Finite Group Databasebeta

← #13149540

Subgroups — #13149540

Order 1728.

1116 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 153 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×#88650C212yesmaximal, complemented
2#11721761C211yesmaximal, characteristic, complemented
3C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
4#11721156C212yesmaximal, complemented
5C4×#6198C211yesmaximal, characteristic, complemented
6#11721117C211yesmaximal, characteristic, complemented
7#11721100C214yesmaximal, complemented
8#11721096C212yesmaximal, complemented
9#10594515—32nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272C2212yes—
12C2×C4×(C3⋊D3)⋊C3C2214yes—
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×#6200C2214yes—
15C2×#6198C2211yescharacteristic
16C2×#6196C2212yes—
17#88650—28no—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19#88542C2212yes—
20C4×#1084C2211yescharacteristic
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22#3750⋊C3C2214yes—
23#88451C2214yes—
24#88450C2212yes—
25#88449C2211yescharacteristic
26#88449C2212yes—
27C2×#3798—34no—
28C4⋊C4×C3⋊D3D312yescomplemented
29C3×#893—32no—
30C3×#893—61no—
31C3×D3×C4⋊C4—32no—
32#63918—32no—
33#63916—32no—
34#63908—32no—
35#63905—34no—
36C4×#626—32no—
37#63853—34no—
38#63851—34no—
39C22×#272C2312yes—
40C22×(C3⋊D3)⋊C3C2312yes—
41C2×C4×H(3,𝔽3)C2312yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084C2311yescharacteristic
44C2×#1084C2311yescharacteristic
45C4⋊#272—22no—
46C4×#272—24no—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C2×(C3⋊Dic3)⋊C3C2312yes—
49C4×(C3⋊D3)⋊C3—28no—
50C4×(C3⋊D3)⋊C3—24no—
51#6200—28no—
52#6198D414yescomplemented
53#6196—24no—
54#5551—91no—
55#5507—92no—
56C2×C4×C3⋊D3C2×D314yes—
57C2×C4×C3⋊D3C2×D312yes—
58C2×C3×D12—32no—
59C2×C3×D12—61no—
60C2×C3×C4×D3—34no—
61C2×C3×C4×D3—32no—
62C2×C3×C4×D3—34no—
63C2×C3×C4×D3—62no—
64C2×#627—34no—
65C2×#626—32no—
66C2×#626—34no—
67C2×#625—34no—
68#3798—616no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2×D312yes—
71C4⋊(C3⋊Dic3)C2×D312yes—
72#3750C2×D314yes—
73C3×(C2×D3)⋊C4—34no—
74C3×(C2×D3)⋊C4—62no—
75C3×C4⋊Dic3—32no—
76C3×Dic3⋊C4—34no—
77C3×C4×Dic3—32no—
78C3×C4×Dic3—61no—
79Dic3⋊Dic3—32no—
80Dic3⋊Dic3—34no—
81#3722—32no—
82(C2×D3)⋊Dic3—34no—
83(C2×D3)⋊Dic3—34no—
84(C2×D3)⋊Dic3—32no—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×#272—22no—
87C2×#272—22no—
88C2×#272—22no—
89C2×(C3⋊D3)⋊C3—22no—
90C2×(C3⋊D3)⋊C3—22no—
91C2×(C3⋊D3)⋊C3C2×D414yes—
92C4×H(3,𝔽3)—22no—
93C4×H(3,𝔽3)#4112yescomplemented
94#1084—21no—
95#1084C2×D412yes—
96(C3⋊Dic3)⋊C3—24no—
97(C3⋊Dic3)⋊C3—22no—
98C2×#235—92no—
99C2×#235—94no—
100C3×C4×D4—91no—
101C2×C3×C4⋊C4—92no—
102#938—92no—
103#930—92no—
104#929—94no—
105C4×#67—92no—
106#924—92no—
107#896—94no—
108#895—92no—
109#894—94no—
110#893—32no—
111#893—61no—
112D3×C4⋊C4—32no—
113D3×C4⋊C4—92no—
114#888—92no—
115#883—94no—
116C4×D12—91no—
117#869—91no—
118C22×C3⋊D3C22×D312yes—
119C22×C3×D3—32no—
120C22×C3×D3—34no—
121C22×C3×D3—62no—
122C2×C32×C4—21no—
123C2×C32×C4—22no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D312yes—
126C2×C3⋊Dic3C22×D312yes—
127C2×C3⋊Dic3C22×D314yes—
128C4×C3⋊D3—28no—
129C4×C3⋊D3—24no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C3×D12—64no—
137C3×D12—64no—
138C3×C4×D3—68no—
139C3×C4×D3—68no—
140C3×C4×D3—64no—
141C3×C4×D3—124no—
142#627—68no—
143#626—38no—
144#626—68no—
145#625—68no—
146#547—271noSylow
147C2×H(3,𝔽3)C22×D411yescharacteristic
148C2×H(3,𝔽3)C2×#4111yescharacteristic
149C2×H(3,𝔽3)2−1+411yescharacteristic
150#272—42no—
151(C3⋊D3)⋊C3—24no—
152C2×C3×D4—91no—
153C22×C3×C4—92no—
154C22×C3×C4—94no—
155C22×C3×C4—92no—
156C2×#67—92no—
157C2×#67—94no—
158C2×#67—94no—
159#235—188no—
160#235—1816no—
161C2×D12—32no—
162C2×D12—61no—
163C2×D12—91no—
164C2×D12—94no—
165C2×C4×D3—34no—
166C2×C4×D3—32no—
167C2×C4×D3—34no—
168C2×C4×D3—62no—
169C2×C4×D3—94no—
170C2×C4×D3—92no—
171C2×Dic6—92no—
172C2×Dic6—94no—
173C3×C4⋊C4—32no—
174C3×C4⋊C4—61no—
175C3×C4⋊C4—92no—
176C3×C4⋊C4—94no—
177C3×C4⋊C4D3211yescharacteristic
178C3×C22⋊C4—92no—
179C3×C42—91no—
180#217—94no—
181#217—92no—
182(C2×D3)⋊C4—34no—
183(C2×D3)⋊C4—62no—
184(C2×D3)⋊C4—92no—
185(C2×D3)⋊C4—92no—
186(C2×D3)⋊C4—94no—
187(C2×D3)⋊C4—94no—
188C4⋊Dic3—32no—
189C4⋊Dic3—92no—
190C4⋊Dic3—92no—
191C4⋊Dic3—91no—
192Dic3⋊C4—34no—
193Dic3⋊C4—94no—
194Dic3⋊C4—94no—
195Dic3⋊C4—92no—
196C4×Dic3—32no—
197C4×Dic3—61no—
198C22×C32—21no—
199C22×C32C23×D312yes—
200C2×C3⋊D3—22no—
201C2×C3⋊D3—22no—
202C2×C3⋊D3D3×D414yes—
203C2×C3×D3—34no—
204C2×C3×D3—62no—
205C2×C3×D3—62no—
206C2×C3×D3—64no—
207C2×C3×D3—64no—
208C2×C3×D3—64no—
209C2×C3×D3—64no—
210C2×C3×D3—122no—
211C2×C3×D3—122no—
212C32×C4—24no—
213C32×C4—22no—
214C32×C4—42no—
215C32×C4#23514yescomplemented
216C3⋊Dic3—22no—
217C3⋊Dic3—24no—
218C3×Dic3—34no—
219C3×Dic3—62no—
220C3×Dic3—62no—
221C3×Dic3—64no—
222C3×Dic3—62no—
223C3×Dic3—121no—
224C2×#41—272no—
225C4⋊Q8—271no—
226#123—274no—
227C22⋊Q8—272no—
228#121—272no—
229C4×D4—91no—
230C4×D4—271no—
231C2×C4⋊C4—92no—
232H(3,𝔽3)#54711yescharacteristic, complemented, Sylow
233C23×C3—92no—
234C23×C3—92no—
235C22×D3—32no—
236C22×D3—34no—
237C22×D3—62no—
238C22×D3—92no—
239C3×D4—182no—
240C2×C3×C4—32no—
241C2×C3×C4—34no—
242C2×C3×C4—61no—
243C2×C3×C4—62no—
244C2×C3×C4—91no—
245C2×C3×C4—92no—
246C2×C3×C4—94no—
247C2×C3×C4—91no—
248C2×C3×C4—184no—
249C2×C3×C4—184no—
250C2×C3×C4—188no—
251C2×C3×C4C2×D3212yes—
252C2×C3×C4C2×D3211yescharacteristic
253#67—98no—
254#67—188no—
255#67—188no—
256C2×Dic3—32no—
257C2×Dic3—32no—
258C2×Dic3—32no—
259C2×Dic3—34no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—92no—
263C2×Dic3—94no—
264D12—64no—
265D12—64no—
266D12—94no—
267D12—188no—
268C4×D3—68no—
269C4×D3—68no—
270C4×D3—64no—
271C4×D3—124no—
272C4×D3—184no—
273C4×D3—188no—
274Dic6—188no—
275Dic6—184no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32C2×#23512yes—
280C2×C32C2×D3×D412yes—
281C2×C32#100712yes—
282C3⋊D3—24no—
283C3×D3—64no—
284C3×D3—124no—
285C3×D3—124no—
286#41—548no—
287C2×Q8—272no—
288C2×D4—91no—
289C2×D4—271no—
290C2×D4—274no—
291C22×C4—92no—
292C22×C4—94no—
293C22×C4—92no—
294C4⋊C4—92no—
295C4⋊C4—94no—
296C4⋊C4—271no—
297C4⋊C4—272no—
298C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C22⋊C4—92no—
300C22⋊C4—272no—
301C22⋊C4—274no—
302C42—91no—
303C22×C3—32no—
304C22×C3—61no—
305C22×C3—94no—
306C22×C3—182no—
307C22×C3—182no—
308C22×C3—182no—
309C22×C3—182no—
310C22×C3—182no—
311C22×C3C22×D3211yescharacteristic
312C2×D3—34no—
313C2×D3—62no—
314C2×D3—62no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—64no—
318C2×D3—64no—
319C2×D3—94no—
320C2×D3—122no—
321C2×D3—122no—
322C2×D3—182no—
323C2×D3—182no—
324C3×C4—22no—
325C3×C4—34no—
326C3×C4—64no—
327C3×C4—62no—
328C3×C4—92no—
329C3×C4—122no—
330C3×C4—184no—
331C3×C4—182no—
332C3×C4—181no—
333C3×C4#379812yes—
334Dic3—34no—
335Dic3—62no—
336Dic3—62no—
337Dic3—62no—
338Dic3—64no—
339Dic3—121no—
340Dic3—184no—
341Dic3—182no—
342C32—21no—
343C32#550712yescomplemented
344C23—92no—
345C23—92no—
346Q8—544no—
347D4—182no—
348D4—274no—
349D4—548no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—92no—
353C2×C4—94no—
354C2×C4—184no—
355C2×C4—188no—
356C2×C4—184no—
357C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
358C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
359C2×C3—32no—
360C2×C3—32no—
361C2×C3—32no—
362C2×C3—61no—
363C2×C3—61no—
364C2×C3—61no—
365C2×C3—184no—
366C2×C3—362no—
367C2×C3C2×#379811yescharacteristic
368C2×C3D32×D411yescharacteristic
369C2×C3#6431611yescharacteristic
370D3—64no—
371D3—124no—
372D3—124no—
373D3—184no—
374C22—94no—
375C22—182no—
376C22—182no—
377C22—182no—
378C22—182no—
379C22—182no—
380C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C4—22no—
382C4—92no—
383C4—184no—
384C4—181no—
385C4—182no—
386C4#8865012yescomplemented
387C3—32no—
388C3—61no—
389C3#1059451211yescharacteristic
390C2—184no—
391C2—362no—
392C2C2×#8865011yescharacteristic
393C2D4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
394C2#1172251711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#547
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88650C2semidirect
2. #11721761C2semidirect
3. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
4. #11721156C2semidirect
5. C4×#6198C2semidirect
6. #11721117C2semidirect
7. #11721100C2semidirect
8. #11721096C2semidirect
28. C4⋊C4×C3⋊D3D3semidirect
52. #6198D4semidirect
93. C4×H(3,𝔽3)#41semidirect
215. C32×C4#235semidirect
232. H(3,𝔽3)#547semidirect
343. C32#5507semidirect
386. C4#88650semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
#117217612
C4⋊C4×(C3⋊D3)⋊C32
#117211562
C4×#61982
#117211172
#117211002
#117210962
#105945153