Finite Group Databasebeta

← #13159173

Subgroups — #13159173

Order 1728.

924 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C32×#895C211yesmaximal, characteristic, complemented
2#11723515C211yesmaximal, characteristic, complemented
3C2×#88789C211yesmaximal, characteristic, complemented
4C2×#88786C211yesmaximal, characteristic, complemented
5#11722178C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592738—31nomaximal
9#10590504—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×D12—21no—
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6290—21no—
15C33×C4⋊C4C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×(C2×D3)⋊C4—21no—
20#88803—21no—
21#88789—22no—
22#88786—22no—
23C32×#121—31no—
24C2×#3753—31no—
25#64139—31no—
26C2×#3748—31no—
27C3×#895D314yescomplemented
28#63908—34no—
29C2×#3713—34no—
30C2×#3711—34no—
31#63659—31no—
32#63636D311yescharacteristic, complemented
33#63635—31no—
34#63635—34no—
35#63635—64no—
36C3×#809—34no—
37#63563—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3—21no—
40C22×C32×D3D411yescharacteristic
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C2×C32×Dic3D411yescharacteristic
47C32×D12—21no—
48C32×D12—21no—
49C32×D12—42no—
50C32×C4×D3—21no—
51C32×C4×D3—21no—
52#6290—42no—
53C3⋊(C3⋊(C3⋊C8))D412yescomplemented
54C3×C3⋊(C3⋊C8)—21no—
55#5039—94no—
56#4782—91no—
57C2×C32×D4—31no—
58C2×C32×D4—61no—
59C22×C32×C4—31no—
60C2×#638—61no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×D12—24no—
63C2×C3×D12—34no—
64C2×C3×D12C2×D314yes—
65C2×C3×C4×D3C2×D314yes—
66C2×#625—64no—
67C32×C4⋊C4—24no—
68C32×C4⋊C4C2×D314yes—
69C32×C4⋊C4C2×D311yescharacteristic
70C32×C22⋊C4—61no—
71#3753—62no—
72#3752—61no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—34no—
75C4⋊(C3⋊Dic3)—64no—
76#3748—62no—
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—34no—
79C2×C3⋊(C3⋊C8)—64no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×(C2×D3)⋊C4—24no—
82C2×C3×C3⋊C8—34no—
83(C2×D3)⋊Dic3—64no—
84#3713—68no—
85#3711—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3—21no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C33×C4—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C32×Dic3—21no—
101C3×#121—34no—
102#938—91no—
103C2×#213—94no—
104C2×#213—91no—
105#930—94no—
106C2×#208—94no—
107C2×ΓL(1,𝔽25)—91no—
108#895C3⋊D311yescharacteristic, complemented
109C3×#102—91no—
110#832—94no—
111#821—91no—
112#809—34no—
113#808—94no—
114#808—91no—
115#808—184no—
116C23×C32—31no—
117C23×C32—61no—
118C22×C3⋊D3—21no—
119C22×C3×D3—24no—
120C22×C3×D3—64no—
121C22×C3×D3#6714yes—
122C32×D4—61no—
123C32×D4—61no—
124C32×D4—122no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—31no—
129C2×C32×C4—44no—
130C2×C32×C4—61no—
131C2×C32×C4—61no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—122no—
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—124no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C2×C3×Dic3#6714yes—
141C3×D12—24no—
142C3×D12—24no—
143C3×D12—48no—
144C3×D12—64no—
145C3×D12—64no—
146C3×C4×D3—24no—
147C3×C4×D3—24no—
148#625—128no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153C3×C3⋊C8—64no—
154#469—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33AGL(1,ℤ/8)11yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3×C3⋊D3—81no—
159C32×D3—41no—
160C32×D3—81no—
161C2×C3×D4—34no—
162C2×C3×D4—64no—
163C2×C3×D4—91no—
164C22×C3×C4—34no—
165C2×#67—184no—
166C2×#67—181no—
167C2×D12—21no—
168C2×D12—34no—
169C2×D12C2×C3⋊D311yescharacteristic
170C2×C4×D3C2×C3⋊D311yescharacteristic
171C2×C3×C8—91no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4D3214yes—
174C3×C4⋊C4C2×C3⋊D311yescharacteristic
175C3×C22⋊C4—64no—
176#217—181no—
177#213—182no—
178#213—188no—
179(C2×D3)⋊C4—21no—
180(C2×D3)⋊C4—184no—
181C4⋊Dic3—94no—
182C4⋊Dic3—91no—
183C4⋊Dic3—184no—
184#208—188no—
185C2×C3⋊C8—34no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—184no—
189ΓL(1,𝔽25)—182no—
190C22×C32—24no—
191C22×C32—61no—
192C22×C32—61no—
193C22×C32—61no—
194C22×C32—121no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32D3×D411yescharacteristic
198C22×C32C2×#6714yes—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3×D3—24no—
203C2×C3×D3—24no—
204C2×C3×D3—24no—
205C2×C3×D3—44no—
206C2×C3×D3—44no—
207C2×C3×D3—44no—
208C2×C3×D3—124no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—41no—
214C32×C4—44no—
215C32×C4—61no—
216C32×C4—84no—
217C32×C4#23511yescharacteristic
218C32×C4D3×D414yes—
219C32×C4D3×D411yescharacteristic
220C32×C4#23714yes—
221C3⋊Dic3—121no—
222C3⋊Dic3—124no—
223C3⋊Dic3—244no—
224C3×Dic3—24no—
225C2×D8—271no—
226C2×M16—271no—
227#121—31no—
228#121—271no—
229#106—271no—
230#102—91no—
231#102—271no—
232C33#46911yescharacteristic, complemented, Sylow
233C23×C3—34no—
234C23×C3—64no—
235C23×C3—181no—
236C22×D3—21no—
237C22×D3—64no—
238C22×D3#63811yescharacteristic
239C3×D4—64no—
240C3×D4—64no—
241C3×D4—128no—
242C3×D4—181no—
243C3×D4—181no—
244C2×C3×C4—24no—
245C2×C3×C4—21no—
246C2×C3×C4—24no—
247C2×C3×C4—34no—
248C2×C3×C4—44no—
249C2×C3×C4—64no—
250C2×C3×C4—64no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—362no—
254#67—368no—
255C2×Dic3—184no—
256C2×Dic3—181no—
257C2×Dic3—364no—
258C2×Dic3#63811yescharacteristic
259D12—21no—
260D12—21no—
261D12—42no—
262D12—64no—
263D12—64no—
264C4×D3—21no—
265C4×D3—21no—
266C3×C8—181no—
267C3⋊C8—64no—
268C3⋊C8—98no—
269C3⋊C8—92no—
270C3⋊C8—188no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32—121no—
275C2×C32—241no—
276C2×C32#88411yescharacteristic
277C2×C32#91111yescharacteristic
278C2×C32#91411yescharacteristic
279C2×C32#93214yes—
280C2×C32#93914yes—
281C2×C32#94914yes—
282C3⋊D3—81no—
283C3×D3—44no—
284C3×D3—84no—
285C3×D3—244no—
286C2×D4—31no—
287C2×D4—61no—
288C2×D4—91no—
289C2×D4—541no—
290C22×C4—31no—
291D8—542no—
292M16—542no—
293C2×C8—91no—
294C2×C8—271no—
295C4⋊C4—271no—
296C4⋊C4D3×C3⋊D311yescharacteristic
297C22⋊C4—61no—
298C22⋊C4—541no—
299C22×C3—24no—
300C22×C3—64no—
301C22×C3—64no—
302C22×C3—64no—
303C22×C3—124no—
304C22×C3—124no—
305C22×C3—124no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3D3×#6714yes—
310C22×C3C2×#63811yescharacteristic
311C2×D3—21no—
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—124no—
318C2×D3—124no—
319C2×D3—124no—
320C3×C4—24no—
321C3×C4—24no—
322C3×C4—41no—
323C3×C4—44no—
324C3×C4—64no—
325C3×C4—84no—
326C3×C4#379514yes—
327C3×C4#380214yes—
328C3×C4D4×C3⋊D311yescharacteristic
329C3×C4#383011yescharacteristic
330Dic3—21no—
331Dic3—361no—
332Dic3—364no—
333Dic3—724no—
334C32—24no—
335C32#478211yescharacteristic, complemented
336C32#503914yescomplemented
337C23—31no—
338C23—61no—
339C23—181no—
340D4—61no—
341D4—61no—
342D4—122no—
343D4—181no—
344D4—181no—
345D4—1082no—
346C2×C4—21no—
347C2×C4—31no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—541no—
351C2×C4C2×D3×C3⋊D311yescharacteristic
352C8—181no—
353C8—272no—
354C2×C3—24no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—124no—
358C2×C3—244no—
359C2×C3—721no—
360C2×C3#6388514yes—
361C2×C3#6393214yes—
362C2×C3#6393914yes—
363C2×C3#6414111yescharacteristic
364C2×C3#6414811yescharacteristic
365C2×C3#6415811yescharacteristic
366D3—41no—
367D3—81no—
368D3—244no—
369C22—61no—
370C22—61no—
371C22—61no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—361no—
376C22—361no—
377C22—361no—
378C22D3×#63811yescharacteristic
379C4—41no—
380C4—61no—
381C4—1081no—
382C4#8901411yescharacteristic
383C4#8902611yescharacteristic
384C3—24no—
385C3#1059050414yescomplemented
386C3#1059273811yescharacteristic, complemented
387C2—121no—
388C2—241no—
389C2—721no—
390C2#1172347411yescharacteristic
391C2#1172355511yescharacteristic
392C2#1172356811yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#469
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#895C2semidirect
2. #11723515C2semidirect
3. C2×#88789C2semidirect
4. C2×#88786C2semidirect
5. #11722178C2semidirect
6. C3×#63636C2semidirect
7. #11722032C2semidirect
27. C3×#895D3semidirect
32. #63636D3semidirect
53. C3⋊(C3⋊(C3⋊C8))D4semidirect
108. #895C3⋊D3semidirect
232. C33#469semidirect
335. C32#4782semidirect
336. C32#5039semidirect
385. C3#10590504semidirect
386. C3#10592738semidirect

Maximal subgroups

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

subgroupindex
C32×#8952
#117235152
C2×#887892
C2×#887862
#117221782
C3×#636362
#117220322
#105927383
#105905043