Finite Group Databasebeta

← #13161238

Subgroups — #13161238

Order 1728.

1040 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 172 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×#88862C211yesmaximal, characteristic, complemented
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic, complemented
4#11723524C211yesmaximal, characteristic
5#11722051C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592862—31nomaximal
9#10591617—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6291—21no—
14C2×#6291—21no—
15#88862—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4×C3⋊Dic3—21no—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C32×#215—22no—
22C2×C32×C3⋊C8C2211yescharacteristic
23C2×C32×SD16—31no—
24Q8×C3⋊Dic3D311yescharacteristic, complemented
25C2×#3755—31no—
26C2×#3755—34no—
27C2×#3755—64no—
28#64149—31no—
29C3×Q8×Dic3—34no—
30C2×C3×#215D314yescomplemented
31#63911—34no—
32#63659—31no—
33#63648—31no—
34#63641—31no—
35Dic3⋊(C3⋊C8)—34no—
36#63564—34no—
37#63562—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C32×D3—21no—
40C33×Q8—21no—
41C33×Q8D412yescomplemented
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C32×D12—21no—
50C32×D12—21no—
51#6291—42no—
52#6291—42no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54C32×C3⋊C8—21no—
55#5163—94no—
56#5027—91no—
57C2×C32×Q8—24no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C2×D314yes—
60C2×C32×D4—31no—
61C2×#638—61no—
62C2×#638—61no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—34no—
65C2×C4⋊(C3⋊D3)—64no—
66C2×C3×D12C2×D314yes—
67C2×#626—64no—
68C2×#626—64no—
69C32×SD16—62no—
70C2×C32×C8—31no—
71#3755—616no—
72#3755—62no—
73#3755—68no—
74C4⋊(C3⋊Dic3)—21no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C4×C3⋊Dic3—21no—
77C4×C3⋊Dic3C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—34no—
80C2×C3⋊(C3⋊C8)—64no—
81C3×#215—28no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3—64no—
84C3×C4×Dic3—34no—
85C3×C4×Dic3—64no—
86C2×C3×C3⋊C8C2×D314yes—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3⋊(C3⋊D3)—41no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C33×C4—22no—
95C33×C4C2×D411yescharacteristic
96C33×C4#4111yescharacteristic
97C3×C3⋊Dic3—21no—
98C3×C3⋊Dic3—41no—
99C3×C3⋊Dic3SD1612yescomplemented
100C2×C3×SD16—34no—
101C3×C4×Q8—91no—
102Q8×Dic3—34no—
103C2×#215—94no—
104C2×#215—91no—
105C2×#215—184no—
106C2×#215C3⋊D311yescharacteristic, complemented
107#940—94no—
108#874—91no—
109#832—94no—
110#821—94no—
111Dic3⋊C8—94no—
112#809—91no—
113#809—91no—
114C4⋊(C3⋊C8)—91no—
115C23×C32—61no—
116C22×C3⋊D3—61no—
117C22×C3⋊D3—64no—
118C22×C3⋊D3—124no—
119C22×C3×D3—24no—
120C32×Q8—21no—
121C32×Q8—28no—
122C32×Q8—24no—
123C32×Q8—44no—
124C32×Q8D1212yescomplemented
125C32×Q8#6718yescomplemented
126C32×D4—61no—
127C32×D4—61no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—44no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—122no—
135#638—122no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3#6711yescharacteristic
138C2×C3⋊Dic3#6711yescharacteristic
139C4⋊(C3⋊D3)—68no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—61no—
142C4⋊(C3⋊D3)—64no—
143C4⋊(C3⋊D3)—64no—
144C4⋊(C3⋊D3)—124no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—64no—
148C3×D12—24no—
149C3×D12—24no—
150#626—128no—
151#626—128no—
152C32×C8—61no—
153C3⋊(C3⋊C8)—68no—
154C3⋊(C3⋊C8)—61no—
155C3⋊(C3⋊C8)—64no—
156C3×C3⋊C8—24no—
157#460—271noSylow
158C2×C33#12111yescharacteristic
159C2×C33C2×SD1611yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C3⋊(C3⋊D3)—81no—
162C32×D3—81no—
163C2×C3×Q8—24no—
164C2×C3×Q8D3214yes—
165C2×C3×Q8C2×C3⋊D311yescharacteristic
166C2×C3×D4—34no—
167C2×#67—184no—
168C2×#67—184no—
169C2×D12—94no—
170C2×D12—91no—
171C2×D12—184no—
172C2×D12—181no—
173C2×D12—181no—
174C2×D12C2×C3⋊D311yescharacteristic
175C3×SD16—68no—
176C2×C3×C8—34no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C42—91no—
180C3×C42—181no—
181#215—22no—
182#215—1816no—
183#215—188no—
184#215—182no—
185C4⋊Dic3—34no—
186C4⋊Dic3—64no—
187C4×Dic3—34no—
188C4×Dic3—64no—
189C2×C3⋊C8—94no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—184no—
192C2×C3⋊C8C2×C3⋊D311yescharacteristic
193C22×C32—24no—
194C22×C32—121no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32D3×D414yes—
198C22×C32C2×#6711yescharacteristic
199C2×C3⋊D3—128no—
200C2×C3⋊D3—121no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—121no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—244no—
207C2×C3⋊D3—244no—
208C2×C3×D3—44no—
209C2×C3×D3—44no—
210C2×C3×D3—44no—
211C32×C4—22no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—28no—
215C32×C4—48no—
216C32×C4C2×D1211yescharacteristic
217C32×C4#23511yescharacteristic
218C32×C4#23714yes—
219C32×C4C2×#6714yes—
220C3⋊Dic3—21no—
221C3⋊Dic3—41no—
222C3⋊Dic3#21512yescomplemented
223C3×Dic3—38no—
224C3×Dic3—64no—
225C3×Dic3—124no—
226C2×SD16—31no—
227C2×SD16—271no—
228#127—271no—
229C4×Q8—91no—
230C4⋊C8—271no—
231#102—271no—
232#102—271no—
233C33#46011yescharacteristic, complemented, Sylow
234C23×C3—64no—
235C22×D3—21no—
236C22×D3—184no—
237C22×D3—181no—
238C22×D3—364no—
239C3×Q8—24no—
240C3×Q8—21no—
241C3×Q8—28no—
242C3×Q8—44no—
243C3×Q8#62618yescomplemented
244C3×Q8#63812yescomplemented
245C3×D4—64no—
246C3×D4—64no—
247C2×C3×C4—24no—
248C2×C3×C4—21no—
249C2×C3×C4—24no—
250C2×C3×C4—44no—
251C2×C3×C4—91no—
252C2×C3×C4—91no—
253C2×C3×C4—181no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C22×C3⋊D311yescharacteristic
256#67—368no—
257#67—368no—
258C2×Dic3—34no—
259C2×Dic3—34no—
260C2×Dic3—64no—
261D12—21no—
262D12—21no—
263D12—188no—
264D12—184no—
265D12—184no—
266D12—181no—
267D12—181no—
268D12—364no—
269D12—362no—
270D12—362no—
271C3×C8—64no—
272C3⋊C8—21no—
273C3⋊C8—188no—
274C3⋊C8—184no—
275C3⋊C8—181no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—241no—
280C2×C32D3×SD1614yes—
281C2×C32#91414yes—
282C2×C32#93011yescharacteristic
283C2×C32#93914yes—
284C2×C32C2×#21511yescharacteristic
285C2×C32#94911yescharacteristic
286C3⋊D3—248no—
287C3⋊D3—244no—
288C3⋊D3—241no—
289C3×D3—84no—
290C2×Q8D3×C3⋊D311yescharacteristic
291C2×D4—31no—
292C2×D4—271no—
293C2×D4—541no—
294C2×D4—541no—
295SD16—62no—
296SD16—542no—
297C2×C8—31no—
298C2×C8—271no—
299C4⋊C4—91no—
300C4⋊C4—181no—
301C42—91no—
302C42—181no—
303C22×C3—24no—
304C22×C3—124no—
305C22×C3—124no—
306C22×C3—124no—
307C22×C3D3×#6714yes—
308C22×C3D4×C3⋊D311yescharacteristic
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—368no—
313C2×D3—361no—
314C2×D3—361no—
315C2×D3—361no—
316C2×D3—364no—
317C2×D3—364no—
318C2×D3—364no—
319C2×D3—724no—
320C2×D3—724no—
321C3×C4—22no—
322C3×C4—28no—
323C3×C4—24no—
324C3×C4—24no—
325C3×C4—48no—
326C3×C4—92no—
327C3×C4—181no—
328C3×C4—361no—
329C3×C4#380414yes—
330C3×C4C2×#62614yes—
331C3×C4#383011yescharacteristic
332C3×C4C2×#63811yescharacteristic
333Dic3—38no—
334Dic3—64no—
335Dic3—124no—
336C32—24no—
337C32#502711yescharacteristic, complemented
338C32#516314yescomplemented
339C23—61no—
340C23—541no—
341Q8—21no—
342Q8#629112yescomplemented
343D4—61no—
344D4—61no—
345D4—541no—
346D4—541no—
347D4—1082no—
348D4—1082no—
349C2×C4—21no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—181no—
353C2×C4C2×D3×C3⋊D311yescharacteristic
354C8—61no—
355C8—541no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—244no—
360C2×C3D3×#21514yes—
361C2×C3#6393914yes—
362C2×C3#6397814yes—
363C2×C3SD16×C3⋊D311yescharacteristic
364C2×C3#6412311yescharacteristic
365C2×C3#6414811yescharacteristic
366D3—81no—
367D3—728no—
368D3—721no—
369D3—724no—
370C22—121no—
371C22—121no—
372C22—121no—
373C22—1081no—
374C22—1081no—
375C22—1081no—
376C22C3⋊D3×#6711yescharacteristic
377C4—22no—
378C4—92no—
379C4—181no—
380C4—361no—
381C4#8902811yescharacteristic
382C4C2×#629111yescharacteristic
383C3—24no—
384C3#1059161714yescomplemented
385C3#1059286211yescharacteristic, complemented
386C2—241no—
387C2—2161no—
388C2C3⋊D3×#21511yescharacteristic
389C2#1172357011yescharacteristic
390C2#1172363511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723524C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#460
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88862C2semidirect
2. C3×Q8×C3⋊Dic3C2semidirect
3. C2×C32×#215C2semidirect
5. #11722051C2semidirect
6. #11722036C2semidirect
7. #11722031C2semidirect
24. Q8×C3⋊Dic3D3semidirect
30. C2×C3×#215D3semidirect
41. C33×Q8D4semidirect
99. C3×C3⋊Dic3SD16semidirect
106. C2×#215C3⋊D3semidirect
124. C32×Q8D12semidirect
125. C32×Q8#67semidirect
222. C3⋊Dic3#215semidirect
233. C33#460semidirect
243. C3×Q8#626semidirect
244. C3×Q8#638semidirect
337. C32#5027semidirect
338. C32#5163semidirect
342. Q8#6291semidirect
384. C3#10591617semidirect
385. C3#10592862semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C3×Q8×C3⋊Dic32
C2×C32×#2152
#117235242
#117220512
#117220362
#117220312
#105928623
#105916173