Finite Group Databasebeta

← #13138032

Subgroups — #13138032

Order 1728.

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

normal subgroups only

This group has 476 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#64104⋊C3C211yesmaximal, characteristic
2#11721107C211yesmaximal, characteristic, complemented
3C2×#88438C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
5#11720532C211yesmaximal, characteristic, complemented
6#63635⋊C3C211yesmaximal, characteristic, complemented
7#11720318C211yesmaximal, characteristic, complemented
8#10590523—31nomaximal
9#10590514—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88448—21no—
21#88438—22no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64104D311yescharacteristic
24C3×#895—31no—
25#63871—31no—
26#63856—31no—
27C2×#3717—31no—
28C2×#3715—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31#63635D311yescharacteristic, complemented
32C3×#810—31no—
33C3×#810—31no—
34C3×#810—61no—
35C3×#808—31no—
36#63564—31no—
37#63563—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43#6229—21no—
44#6229—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—42no—
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5041—91no—
55#4864—91no—
56#4807—91no—
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×D12—61no—
62C2×C3×C4×D3—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—61no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C2×D311yescharacteristic
69#3752—21no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×(C2×D3)⋊C4—61no—
73C3×C4⋊Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78#3724—61no—
79Dic3⋊Dic3—61no—
80#3722—61no—
81(C2×D3)⋊Dic3—61no—
82#3717—62no—
83#3715—62no—
84#3710—62no—
85D3×C3⋊C8—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3SD1612yescomplemented
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96#1084—41no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99C3×#121—91no—
100#946—91no—
101C2×#215—91no—
102C2×#214—91no—
103#924—91no—
104C22×C3⋊C8—91no—
105C2×#204—91no—
106C2×D3×C8—91no—
107C2×D3×C8—91no—
108#897—91no—
109#895—31no—
110#895—91no—
111C3×#106—91no—
112C3×#103—91no—
113#832—91no—
114#821—91no—
115#810—31no—
116#810—31no—
117#810—61no—
118#809—91no—
119#808—31no—
120#808—91no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3#6711yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—61no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3#6711yescharacteristic
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—42no—
136C4×C3⋊D3—21no—
137C4×C3⋊D3—21no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—121no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—122no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3×Dic6—62no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—121no—
154C3⋊(C3⋊C8)D1212yescomplemented
155C3×C3⋊C8—32no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—121no—
159#465—271noSylow
160C2×H(3,𝔽3)#12111yescharacteristic
161C2×H(3,𝔽3)C2×SD1611yescharacteristic
162C2×H(3,𝔽3)#13511yescharacteristic
163(C3⋊D3)⋊C3—22no—
164(C3⋊D3)⋊C3—81no—
165C2×C3×Q8—91no—
166C2×C3×D4—91no—
167C2×C3×D4—181no—
168C22×C3×C4—91no—
169C2×D12—31no—
170C2×D12—61no—
171C2×D12—91no—
172C2×D12—181no—
173C2×C4×D3—31no—
174C2×C4×D3—91no—
175C2×Dic6—31no—
176C2×Dic6—31no—
177C2×Dic6—61no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4—31no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4—91no—
184C3×C4⋊C4D3211yescharacteristic
185C3×C22⋊C4—181no—
186#217—181no—
187#215—182no—
188#214—182no—
189(C2×D3)⋊C4—61no—
190(C2×D3)⋊C4—181no—
191(C2×D3)⋊C4—181no—
192(C2×D3)⋊C4—181no—
193C4⋊Dic3—31no—
194C4⋊Dic3—91no—
195C4⋊Dic3—181no—
196Dic3⋊C4—181no—
197Dic3⋊C4—181no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—61no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—182no—
204#204—182no—
205D3×C8—182no—
206D3×C8—182no—
207C22×C32—21no—
208C22×C32D3×D411yescharacteristic
209C22×C32C2×#6711yescharacteristic
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—41no—
214C2×C3⋊D3—41no—
215C2×C3⋊D3#21412yescomplemented
216C2×C3×D3—32no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—81no—
227C32×C4C2×D1211yescharacteristic
228C32×C4D3×D411yescharacteristic
229C32×C4#23711yescharacteristic
230C32×C4#23911yescharacteristic
231C3⋊Dic3—21no—
232C3⋊Dic3—41no—
233C3×Dic3—61no—
234C3×Dic3—122no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C2×SD16—271no—
239C22×C8—271no—
240C22⋊Q8—271no—
241#121—91no—
242#106—91no—
243#103—91no—
244#102—271no—
245H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
246C23×C3—91no—
247C23×C3—181no—
248C22×D3—31no—
249C22×D3—61no—
250C22×D3—91no—
251C22×D3—181no—
252C3×Q8—181no—
253C3×Q8—181no—
254C3×D4—181no—
255C3×D4—181no—
256C3×D4—362no—
257C2×C3×C4—21no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—91no—
264C2×C3×C4—121no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4C2×D3211yescharacteristic
270C2×Dic3—31no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—91no—
275C2×Dic3—121no—
276C2×Dic3—181no—
277D12—61no—
278D12—61no—
279D12—122no—
280D12—181no—
281D12—181no—
282D12—362no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—181no—
286C4×D3—181no—
287Dic6—61no—
288Dic6—61no—
289Dic6—61no—
290Dic6—61no—
291Dic6—62no—
292Dic6—121no—
293C3×C8—92no—
294C3×C8—181no—
295C3⋊C8—32no—
296C3⋊C8—61no—
297C3⋊C8—61no—
298C3⋊C8—92no—
299C3⋊C8—121no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32#89511yescharacteristic
304C2×C32D3×SD1611yescharacteristic
305C2×C32#91611yescharacteristic
306C2×C32C2×#21411yescharacteristic
307C2×C32#93911yescharacteristic
308C2×C32#95011yescharacteristic
309C3⋊D3—22no—
310C3⋊D3—81no—
311C3×D3—62no—
312C3×D3—241no—
313C2×Q8—91no—
314C2×D4—91no—
315C2×D4—181no—
316C22×C4—91no—
317SD16—542no—
318C2×C8—91no—
319C2×C8—91no—
320C2×C8—542no—
321C4⋊C4—91no—
322C4⋊C4—541no—
323C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
324C22⋊C4—181no—
325C22⋊C4—541no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—92no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3D3×#6711yescharacteristic
336C2×D3—32no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—92no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—181no—
344C2×D3—181no—
345C2×D3—361no—
346C2×D3—361no—
347C2×D3—361no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—41no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—181no—
358C3×C4—241no—
359C3×C4—361no—
360C3×C4—361no—
361C3×C4#379611yescharacteristic
362C3×C4D3×D1211yescharacteristic
363Dic3—61no—
364Dic3—121no—
365Dic3—121no—
366Dic3—122no—
367Dic3—121no—
368Dic3—181no—
369Dic3—361no—
370C32—21no—
371C32#486411yescharacteristic, complemented
372C32#504111yescharacteristic, complemented
373C23—91no—
374C23—181no—
375Q8—181no—
376Q8—181no—
377D4—181no—
378D4—181no—
379D4—362no—
380C2×C4—21no—
381C2×C4—91no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4—181no—
386C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
387C8—92no—
388C8—181no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—61no—
396C2×C3—61no—
397C2×C3—61no—
398C2×C3—182no—
399C2×C3—721no—
400C2×C3#6388611yescharacteristic
401C2×C3D3×#21411yescharacteristic
402C2×C3#6394711yescharacteristic
403D3—62no—
404D3—182no—
405D3—241no—
406D3—721no—
407C22—92no—
408C22—181no—
409C22—181no—
410C22—361no—
411C22—361no—
412C22—361no—
413C22#8867611yescharacteristic
414C4—41no—
415C4—181no—
416C4—361no—
417C4—361no—
418C4#8864911yescharacteristic
419C4#8865411yescharacteristic
420C3—31no—
421C3—31no—
422C3—61no—
423C3#1059057811yescharacteristic
424C2—182no—
425C2—721no—
426C2#1172112711yescharacteristic
427C2#1172119811yescharacteristic
428C2#1172123911yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#64104⋊C3C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#465
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721107C2semidirect
3. C2×#88438C2semidirect
4. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
5. #11720532C2semidirect
6. #63635⋊C3C2semidirect
7. #11720318C2semidirect
31. #63635D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
92. C2×(C3⋊D3)⋊C3SD16semidirect
154. C3⋊(C3⋊C8)D12semidirect
215. C2×C3⋊D3#214semidirect
245. H(3,𝔽3)#465semidirect
371. C32#4864semidirect
372. C32#5041semidirect

Maximal subgroups

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

subgroupindex
#64104⋊C32
#117211072
C2×#884382
C2×((C3⋊D3)⋊C3)⋊C82
#117205322
#63635⋊C32
#117203182
#105905233
#105905143