3×D3—122no—
348#41—548no—
349C2×Q8—92no—
350C2×D4—91no—
351C2×D4—91no—
352C2×D4—274no—
353C22×C4—92no—
354C22×C4—92no—
355C22×C4—274no—
356C4⋊C4—91no—
357C4⋊C4—92no—
358C4⋊C4—91no—
359C4⋊C4—92no—
360C4⋊C4—274no—
361C22⋊C4—92no—
362C22⋊C4—92no—
363C22⋊C4—274no—
364C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—61no—
368C22×C3—94no—
369C22×C3—182no—
370C22×C3—182no—
371C22×C3—182no—
372C22×C3—182no—
373C22×C3—182no—
374C22×C3C22×D3211yescharacteristic
375C2×D3—34no—
376C2×D3—34no—
377C2×D3—64no—
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—62no—
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—122no—
386C2×D3—122no—
387C2×D3—182no—
388C2×D3—182no—
389C2×D3—182no—
390C3×C4—21no—
391C3×C4—32no—
392C3×C4—32no—
393C3×C4—32no—
394C3×C4—32no—
395C3×C4—61no—
396C3×C4—61no—
397C3×C4—62no—
398C3×C4—62no—
399C3×C4—121no—
400C3×C4—182no—
401C3×C4—184no—
402C3×C4—182no—
403C3×C4#379912yes—
404C3×C4#380212yes—
405Dic3—62no—
406Dic3—62no—
407Dic3—64no—
408Dic3—62no—
409Dic3—122no—
410Dic3—184no—
411Dic3—182no—
412C32—21no—
413C32#555111yescharacteristic, complemented
414C32#572611yescharacteristic, complemented
415C23—92no—
416C23—92no—
417Q8—184no—
418D4—94no—
419D4—182no—
420D4—548no—
421C2×C4—92no—
422C2×C4—92no—
423C2×C4—94no—
424C2×C4—184no—
425C2×C4—184no—
426C2×C4—548no—
427C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
428C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
429C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—61no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—184no—
440C2×C3—362no—
441C2×C3#6429711yescharacteristic
442C2×C3C2×#379911yescharacteristic
443C2×C3C2×#380211yescharacteristic
444D3—64no—
445D3—64no—
446D3—124no—
447D3—122no—
448D3—362no—
449C22—94no—
450C22—182no—
451C22—182no—
452C22—182no—
453C22—182no—
454C22—182no—
455C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
456C4—21no—
457C4—182no—
458C4—182no—
459C4—184no—
460C4#8865212yescomplemented
461C4#8865512yes—
462C3—31no—
463C3—31no—
464C3—61no—
465C3#1059386911yescharacteristic
466C2—184no—
467C2—362no—
468C2#1172248211yescharacteristic
469C2C2×#8865211yescharacteristic
470C2C2×#8865511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#547
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88652C2semidirect
2. C4×C4⋊#272C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #64078⋊C3C2semidirect
5. #11721159C2semidirect
6. #11721152C2semidirect
7. #11721106C2semidirect
8. #11721102C2semidirect
30. C4×C4⋊(C3⋊D3)D3semidirect
31. #64078D3semidirect
51. C4⋊#272D4semidirect
111. C4×H(3,𝔽3)#41semidirect
262. C32×C4#235semidirect
265. C32×C4#239semidirect
281. H(3,𝔽3)#547semidirect
413. C32#5551semidirect
414. C32#5726semidirect
460. C4#88652semidirect

Maximal subgroups

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

subgroupindex
C2×#886522
C4×C4⋊#2722
C4×C4⋊((C3⋊D3)⋊C3)2
#64078⋊C32
#117211592
#117211522
#117211062
#117211022
#105938963
#105938953