Finite Group Databasebeta

← #13147825

Subgroups — #13147825

Order 1728.

1162 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 1162 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×#88648C212yesmaximal, complemented
2C4×#6229C211yesmaximal, characteristic, complemented
3#11721385C211yesmaximal, characteristic, complemented
4C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
5#11721144C212yesmaximal, complemented
6#11721141C212yesmaximal, complemented
7#11721136C212yesmaximal, complemented
8#11721126C214yesmaximal, complemented
9#10593902—31nomaximal
10#10593900—31nomaximal
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
15C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×#6198C2214yes—
18C2×#6198C2214yes—
19C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
20#88648—28no—
21C42×H(3,𝔽3)C2211yescharacteristic
22C4⋊#1084C2211yescharacteristic
23#88540C2212yes—
24C4×#1084C2212yes—
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27#88450C2214yes—
28C2×#3799—32no—
29C2×#3796—32no—
30#64083D311yescharacteristic, complemented
31C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
32C3×#874—31no—
33C3×C4×D12—31no—
34C3×C4×Dic6—31no—
35C3×C4×Dic6—31no—
36C3×C4×Dic6—61no—
37#63916—32no—
38#63911—32no—
39#63911—32no—
40#63910—34no—
41#63909—32no—
42#63894—32no—
43Dic3×D12—32no—
44#63886—34no—
45C22×(C3⋊D3)⋊C3C2314yes—
46C22×(C3⋊D3)⋊C3C2312yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×#1084C2312yes—
51C2×#1084C2312yes—
52#6229D414yescomplemented
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C4⋊((C3⋊D3)⋊C3)—24no—
55C4⋊((C3⋊D3)⋊C3)—24no—
56C4⋊((C3⋊D3)⋊C3)—22no—
57C4⋊((C3⋊D3)⋊C3)—42no—
58C4×(C3⋊D3)⋊C3—24no—
59#6198—28no—
60#6198—44no—
61((C3⋊D3)⋊C3)⋊C4—28no—
62#5718—91no—
63#5577—91no—
64#5553—91no—
65C2×C4⋊(C3⋊D3)C2×D312yes—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4⋊(C3⋊D3)C2×D312yes—
68C2×C4⋊(C3⋊D3)C2×D312yes—
69C2×C4×C3⋊D3C2×D312yes—
70C2×C3×D12—32no—
71C2×C3×D12—32no—
72C2×C3×D12—31no—
73C2×C3×D12—32no—
74C2×C3×C4×D3—32no—
75C2×C3×Dic6—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—61no—
78C2×#626—34no—
79C2×#626—34no—
80C2×#626—34no—
81C2×#626—34no—
82C2×#624—34no—
83C2×D3×Dic3—34no—
84#3799—68no—
85#3796—68no—
86C32×C42—21no—
87C32×C42C2×D311yescharacteristic
88C32×C42C2×D311yescharacteristic
89#3752C2×D312yes—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—61no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—62no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—32no—
101C3×C4×Dic3—62no—
102#3722—34no—
103(C2×D3)⋊Dic3—34no—
104C22×H(3,𝔽3)C2411yescharacteristic
105C2×(C3⋊D3)⋊C3—24no—
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3—24no—
108C2×(C3⋊D3)⋊C3—22no—
109C2×(C3⋊D3)⋊C3—22no—
110C2×(C3⋊D3)⋊C3—24no—
111C4×H(3,𝔽3)—21no—
112C4×H(3,𝔽3)C2×D412yes—
113C4×H(3,𝔽3)#4112yescomplemented
114#1084—22no—
115#1084C2×D414yes—
116(C3⋊Dic3)⋊C3—22no—
117C2×#239—92no—
118C2×#237—92no—
119C2×#235—92no—
120C3×#127—91no—
121C3×C4×Q8—91no—
122C3×C4×D4—91no—
123#940—92no—
124#940—92no—
125#939—94no—
126#938—92no—
127D4×Dic3—92no—
128#930—92no—
129#895—94no—
130#895—92no—
131#893—92no—
132#884—94no—
133#874—31no—
134#874—91no—
135#874—92no—
136C4×D12—31no—
137C4×D12—91no—
138C4×D12—92no—
139C4×Dic6—31no—
140C4×Dic6—31no—
141C4×Dic6—61no—
142C22×C3⋊D3C22×D314yes—
143C22×C3⋊D3C22×D312yes—
144C22×C3×D3—34no—
145C22×C3×D3—32no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3C22×D312yes—
156C4⋊(C3⋊D3)—22no—
157C4⋊(C3⋊D3)—24no—
158C4⋊(C3⋊D3)—24no—
159C4⋊(C3⋊D3)—42no—
160C4×C3⋊D3—24no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—62no—
168C3×D12—64no—
169C3×D12—64no—
170C3×D12—62no—
171C3×D12—122no—
172C3×C4×D3—64no—
173C3×Dic6—34no—
174C3×Dic6—34no—
175C3×Dic6—64no—
176#626—68no—
177#626—68no—
178#626—124no—
179#626—124no—
180#624—68no—
181D3×Dic3—68no—
182#550—271noSylow
183C2×H(3,𝔽3)C22×D411yescharacteristic
184C2×H(3,𝔽3)C2×#4111yescharacteristic
185C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
186(C3⋊D3)⋊C3—42no—
187(C3⋊D3)⋊C3—44no—
188C2×C3×Q8—91no—
189C2×C3×D4—92no—
190C2×C3×D4—92no—
191C2×C3×D4—91no—
192C2×C3×D4—92no—
193C22×C3×C4—92no—
194C2×#67—94no—
195C2×#67—94no—
196C2×#67—94no—
197C2×#67—94no—
198C22×Dic3—94no—
199#239—188no—
200#237—188no—
201#235—188no—
202C2×D12—31no—
203C2×D12—32no—
204C2×D12—32no—
205C2×D12—32no—
206C2×D12—91no—
207C2×D12—92no—
208C2×D12—92no—
209C2×D12—92no—
210C2×D12—94no—
211C2×D12—94no—
212C2×C4×D3—32no—
213C2×C4×D3—92no—
214C2×C4×D3—94no—
215C2×C4×D3—94no—
216C2×Dic6—31no—
217C2×Dic6—31no—
218C2×Dic6—61no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—92no—
221C3×C4⋊C4—91no—
222C3×C22⋊C4—92no—
223C3×C42—31no—
224C3×C42—31no—
225C3×C42—61no—
226C3×C42—92no—
227C3×C42D3211yescharacteristic
228#217—94no—
229(C2×D3)⋊C4—32no—
230(C2×D3)⋊C4—92no—
231(C2×D3)⋊C4—94no—
232(C2×D3)⋊C4—94no—
233C4⋊Dic3—31no—
234C4⋊Dic3—31no—
235C4⋊Dic3—31no—
236C4⋊Dic3—61no—
237C4⋊Dic3—91no—
238Dic3⋊C4—32no—
239Dic3⋊C4—32no—
240Dic3⋊C4—62no—
241C4×Dic3—32no—
242C4×Dic3—32no—
243C4×Dic3—62no—
244C22×C32—21no—
245C22×C32C23×D311yescharacteristic
246C22×C32C23×D311yescharacteristic
247C2×C3⋊D3—22no—
248C2×C3⋊D3—24no—
249C2×C3⋊D3—22no—
250C2×C3⋊D3—24no—
251C2×C3⋊D3—24no—
252C2×C3⋊D3—22no—
253C2×C3×D3—64no—
254C2×C3×D3—64no—
255C2×C3×D3—62no—
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C2×C3×D3—64no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—22no—
262C32×C4—22no—
263C32×C4—41no—
264C32×C4#23512yescomplemented
265C32×C4D3×D412yes—
266C32×C4D3×D412yes—
267C32×C4#23712yescomplemented
268C3⋊Dic3—22no—
269C3×Dic3—34no—
270C3×Dic3—34no—
271C3×Dic3—64no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C3×Dic3—122no—
276C2×#41—272no—
277#127—91no—
278#127—272no—
279#121—274no—
280#121—272no—
281C4×Q8—91no—
282C4×D4—91no—
283C4×D4—272no—
284H(3,𝔽3)#55011yescharacteristic, complemented, Sylow
285C23×C3—94no—
286C23×C3—92no—
287C22×D3—32no—
288C22×D3—34no—
289C22×D3—94no—
290C22×D3—92no—
291C3×Q8—94no—
292C3×D4—184no—
293C3×D4—182no—
294C3×D4—184no—
295C3×D4—362no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—92no—
306C2×C3×C4—92no—
307C2×C3×C4—92no—
308C2×C3×C4—184no—
309C2×C3×C4C2×D3211yescharacteristic
310C2×C3×C4C2×D3211yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312#67—188no—
313#67—188no—
314#67—364no—
315#67—364no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—32no—
320C2×Dic3—32no—
321C2×Dic3—62no—
322C2×Dic3—62no—
323C2×Dic3—92no—
324C2×Dic3—188no—
325D12—64no—
326D12—62no—
327D12—64no—
328D12—122no—
329D12—184no—
330D12—182no—
331D12—184no—
332D12—188no—
333D12—362no—
334D12—364no—
335C4×D3—64no—
336C4×D3—184no—
337C4×D3—188no—
338C4×D3—188no—
339Dic6—34no—
340Dic6—34no—
341Dic6—64no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32C2×#23511yescharacteristic
346C2×C32C2×D3×D411yescharacteristic
347C2×C32C2×D3×D411yescharacteristic
348C2×C32C2×#23711yescharacteristic
349C2×C32#100411yescharacteristic
350C2×C32#100911yescharacteristic
351C3⋊D3—44no—
352C3⋊D3—42no—
353C3×D3—122no—
354C3×D3—124no—
355#41—548no—
356C2×Q8—91no—
357C2×D4—91no—
358C2×D4—92no—
359C2×D4—92no—
360C2×D4—92no—
361C2×D4—274no—
362C2×D4—274no—
363C22×C4—92no—
364C22×C4—274no—
365C4⋊C4—91no—
366C4⋊C4—92no—
367C4⋊C4—91no—
368C22⋊C4—92no—
369C22⋊C4—274no—
370C42—92no—
371C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—61no—
375C22×C3—184no—
376C22×C3—182no—
377C22×C3—182no—
378C22×C3—184no—
379C22×C3—182no—
380C22×C3—184no—
381C22×C3C22×D3211yescharacteristic
382C2×D3—62no—
383C2×D3—64no—
384C2×D3—62no—
385C2×D3—64no—
386C2×D3—64no—
387C2×D3—62no—
388C2×D3—184no—
389C2×D3—182no—
390C2×D3—184no—
391C2×D3—182no—
392C2×D3—184no—
393C2×D3—182no—
394C3×C4—21no—
395C3×C4—32no—
396C3×C4—32no—
397C3×C4—32no—
398C3×C4—32no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—62no—
402C3×C4—62no—
403C3×C4—94no—
404C3×C4—121no—
405C3×C4—182no—
406C3×C4—182no—
407C3×C4#379512yes—
408C3×C4#380212yes—
409Dic3—34no—
410Dic3—34no—
411Dic3—64no—
412Dic3—62no—
413Dic3—62no—
414Dic3—62no—
415Dic3—122no—
416Dic3—182no—
417C32—21no—
418C32#555311yescharacteristic, complemented
419C32#571811yescharacteristic, complemented
420C23—92no—
421C23—94no—
422Q8—94no—
423D4—182no—
424D4—184no—
425D4—184no—
426D4—362no—
427D4—548no—
428D4—1084no—
429C2×C4—92no—
430C2×C4—92no—
431C2×C4—92no—
432C2×C4—184no—
433C2×C4—548no—
434C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
435C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
436C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—61no—
444C2×C3—61no—
445C2×C3—61no—
446C2×C3—362no—
447C2×C3—364no—
448C2×C3C2×#379511yescharacteristic
449C2×C3C2×#380211yescharacteristic
450C2×C3#6430711yescharacteristic
451D3—122no—
452D3—124no—
453D3—364no—
454D3—362no—
455C22—182no—
456C22—184no—
457C22—182no—
458C22—184no—
459C22—184no—
460C22—182no—
461C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
462C4—21no—
463C4—94no—
464C4—182no—
465C4—182no—
466C4#8864812yescomplemented
467C4#8865512yes—
468C3—31no—
469C3—31no—
470C3—61no—
471C3#1059390611yescharacteristic
472C2—362no—
473C2—364no—
474C2C2×#8864811yescharacteristic
475C2C2×#8865511yescharacteristic
476C2#1172249211yescharacteristic
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)#550
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×#88648C2semidirect
2. C4×#6229C2semidirect
3. #11721385C2semidirect
4. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
5. #11721144C2semidirect
6. #11721141C2semidirect
7. #11721136C2semidirect
8. #11721126C2semidirect
30. #64083D3semidirect
31. C4×C4⋊(C3⋊D3)D3semidirect
52. #6229D4semidirect
113. C4×H(3,𝔽3)#41semidirect
264. C32×C4#235semidirect
267. C32×C4#237semidirect
284. H(3,𝔽3)#550semidirect
418. C32#5553semidirect
419. C32#5718semidirect
466. C4#88648semidirect

Maximal subgroups

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

subgroupindex
C2×#886482
C4×#62292
#117213852
C4×C4⋊((C3⋊D3)⋊C3)2
#117211442
#117211412
#117211362
#117211262
#105939023
#105939003