Finite Group Databasebeta

← #13147813

Subgroups — #13147813

Order 1728.

916 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#11721718C211yesmaximal, characteristic, complemented
2#11721385C211yesmaximal, characteristic, complemented
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721159C212yesmaximal, complemented
5#11721136C212yesmaximal, complemented
6#11721127C214yesmaximal, complemented
7#11721125C214yesmaximal, complemented
8#10593989—31nomaximal
9#10593949—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
13C2×#6200C2214yes—
14C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
15C42×H(3,𝔽3)C2211yescharacteristic
16#88542C2212yes—
17#88540C2212yes—
18C4×#1084C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20#3750⋊C3C2214yes—
21#88450C2214yes—
22#88450C2214yes—
23#64083D311yescharacteristic, complemented
24#64079D311yescharacteristic, complemented
25C3×#874—31no—
26C3×#872—31no—
27C3×#872—31no—
28C3×#872—61no—
29C3×#870—31no—
30#63910—34no—
31#63909—32no—
32#63908—32no—
33#63897—34no—
34#63894—32no—
35Dic3×D12—32no—
36#63885—34no—
37#63884—34no—
38C22×#272C2311yescharacteristic
39C22×(C3⋊D3)⋊C3C2314yes—
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084C2312yes—
43C2×#1084C2311yescharacteristic
44C4×#272—22no—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46C4⋊((C3⋊D3)⋊C3)—24no—
47C4⋊((C3⋊D3)⋊C3)—44no—
48#6200—44no—
49((C3⋊D3)⋊C3)⋊C4—44no—
50#5719—91no—
51#5697—91no—
52#5535—91no—
53C2×C4⋊(C3⋊D3)C2×D312yes—
54C2×C4⋊(C3⋊D3)C2×D314yes—
55C2×C3×D12—32no—
56C2×C3×D12—34no—
57C2×C3×C4×D3—31no—
58C2×C3×C4×D3—31no—
59C2×C3×C4×D3—61no—
60C2×#626—34no—
61C2×#625—34no—
62C2×#624—34no—
63C2×D3×Dic3—34no—
64C32×C42—21no—
65C32×C42C2×D311yescharacteristic
66C32×C42C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D312yes—
68#3750C2×D314yes—
69C3×(C2×D3)⋊C4—32no—
70C3×(C2×D3)⋊C4—32no—
71C3×(C2×D3)⋊C4—62no—
72C3×C4⋊Dic3—32no—
73C3×Dic3⋊C4—32no—
74C3×Dic3⋊C4—32no—
75C3×Dic3⋊C4—34no—
76C3×Dic3⋊C4—62no—
77C3×C4×Dic3—31no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—61no—
80#3722—34no—
81#3722—34no—
82(C2×D3)⋊Dic3—34no—
83(C2×D3)⋊Dic3—34no—
84C22×H(3,𝔽3)C2411yescharacteristic
85C2×#272—21no—
86C2×#272—22no—
87C2×(C3⋊D3)⋊C3—24no—
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3—24no—
90C4×H(3,𝔽3)—22no—
91C4×H(3,𝔽3)#4112yes—
92#1084—21no—
93#1084—22no—
94(C3⋊Dic3)⋊C3—24no—
95C3×#127—91no—
96C3×#125—91no—
97C3×#117—91no—
98#939—94no—
99#938—92no—
100#935—94no—
101D4×Dic3—92no—
102#930—92no—
103#895—94no—
104#895—92no—
105#894—94no—
106#894—94no—
107#893—92no—
108#884—94no—
109#874—31no—
110#874—91no—
111C4×D12—92no—
112#872—31no—
113#872—31no—
114#872—61no—
115#870—31no—
116#870—91no—
117C22×C3⋊D3C22×D314yes—
118C22×C3×D3—34no—
119C22×C3×D3—31no—
120C22×C3×D3—31no—
121C22×C3×D3—61no—
122C2×C32×C4—21no—
123C2×C32×C4—22no—
124C2×C32×C4C22×D312yes—
125C2×C32×C4C22×D312yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3C22×D314yes—
129C4⋊(C3⋊D3)—24no—
130C4⋊(C3⋊D3)—44no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—62no—
138C3×D12—64no—
139C3×D12—124no—
140C3×C4×D3—64no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143#626—124no—
144#625—124no—
145#624—124no—
146D3×Dic3—124no—
147#540—271noSylow
148C2×H(3,𝔽3)C2×#4111yescharacteristic
149C2×H(3,𝔽3)H(5,𝔽2)12yes—
150#272—41no—
151(C3⋊D3)⋊C3—44no—
152C2×C3×D4—92no—
153C2×C3×D4—94no—
154C22×C3×C4—91no—
155C2×#67—94no—
156C2×#67—94no—
157C22×Dic3—94no—
158C2×D12—32no—
159C2×D12—34no—
160C2×D12—92no—
161C2×D12—94no—
162C2×D12—94no—
163C2×C4×D3—31no—
164C2×C4×D3—31no—
165C2×C4×D3—61no—
166C2×C4×D3—94no—
167C2×C4×D3—94no—
168C3×C4⋊C4—92no—
169C3×C4⋊C4—94no—
170C3×C4⋊C4—92no—
171C3×C22⋊C4—92no—
172C3×C42—31no—
173C3×C42—31no—
174C3×C42—61no—
175C3×C42—91no—
176C3×C42D3211yescharacteristic
177#217—94no—
178#217—94no—
179(C2×D3)⋊C4—32no—
180(C2×D3)⋊C4—32no—
181(C2×D3)⋊C4—62no—
182(C2×D3)⋊C4—94no—
183(C2×D3)⋊C4—94no—
184(C2×D3)⋊C4—94no—
185(C2×D3)⋊C4—94no—
186C4⋊Dic3—32no—
187C4⋊Dic3—92no—
188Dic3⋊C4—32no—
189Dic3⋊C4—32no—
190Dic3⋊C4—34no—
191Dic3⋊C4—62no—
192Dic3⋊C4—94no—
193C4×Dic3—31no—
194C4×Dic3—31no—
195C4×Dic3—61no—
196C22×C32—21no—
197C22×C32C23×D311yescharacteristic
198C22×C32C23×D311yescharacteristic
199C2×C3⋊D3—24no—
200C2×C3⋊D3—24no—
201C2×C3⋊D3—24no—
202C2×C3×D3—62no—
203C2×C3×D3—61no—
204C2×C3×D3—62no—
205C2×C3×D3—64no—
206C2×C3×D3—61no—
207C2×C3×D3—64no—
208C2×C3×D3—64no—
209C2×C3×D3—62no—
210C2×C3×D3—122no—
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—22no—
214C32×C4—42no—
215C32×C4#23712yes—
216C32×C4#23912yes—
217C3⋊Dic3—24no—
218C3×Dic3—62no—
219C3×Dic3—61no—
220C3×Dic3—64no—
221C3×Dic3—62no—
222C3×Dic3—61no—
223C3×Dic3—62no—
224C3×Dic3—122no—
225#127—91no—
226#125—91no—
227#123—274no—
228#121—274no—
229#121—272no—
230C4×D4—272no—
231#117—91no—
232H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
233C23×C3—94no—
234C23×C3—91no—
235C22×D3—31no—
236C22×D3—31no—
237C22×D3—34no—
238C22×D3—61no—
239C22×D3—94no—
240C3×D4—184no—
241C3×D4—364no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—61no—
247C2×C3×C4—62no—
248C2×C3×C4—94no—
249C2×C3×C4—91no—
250C2×C3×C4—92no—
251C2×C3×C4—182no—
252C2×C3×C4C2×D3212yes—
253C2×C3×C4C2×D3211yescharacteristic
254#67—364no—
255#67—364no—
256C2×Dic3—32no—
257C2×Dic3—32no—
258C2×Dic3—31no—
259C2×Dic3—31no—
260C2×Dic3—34no—
261C2×Dic3—61no—
262C2×Dic3—62no—
263C2×Dic3—94no—
264C2×Dic3—364no—
265D12—64no—
266D12—124no—
267D12—184no—
268D12—364no—
269D12—364no—
270C4×D3—62no—
271C4×D3—62no—
272C4×D3—64no—
273C4×D3—364no—
274C4×D3—364no—
275C2×C32—21no—
276C2×C32—22no—
277C2×C32C2×#23711yescharacteristic
278C2×C32#100412yes—
279C2×C32C2×#23911yescharacteristic
280C2×C32#100912yes—
281C3⋊D3—44no—
282C3×D3—122no—
283C3×D3—124no—
284C3×D3—121no—
285C3×D3—121no—
286C2×D4—92no—
287C2×D4—94no—
288C2×D4—274no—
289C22×C4—91no—
290C22×C4—274no—
291C4⋊C4—92no—
292C4⋊C4—92no—
293C4⋊C4—94no—
294C22⋊C4—92no—
295C22⋊C4—274no—
296C22⋊C4—274no—
297C42—91no—
298C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C22×C3—31no—
300C22×C3—31no—
301C22×C3—61no—
302C22×C3—184no—
303C22×C3—184no—
304C22×C3—184no—
305C22×C3—181no—
306C22×C3—182no—
307C22×C3C22×D3211yescharacteristic
308C2×D3—61no—
309C2×D3—61no—
310C2×D3—62no—
311C2×D3—62no—
312C2×D3—62no—
313C2×D3—64no—
314C2×D3—64no—
315C2×D3—64no—
316C2×D3—122no—
317C2×D3—184no—
318C2×D3—184no—
319C2×D3—184no—
320C3×C4—22no—
321C3×C4—32no—
322C3×C4—32no—
323C3×C4—62no—
324C3×C4—62no—
325C3×C4—62no—
326C3×C4—122no—
327C3×C4—184no—
328C3×C4—181no—
329C3×C4—182no—
330C3×C4#379612yes—
331Dic3—61no—
332Dic3—61no—
333Dic3—62no—
334Dic3—62no—
335Dic3—62no—
336Dic3—64no—
337Dic3—122no—
338Dic3—184no—
339C32—21no—
340C32#569711yescharacteristic, complemented
341C32#571911yescharacteristic, complemented
342C23—91no—
343C23—94no—
344D4—184no—
345D4—364no—
346D4—1084no—
347C2×C4—92no—
348C2×C4—91no—
349C2×C4—94no—
350C2×C4—182no—
351C2×C4—1084no—
352C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
353C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—32no—
357C2×C3—32no—
358C2×C3—61no—
359C2×C3—62no—
360C2×C3—364no—
361C2×C3—361no—
362C2×C3C2×#379611yescharacteristic
363C2×C3#6430712yes—
364D3—121no—
365D3—121no—
366D3—122no—
367D3—124no—
368D3—364no—
369C22—181no—
370C22—184no—
371C22—184no—
372C22—184no—
373C22—182no—
374C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
375C4—22no—
376C4—181no—
377C4—182no—
378C4—184no—
379C4#8864912yes—
380C3—31no—
381C3—31no—
382C3—61no—
383C3#1059395011yescharacteristic
384C2—361no—
385C2—364no—
386C2C2×#8864911yescharacteristic
387C2#1172249212yes—
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)#540
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. #11721718C2semidirect
2. #11721385C2semidirect
3. #64079⋊C3C2semidirect
4. #11721159C2semidirect
5. #11721136C2semidirect
6. #11721127C2semidirect
7. #11721125C2semidirect
23. #64083D3semidirect
24. #64079D3semidirect
232. H(3,𝔽3)#540semidirect
340. C32#5697semidirect
341. C32#5719semidirect

Maximal subgroups

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

subgroupindex
#117217182
#117213852
#64079⋊C32
#117211592
#117211362
#117211272
#117211252
#105939893
#105939493