Finite Group Databasebeta

← #13140896

Subgroups — #13140896

Order 1728.

434 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 73 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#64156⋊C3C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3#11721082C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic
5#11720583C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic
8#10591579—31nomaximal
9#10591577—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3756⋊C3—22no—
16#3752⋊C3—21no—
17#3752⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#88447—21no—
22#88431—22no—
23#64156D311yescharacteristic
24C2×#3756D311yescharacteristic, complemented
25C3×#947—31no—
26C2×C3×#216—31no—
27#63844—31no—
28#63843—31no—
29C2×#3717—31no—
30C2×#3716—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#835—61no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63564—31no—
37#63562—31no—
38C22×(C3⋊D3)⋊C3—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5185—91no—
54#5054—91no—
55#4798—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×Dic6—31no—
63#3756—22no—
64#3752—21no—
65#3752—21no—
66C4×C3⋊Dic3C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×#216—62no—
69C3×(C2×D3)⋊C4—61no—
70C3×(C2×D3)⋊C4—61no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C3×C4×Dic3—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—61no—
79#3724—61no—
80Dic3⋊Dic3—61no—
81Dic32—61no—
82Dic32—61no—
83#3717—62no—
84#3716—62no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×(C3⋊D3)⋊C3—41no—
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92#1084—41no—
93(C3⋊Dic3)⋊C3—22no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—21no—
96C2×C3×Q16—91no—
97C3×#124—91no—
98#947—31no—
99#947—91no—
100Q8×Dic3—91no—
101C2×#216—31no—
102C2×#216—91no—
103C2×#215—91no—
104C2×#214—91no—
105C2×#204—91no—
106#887—91no—
107C4×Dic6—91no—
108C3×#103—91no—
109#835—31no—
110#835—31no—
111#835—61no—
112#832—91no—
113#821—91no—
114Dic3⋊C8—91no—
115Dic3⋊C8—91no—
116#809—91no—
117C4⋊(C3⋊C8)—91no—
118C22×C3⋊D3—21no—
119C22×C3×D3—61no—
120C32×Q8—22no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—41no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3#6711yescharacteristic
134C2×C3⋊Dic3#6711yescharacteristic
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137#634—21no—
138#634#6712yes—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—121no—
145C3×D12—61no—
146C3×D12—61no—
147C3×Dic6—32no—
148C3×Dic6—61no—
149C3⋊(C3⋊C8)—21no—
150C3×C3⋊C8—61no—
151C3×C3⋊C8—61no—
152C3×C3⋊C8—61no—
153C3×C3⋊C8—121no—
154#464—271noSylow
155C2×H(3,𝔽3)#12111yescharacteristic
156C2×H(3,𝔽3)#13511yescharacteristic
157C2×H(3,𝔽3)#13711yescharacteristic
158(C3⋊D3)⋊C3—81no—
159C2×C3×Q8—31no—
160C2×C3×Q8—31no—
161C2×C3×Q8—61no—
162C2×C3×Q8—91no—
163C2×C3×Q8D3211yescharacteristic
164C2×C3×D4—91no—
165C2×D12—31no—
166C2×D12—91no—
167C2×Dic6—31no—
168C2×Dic6—91no—
169C3×Q16—182no—
170C2×C3×C8—91no—
171C2×C3×C8—91no—
172C3×C4⋊C4—91no—
173C3×C22⋊C4—181no—
174C3×C22⋊C4—181no—
175C3×C42—91no—
176#216—62no—
177#216—182no—
178#215—182no—
179#214—182no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—61no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—181no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—61no—
187C4⋊Dic3—181no—
188Dic3⋊C4—181no—
189Dic3⋊C4—181no—
190C4×Dic3—31no—
191C4×Dic3—91no—
192C4×Dic3—181no—
193C4×Dic3—181no—
194C4×Dic3—181no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—61no—
199C2×C3⋊C8—91no—
200#204—182no—
201C22×C32—21no—
202C22×C32D3×D411yescharacteristic
203C22×C32C2×#6711yescharacteristic
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—42no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4D3×D411yescharacteristic
216C32×C4#23711yescharacteristic
217C32×C4#23711yescharacteristic
218C32×C4C2×#6711yescharacteristic
219C3⋊Dic3—22no—
220C3⋊Dic3—21no—
221C3⋊Dic3—21no—
222C3×Dic3—61no—
223C3×Dic3—61no—
224C3×Dic3—62no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—241no—
228C2×Q16—91no—
229C2×SD16—271no—
230#124—91no—
231C4×Q8—271no—
232C4⋊C8—271no—
233#103—91no—
234#102—271no—
235H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
236C23×C3—181no—
237C22×D3—61no—
238C22×D3—181no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—61no—
242C3×Q8—61no—
243C3×Q8—61no—
244C3×Q8—61no—
245C3×Q8—62no—
246C3×Q8—92no—
247C3×Q8—121no—
248C3×Q8—181no—
249C3×D4—181no—
250C3×D4—181no—
251C2×C3×C4—21no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—91no—
258C2×C3×C4—91no—
259C2×C3×C4—121no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4C2×D3211yescharacteristic
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—91no—
269C2×Dic3—91no—
270C2×Dic3—121no—
271C2×Dic3—181no—
272D12—61no—
273D12—61no—
274D12—181no—
275D12—181no—
276Dic6—32no—
277Dic6—61no—
278Dic6—92no—
279Dic6—181no—
280C3×C8—181no—
281C3×C8—181no—
282C3⋊C8—61no—
283C3⋊C8—61no—
284C3⋊C8—61no—
285C3⋊C8—121no—
286C3⋊C8—181no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32#91811yescharacteristic
291C2×C32#91911yescharacteristic
292C2×C32#93811yescharacteristic
293C2×C32#93911yescharacteristic
294C2×C32#95011yescharacteristic
295C2×C32#95111yescharacteristic
296C3⋊D3—81no—
297C3×D3—241no—
298C2×Q8—91no—
299C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C2×D4—91no—
301Q16—182no—
302SD16—542no—
303C2×C8—91no—
304C2×C8—91no—
305C4⋊C4—91no—
306C4⋊C4—541no—
307C22⋊C4—181no—
308C22⋊C4—181no—
309C42—91no—
310C42—541no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—61no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3D3×#6711yescharacteristic
318C2×D3—121no—
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—361no—
322C2×D3—361no—
323C2×D3—361no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—41no—
329C3×C4—61no—
330C3×C4—61no—
331C3×C4—121no—
332C3×C4—121no—
333C3×C4—122no—
334C3×C4—181no—
335C3×C4—181no—
336C3×C4—182no—
337C3×C4—361no—
338C3×C4#380511yescharacteristic
339C3×C4D3×#6711yescharacteristic
340Dic3—61no—
341Dic3—61no—
342Dic3—62no—
343Dic3—121no—
344Dic3—121no—
345Dic3—182no—
346Dic3—181no—
347Dic3—181no—
348Dic3—241no—
349C32—21no—
350C32#505411yescharacteristic, complemented
351C32#518511yescharacteristic, complemented
352C23—181no—
353Q8—21no—
354Q8—21no—
355Q8—92no—
356Q8—181no—
357D4—181no—
358D4—181no—
359C2×C4—21no—
360C2×C4—91no—
361C2×C4—91no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
365C8—181no—
366C8—181no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—61no—
376C2×C3—721no—
377C2×C3#6394311yescharacteristic
378C2×C3#6394911yescharacteristic
379C2×C3#6397011yescharacteristic
380D3—241no—
381D3—721no—
382C22—361no—
383C22—361no—
384C22—361no—
385C22#8867611yescharacteristic
386C4—41no—
387C4—181no—
388C4—181no—
389C4—182no—
390C4—361no—
391C4#8867211yescharacteristic
392C4#8867611yescharacteristic
393C3—31no—
394C3—31no—
395C3—61no—
396C3#1059157811yescharacteristic
397C2—721no—
398C2#1172123511yescharacteristic
399C2#1172123711yescharacteristic
400C2#1172131311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
3. #11721082C2semidirect
5. #11720583C2semidirect
6. #11720311C2semidirect
24. C2×#3756D3semidirect
235. H(3,𝔽3)#464semidirect
350. C32#5054semidirect
351. C32#5185semidirect

Maximal subgroups

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

subgroupindex
#64156⋊C32
C2×#3756⋊C32
#117210822
C2×#884312
#117205832
#117203112
#117203082
#105915793
#105915773