Finite Group Databasebeta

← #13138957

Subgroups — #13138957

Order 1728.

495 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88528C211yesmaximal, characteristic
2#11721149C211yesmaximal, characteristic
3#11721083C211yesmaximal, characteristic, complemented
4#63644⋊C3C211yesmaximal, characteristic, complemented
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic
8#10590775—31nomaximal
9Dic3×#204—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#6199—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#88528C414yescomplemented
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88449—21no—
21#88447—21no—
22C2×C3×#204—31no—
23C2×C3×#204—31no—
24C2×C3×#204—61no—
25#63898—31no—
26Dic3×D12—31no—
27#63843—31no—
28Dic3×Dic6—31no—
29#63644D311yescharacteristic, complemented
30C8×C3⋊Dic3D311yescharacteristic, complemented
31C3×#817—31no—
32C3×C8×Dic3—31no—
33#63566—31no—
34#63565—31no—
35#63563—31no—
36#63562—31no—
37C22×#272—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C4⋊#272—21no—
41C4⋊#272C2×C412yes—
42#6229—21no—
43#6229C2×C412yescomplemented
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48#6199—42no—
49C8×H(3,𝔽3)C2×C412yescomplemented
50(C3⋊(C3⋊C8))⋊C3—21no—
51Dic3×SD16—91no—
52#4852—91no—
53C4×#204—91no—
54C2×C3×D12—31no—
55C2×C3×D12—31no—
56C2×C3×D12—61no—
57C2×C3×Dic6—31no—
58C2×C3×Dic6—31no—
59C2×C3×Dic6—61no—
60C2×D3×Dic3—61no—
61C2×D3×Dic3—61no—
62C2×C32×C8—21no—
63C2×C32×C8C2×D311yescharacteristic
64C2×C32×C8C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4×Dic3—31no—
72C2×C3×C3⋊C8—31no—
73C3×#204—34no—
74C3×#204—34no—
75C3×#204—64no—
76Dic3⋊Dic3—61no—
77Dic3⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79(C2×D3)⋊Dic3—61no—
80Dic32—61no—
81Dic32—61no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×#272—22no—
84C2×#272—41no—
85C2×#272—41no—
86C4×H(3,𝔽3)C22×C411yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88#1084—22no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3—41no—
92(C3⋊Dic3)⋊C3SD1612yescomplemented
93C2×C3×SD16—91no—
94Q8×Dic3—91no—
95D4×Dic3—91no—
96C2×#204—31no—
97C2×#204—31no—
98C2×#204—61no—
99#893—91no—
100#887—91no—
101C4×D12—91no—
102C4×Dic6—91no—
103C3×#106—91no—
104C3×C4×C8—91no—
105#835—91no—
106#832—91no—
107#821—91no—
108#817—31no—
109#817—91no—
110#816—91no—
111C8×Dic3—31no—
112C8×Dic3—91no—
113#810—91no—
114#809—91no—
115C22×C3×D3—61no—
116C22×C3×D3—61no—
117C22×C3×D3—121no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3D1211yescharacteristic
124C2×C3⋊Dic3D1211yescharacteristic
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—121no—
132C3×D12—32no—
133C3×D12—32no—
134C3×D12—62no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—121no—
138C3×Dic6—32no—
139C3×Dic6—32no—
140C3×Dic6—62no—
141C3×Dic6—61no—
142C3×Dic6—61no—
143C3×Dic6—121no—
144D3×Dic3—122no—
145D3×Dic3—122no—
146C32×C8—22no—
147C32×C8C4×D312yescomplemented
148C32×C8C4×D312yescomplemented
149C3⋊(C3⋊C8)—21no—
150C3×C3⋊C8—61no—
151C4×SD16—271noSylow
152C2×H(3,𝔽3)C4×D411yescharacteristic
153C2×H(3,𝔽3)C2×SD1611yescharacteristic
154C2×H(3,𝔽3)#13511yescharacteristic
155#272—42no—
156C2×C3×Q8—91no—
157C2×C3×D4—91no—
158C22×Dic3—181no—
159C2×D12—31no—
160C2×D12—31no—
161C2×D12—61no—
162C2×C4×D3—181no—
163C2×C4×D3—181no—
164C2×Dic6—31no—
165C2×Dic6—31no—
166C2×Dic6—61no—
167C3×SD16—94no—
168C2×C3×C8—31no—
169C2×C3×C8—31no—
170C2×C3×C8—61no—
171C2×C3×C8—91no—
172C2×C3×C8D3211yescharacteristic
173C3×C4⋊C4—91no—
174C3×C4⋊C4—91no—
175C3×C42—91no—
176#217—181no—
177(C2×D3)⋊C4—181no—
178(C2×D3)⋊C4—181no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—91no—
182C4⋊Dic3—91no—
183C4⋊Dic3—181no—
184Dic3⋊C4—181no—
185Dic3⋊C4—181no—
186C4×Dic3—31no—
187C4×Dic3—91no—
188C4×Dic3—181no—
189C4×Dic3—181no—
190C4×Dic3—181no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—91no—
193#204—34no—
194#204—34no—
195#204—64no—
196C22×C32—21no—
197C22×C32C2×D1211yescharacteristic
198C22×C32D3×D411yescharacteristic
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C2×C3×D3—122no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—241no—
207C2×C3×D3—241no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4C2×C4×D311yescharacteristic
211C32×C4C2×C4×D311yescharacteristic
212C32×C4#23511yescharacteristic
213C32×C4#23911yescharacteristic
214C3⋊Dic3—21no—
215C3⋊Dic3—41no—
216C3⋊Dic3—41no—
217C3⋊Dic3#20412yescomplemented
218C3×Dic3—32no—
219C3×Dic3—62no—
220C3×Dic3—62no—
221C3×Dic3—61no—
222C3×Dic3—122no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C2×SD16—91no—
226C4×Q8—271no—
227C4×D4—271no—
228#106—91no—
229#103—271no—
230#102—271no—
231C4×C8—91no—
232H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
233C23×C3—181no—
234C22×D3—61no—
235C22×D3—61no—
236C22×D3—121no—
237C3×Q8—92no—
238C3×Q8—181no—
239C3×D4—92no—
240C3×D4—181no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—61no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4C2×D3211yescharacteristic
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—91no—
257C2×Dic3—91no—
258C2×Dic3—121no—
259C2×Dic3—181no—
260C2×Dic3—181no—
261C2×Dic3—362no—
262D12—32no—
263D12—32no—
264D12—62no—
265D12—61no—
266D12—61no—
267D12—121no—
268C4×D3—362no—
269C4×D3—362no—
270Dic6—32no—
271Dic6—32no—
272Dic6—62no—
273Dic6—61no—
274Dic6—61no—
275Dic6—121no—
276C3×C8—32no—
277C3×C8—32no—
278C3×C8—62no—
279C3×C8—181no—
280C3×C8#62412yes—
281C3⋊C8—61no—
282C3⋊C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32C4×D1211yescharacteristic
287C2×C32#89311yescharacteristic
288C2×C32C2×#20411yescharacteristic
289C2×C32#90411yescharacteristic
290C2×C32D3×SD1611yescharacteristic
291C2×C32#91611yescharacteristic
292C3×D3—122no—
293C3×D3—122no—
294C3×D3—242no—
295C2×Q8—91no—
296C2×D4—91no—
297C22×C4—541no—
298SD16—94no—
299C2×C8—91no—
300C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
301C4⋊C4—91no—
302C4⋊C4—91no—
303C4⋊C4—541no—
304C22⋊C4—541no—
305C42—91no—
306C42—541no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—61no—
310C22×C3—182no—
311C22×C3—361no—
312C22×C3—361no—
313C22×C3D3×D1211yescharacteristic
314C2×D3—62no—
315C2×D3—62no—
316C2×D3—122no—
317C2×D3—121no—
318C2×D3—121no—
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—241no—
322C2×D3—241no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—61no—
328C3×C4—61no—
329C3×C4—92no—
330C3×C4—181no—
331C3×C4—182no—
332C3×C4—361no—
333C3×C4—361no—
334C3×C4#379911yescharacteristic
335C3×C4C2×#62411yescharacteristic
336Dic3—32no—
337Dic3—62no—
338Dic3—62no—
339Dic3—61no—
340Dic3—92no—
341Dic3—122no—
342Dic3—121no—
343Dic3—121no—
344Dic3—181no—
345Dic3—361no—
346Dic3—361no—
347C32—21no—
348C32C4×#20411yescharacteristic, complemented
349C32#485211yescharacteristic, complemented
350C23—181no—
351Q8—92no—
352Q8—181no—
353D4—92no—
354D4—181no—
355C2×C4—91no—
356C2×C4—91no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—1082no—
361C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C8—181no—
363C8#619912yescomplemented
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—362no—
374C2×C3D3×#20411yescharacteristic
375C2×C3#6380611yescharacteristic
376C2×C3#6389411yescharacteristic
377D3—122no—
378D3—122no—
379D3—242no—
380C22—182no—
381C22—361no—
382C22—361no—
383C22#8865411yescharacteristic
384C4—92no—
385C4—181no—
386C4—182no—
387C4—361no—
388C4—361no—
389C4#8865211yescharacteristic
390C4C2×#619911yescharacteristic
391C3—31no—
392C3—31no—
393C3—61no—
394C3#1059075811yescharacteristic
395C2—362no—
396C2#1172084611yescharacteristic
397C2#1172085411yescharacteristic
398C2#1172114911yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×C4⋊#272C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×SD16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11721083C2semidirect
4. #63644⋊C3C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
6. #11720326C2semidirect
14. #88528C4semidirect
29. #63644D3semidirect
30. C8×C3⋊Dic3D3semidirect
43. #6229C2×C4semidirect
49. C8×H(3,𝔽3)C2×C4semidirect
92. (C3⋊Dic3)⋊C3SD16semidirect
147. C32×C8C4×D3semidirect
148. C32×C8C4×D3semidirect
217. C3⋊Dic3#204semidirect
232. H(3,𝔽3)C4×SD16semidirect
348. C32C4×#204semidirect
349. C32#4852semidirect
363. C8#6199semidirect

Maximal subgroups

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

subgroupindex
C2×#885282
#117211492
#117210832
#63644⋊C32
C8×(C3⋊Dic3)⋊C32
#117203262
#117203242
#105907753
Dic3×#2043