Finite Group Databasebeta

← #13161189

Subgroups — #13161189

Order 1728.

992 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 146 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×#64155C211yesmaximal, characteristic, complemented
3#11723526C211yesmaximal, characteristic
4C2×#88787C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722056C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592511—31nomaximal
9#10591589—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6292—21no—
14#88862—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88804—21no—
22#88787—22no—
23#64155D311yescharacteristic, complemented
24C2×#3755—31no—
25C2×#3755—34no—
26C2×#3755—64no—
27C2×#3743—31no—
28#64104—31no—
29C3×#946—34no—
30#63909—34no—
31C2×#3711—34no—
32C32×#103—31no—
33#63648—31no—
34#63634—31no—
35C3×#835D314yescomplemented
36#63576—34no—
37#63564—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3D411yescharacteristic
40C33×Q8—21no—
41C33×Q8—21no—
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
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C2×C32×Dic3—21no—
51#6292—42no—
52C3⋊(C3⋊(C3⋊C8))D412yescomplemented
53C32×C3⋊C8—21no—
54#5049—91no—
55#4812—94no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)—31no—
60C2×C4⋊(C3⋊D3)—34no—
61C2×C4⋊(C3⋊D3)—64no—
62C2×C4⋊(C3⋊D3)—61no—
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C2×C3×C4×D3—34no—
65C2×#626—64no—
66C2×C32×C8—31no—
67C32×C4⋊C4—31no—
68#3755—62no—
69#3755—68no—
70#3755—616no—
71#3752—21no—
72#3752—61no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74#3750—21no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3743—62no—
79C3×(C2×D3)⋊C4—64no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3C2×D314yes—
82C3×Dic3⋊C4—64no—
83C2×C3×C3⋊C8C2×D314yes—
84#3722—64no—
85#3711—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—41no—
98C32×Dic3—41no—
99C3×C22⋊Q8—91no—
100#946—34no—
101C2×#215—94no—
102C2×#215—91no—
103C2×#215—184no—
104#930—91no—
105C2×#208—91no—
106C2×ΓL(1,𝔽25)—94no—
107#895—94no—
108C3×#103—34no—
109#835C3⋊D311yescharacteristic, complemented
110#821—94no—
111#809—91no—
112#807—94no—
113#807—91no—
114C22×C3⋊D3—61no—
115C22×C3⋊D3—64no—
116C22×C3⋊D3—124no—
117C22×C3⋊D3#6711yescharacteristic
118C22×C3×D3—34no—
119C32×Q8—28no—
120C32×Q8—24no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—24no—
124C32×Q8—44no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—44no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3#6711yescharacteristic
134C4⋊(C3⋊D3)—68no—
135C4⋊(C3⋊D3)—64no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—124no—
140C4⋊(C3⋊D3)—122no—
141C4×C3⋊D3—21no—
142C4×C3⋊D3—21no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—64no—
146C3×C4×D3—64no—
147C3×C4×D3—64no—
148#626—128no—
149C32×C8—61no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153C3×C3⋊C8—24no—
154#468—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33AGL(1,ℤ/8)11yescharacteristic
157C2×C33#13711yescharacteristic
158C3⋊(C3⋊D3)—81no—
159C3×C3⋊D3—41no—
160C2×C3×Q8—24no—
161C2×C3×Q8D3214yes—
162C2×C3×Q8C2×C3⋊D311yescharacteristic
163C22×C3×C4—91no—
164C2×#67—181no—
165C2×D12—94no—
166C2×D12—91no—
167C2×D12—184no—
168C2×D12—184no—
169C2×C4×D3—34no—
170C2×C3×C8—34no—
171C3×C4⋊C4—34no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4—181no—
174C3×C22⋊C4—181no—
175#215—188no—
176#215—182no—
177#215—1816no—
178(C2×D3)⋊C4—64no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—184no—
181C4⋊Dic3—34no—
182C4⋊Dic3C2×C3⋊D311yescharacteristic
183Dic3⋊C4—64no—
184#208—182no—
185C2×C3⋊C8—94no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—184no—
188C2×C3⋊C8C2×C3⋊D311yescharacteristic
189ΓL(1,𝔽25)—188no—
190C22×C32—24no—
191C22×C32D3×D414yes—
192C22×C32C2×#6711yescharacteristic
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3⋊D3—21no—
196C2×C3⋊D3—128no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—121no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—121no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—244no—
204C2×C3⋊D3—244no—
205C2×C3×D3—64no—
206C2×C3×D3—64no—
207C2×C3×D3—64no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—48no—
211C32×C4—44no—
212C32×C4—41no—
213C32×C4—121no—
214C32×C4#23514yes—
215C32×C4D3×D414yes—
216C32×C4D3×D411yescharacteristic
217C32×C4#23711yescharacteristic
218C3⋊Dic3—21no—
219C3⋊Dic3—41no—
220C3×Dic3—44no—
221C3×Dic3—64no—
222C3×Dic3—124no—
223C2×SD16—271no—
224C2×M16—271no—
225C22⋊Q8—91no—
226#121—271no—
227#107—271no—
228#103—31no—
229#102—271no—
230C33#46811yescharacteristic, complemented, Sylow
231C23×C3—91no—
232C22×D3—34no—
233C22×D3—184no—
234C22×D3—181no—
235C22×D3—364no—
236C3×Q8—28no—
237C3×Q8—24no—
238C3×Q8—24no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—44no—
242C2×C3×C4—24no—
243C2×C3×C4—21no—
244C2×C3×C4—24no—
245C2×C3×C4—44no—
246C2×C3×C4—64no—
247C2×C3×C4—91no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—362no—
254C2×Dic3—21no—
255C2×Dic3—34no—
256C2×Dic3—64no—
257D12—188no—
258D12—184no—
259D12—184no—
260D12—181no—
261D12—181no—
262D12—364no—
263D12—368no—
264C4×D3—64no—
265C4×D3—64no—
266C3×C8—64no—
267C3⋊C8—21no—
268C3⋊C8—98no—
269C3⋊C8—92no—
270C3⋊C8—188no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32#88414yes—
275C2×C32#91414yes—
276C2×C32#91814yes—
277C2×C32#93911yescharacteristic
278C2×C32#94211yescharacteristic
279C2×C32#94911yescharacteristic
280C3⋊D3—41no—
281C3⋊D3—248no—
282C3⋊D3—241no—
283C3⋊D3—244no—
284C3×D3—124no—
285C2×Q8D3×C3⋊D311yescharacteristic
286C2×D4—271no—
287C2×D4—541no—
288C22×C4—91no—
289SD16—542no—
290M16—542no—
291C2×C8—31no—
292C2×C8—271no—
293C4⋊C4—31no—
294C4⋊C4—91no—
295C4⋊C4—181no—
296C22⋊C4—181no—
297C22⋊C4—541no—
298C22×C3—24no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3—181no—
302C22×C3D3×#6714yes—
303C22×C3D4×C3⋊D311yescharacteristic
304C2×D3—64no—
305C2×D3—64no—
306C2×D3—64no—
307C2×D3—368no—
308C2×D3—361no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—364no—
312C2×D3—364no—
313C2×D3—364no—
314C2×D3—724no—
315C2×D3—724no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—48no—
319C3×C4—41no—
320C3×C4—44no—
321C3×C4—124no—
322C3×C4—181no—
323C3×C4—361no—
324C3×C4#380414yes—
325C3×C4#381114yes—
326C3×C4#382811yescharacteristic
327C3×C4D4×C3⋊D311yescharacteristic
328Dic3—41no—
329Dic3—64no—
330Dic3—124no—
331C32—24no—
332C32#481214yescomplemented
333C32#504911yescharacteristic, complemented
334C23—91no—
335C23—541no—
336Q8—21no—
337Q8—21no—
338D4—541no—
339D4—541no—
340D4—1082no—
341C2×C4—21no—
342C2×C4—61no—
343C2×C4—91no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—61no—
349C8—272no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—361no—
354C2×C3#6393914yes—
355C2×C3#6394814yes—
356C2×C3#6397114yes—
357C2×C3#6409311yescharacteristic
358C2×C3#6412311yescharacteristic
359C2×C3#6412711yescharacteristic
360D3—124no—
361D3—728no—
362D3—724no—
363D3—721no—
364C22—181no—
365C22—181no—
366C22—181no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22C3⋊D3×#6711yescharacteristic
371C4—41no—
372C4—121no—
373C4—181no—
374C4—361no—
375C4#8903111yescharacteristic
376C4#8903911yescharacteristic
377C3—24no—
378C3#1059158914yescomplemented
379C3#1059251111yescharacteristic, complemented
380C2—361no—
381C2—2161no—
382C2#1172357011yescharacteristic
383C2#1172358611yescharacteristic
384C2#1172362411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723526C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#468
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×#64155C2semidirect
4. C2×#88787C2semidirect
5. C32×#835C2semidirect
6. #11722056C2semidirect
7. #11722036C2semidirect
23. #64155D3semidirect
35. C3×#835D3semidirect
52. C3⋊(C3⋊(C3⋊C8))D4semidirect
109. #835C3⋊D3semidirect
230. C33#468semidirect
332. C32#4812semidirect
333. C32#5049semidirect
378. C3#10591589semidirect
379. C3#10592511semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C3×#641552
#117235262
C2×#887872
C32×#8352
#117220562
#117220362
#105925113
#105915893