Finite Group Databasebeta

← #13140534

Subgroups — #13140534

Order 1728.

498 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 77 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#64149⋊C3C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic, complemented
3#11721082C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591444—31nomaximal
9#10591439—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17#3754⋊C3—22no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#88447—21no—
22#88431—22no—
23#64149D311yescharacteristic
24C2×#3754D311yescharacteristic, complemented
25C3×#940—31no—
26C2×C3×#214—31no—
27#63844—31no—
28#63843—31no—
29C2×#3717—31no—
30C2×#3716—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—61no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63564—31no—
37#63562—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44#638⋊C3—42no—
45#638⋊C3—42no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#634⋊C3—21no—
52#634⋊C3D412yescomplemented
53#6180—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55#5163—91no—
56#5075—91no—
57#4762—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—21no—
62C2×#638—21no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×#634C2×D311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×#67—61no—
67C2×C3×D12—31no—
68C2×C3×Dic6—31no—
69#3754—22no—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#214—62no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C3×C4×Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—61no—
81#3724—61no—
82Dic3⋊Dic3—61no—
83Dic32—61no—
84Dic32—61no—
85#3717—62no—
86#3716—62no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96#1084—41no—
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3SD1612yescomplemented
100C2×C3×SD16—91no—
101C3×#127—91no—
102Q8×Dic3—91no—
103C2×#215—91no—
104#940—31no—
105#940—91no—
106C2×#214—31no—
107C2×#214—91no—
108C2×#214—91no—
109C2×#204—91no—
110#887—91no—
111C4×Dic6—91no—
112C3×#102—91no—
113#832—31no—
114#832—31no—
115#832—61no—
116#832—91no—
117#821—91no—
118Dic3⋊C8—91no—
119Dic3⋊C8—91no—
120#809—91no—
121C4⋊(C3⋊C8)—91no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C22×C3⋊D3—21no—
126C22×C3×D3—61no—
127C32×D4—22no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—41no—
133C2×C32×C4—21no—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136#638—42no—
137#638—42no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3#6711yescharacteristic
140C2×C3⋊Dic3#6711yescharacteristic
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—21no—
143#634—21no—
144#634#6712yescomplemented
145C3×#67—122no—
146C3×#67—122no—
147C2×C3×Dic3—31no—
148C2×C3×Dic3—31no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—121no—
153C3×D12—61no—
154C3×D12—61no—
155C3×Dic6—32no—
156C3×Dic6—61no—
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—61no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—61no—
161C3×C3⋊C8—121no—
162#460—271noSylow
163C2×H(3,𝔽3)—81no—
164C2×H(3,𝔽3)#12111yescharacteristic
165C2×H(3,𝔽3)C2×SD1611yescharacteristic
166C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
167(C3⋊D3)⋊C3—81no—
168C2×C3×Q8—91no—
169C2×C3×D4—31no—
170C2×C3×D4—31no—
171C2×C3×D4—61no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C2×C3×D4—181no—
175C2×C3×D4D3211yescharacteristic
176C2×#67—61no—
177C2×#67—61no—
178C2×#67—181no—
179C2×#67—181no—
180C2×D12—31no—
181C2×D12—91no—
182C2×Dic6—31no—
183C2×Dic6—91no—
184C3×SD16—182no—
185C2×C3×C8—91no—
186C2×C3×C8—91no—
187C3×C4⋊C4—91no—
188C3×C42—91no—
189#215—182no—
190#214—62no—
191#214—182no—
192#214—182no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—61no—
196C4⋊Dic3—181no—
197Dic3⋊C4—181no—
198Dic3⋊C4—181no—
199C4×Dic3—31no—
200C4×Dic3—91no—
201C4×Dic3—181no—
202C4×Dic3—181no—
203C4×Dic3—181no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—61no—
208C2×C3⋊C8—91no—
209#204—182no—
210C22×C32—21no—
211C22×C32—42no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—81no—
219C22×C32—81no—
220C22×C32D3×D411yescharacteristic
221C22×C32C2×#6711yescharacteristic
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4D3×D411yescharacteristic
231C32×C4#23711yescharacteristic
232C32×C4#23711yescharacteristic
233C32×C4C2×#6711yescharacteristic
234C3⋊Dic3—21no—
235C3⋊Dic3—22no—
236C3⋊Dic3#21412yescomplemented
237C3×Dic3—32no—
238C3×Dic3—61no—
239C3×Dic3—62no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—241no—
243C2×SD16—91no—
244C2×SD16—271no—
245#127—91no—
246C4×Q8—271no—
247C4⋊C8—271no—
248#102—91no—
249#102—271no—
250H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
251C23×C3—21no—
252C23×C3—61no—
253C23×C3—61no—
254C23×C3—121no—
255C23×C3—181no—
256C22×D3—61no—
257C22×D3—181no—
258C3×Q8—92no—
259C3×Q8—181no—
260C3×D4—21no—
261C3×D4—21no—
262C3×D4—61no—
263C3×D4—61no—
264C3×D4—61no—
265C3×D4—61no—
266C3×D4—62no—
267C3×D4—121no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4—362no—
271C3×D4—362no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—61no—
275C2×C3×C4—91no—
276C2×C3×C4—91no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4C2×D3211yescharacteristic
280#67—122no—
281#67—122no—
282#67—362no—
283#67—362no—
284C2×Dic3—31no—
285C2×Dic3—31no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—91no—
290C2×Dic3—91no—
291C2×Dic3—121no—
292C2×Dic3—181no—
293D12—61no—
294D12—61no—
295D12—181no—
296D12—181no—
297Dic6—32no—
298Dic6—61no—
299Dic6—92no—
300Dic6—181no—
301C3×C8—181no—
302C3×C8—181no—
303C3⋊C8—61no—
304C3⋊C8—61no—
305C3⋊C8—61no—
306C3⋊C8—121no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—82no—
312C2×C32—81no—
313C2×C32—81no—
314C2×C32D3×SD1611yescharacteristic
315C2×C32#91411yescharacteristic
316C2×C32#93211yescharacteristic
317C2×C32C2×#21411yescharacteristic
318C2×C32#93811yescharacteristic
319C2×C32#93911yescharacteristic
320C3⋊D3—81no—
321C3×D3—241no—
322C2×Q8—91no—
323C2×D4—91no—
324C2×D4—181no—
325C2×D4—181no—
326C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
327SD16—182no—
328SD16—542no—
329C2×C8—91no—
330C2×C8—91no—
331C4⋊C4—91no—
332C4⋊C4—541no—
333C42—91no—
334C42—541no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—61no—
341C22×C3—121no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—122no—
348C22×C3—241no—
349C22×C3—241no—
350C22×C3—361no—
351C22×C3—361no—
352C22×C3—361no—
353C22×C3D3×#6711yescharacteristic
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—61no—
365C3×C4—61no—
366C3×C4—92no—
367C3×C4—181no—
368C3×C4—182no—
369C3×C4—361no—
370C3×C4#380511yescharacteristic
371C3×C4D3×#6711yescharacteristic
372Dic3—32no—
373Dic3—61no—
374Dic3—62no—
375Dic3—92no—
376Dic3—121no—
377Dic3—121no—
378Dic3—182no—
379Dic3—181no—
380Dic3—241no—
381C32—21no—
382C32#507511yescharacteristic, complemented
383C32#516311yescharacteristic, complemented
384C23—21no—
385C23—181no—
386Q8—92no—
387Q8—181no—
388D4—21no—
389D4—21no—
390D4—181no—
391D4—181no—
392D4—362no—
393D4—362no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
399C8—181no—
400C8—181no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—81no—
411C2×C3—241no—
412C2×C3—241no—
413C2×C3—242no—
414C2×C3—721no—
415C2×C3D3×#21411yescharacteristic
416C2×C3#6393411yescharacteristic
417C2×C3#6397011yescharacteristic
418D3—241no—
419D3—721no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—361no—
424C22—361no—
425C22—361no—
426C22#8867611yescharacteristic
427C4—92no—
428C4—182no—
429C4—181no—
430C4—361no—
431C4#8867211yescharacteristic
432C4#8867611yescharacteristic
433C3—31no—
434C3—31no—
435C3—61no—
436C3#1059143811yescharacteristic
437C2—81no—
438C2—721no—
439C2#1172119611yescharacteristic
440C2#1172119811yescharacteristic
441C2#1172131311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#64149⋊C3C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#3754⋊C3C2semidirect
3. #11721082C2semidirect
4. C2×#88431C2semidirect
5. #11720578C2semidirect
6. #11720311C2semidirect
7. #11720308C2semidirect
24. C2×#3754D3semidirect
52. #634⋊C3D4semidirect
99. (C3⋊Dic3)⋊C3SD16semidirect
144. #634#67semidirect
236. C3⋊Dic3#214semidirect
250. H(3,𝔽3)#460semidirect
382. C32#5075semidirect
383. C32#5163semidirect

Maximal subgroups

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

subgroupindex
#64149⋊C32
C2×#3754⋊C32
#117210822
C2×#884312
#117205782
#117203112
#117203082
#105914443
#105914393