Finite Group Databasebeta

← #13159352

Subgroups — #13159352

Order 1728.

1156 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1156 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
1#11723936C211yesmaximal, characteristic, complemented
2#11723531C211yesmaximal, characteristic, complemented
3C2×#88790C211yesmaximal, characteristic, complemented
4C2×#88788C211yesmaximal, characteristic, complemented
5C3×#63635C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592481—31nomaximal
9#10590508—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6291—21no—
15C33×C4⋊C4C2211yescharacteristic
16#88859—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88790—22no—
22#88788—22no—
23#64147—31no—
24C2×#3753—31no—
25C2×#3743—31no—
26#64104—31no—
27#64104—34no—
28#64104—64no—
29#63916—34no—
30C2×#3714—34no—
31C2×#3712—34no—
32C32×#102—31no—
33#63648—31no—
34#63635D311yescharacteristic, complemented
35C3×#809D314yescomplemented
36C3×#808—34no—
37#63564—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C32×D3—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—42no—
47C4×C3⋊(C3⋊D3)—21no—
48C4×C3⋊(C3⋊D3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C32×D12—21no—
51C32×D12—21no—
52#6291—42no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54C32×C3⋊C8—21no—
55#4865—91no—
56#4782—94no—
57C2×C32×D4—31no—
58C2×#638—61no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C4⋊(C3⋊D3)—34no—
61C2×C4⋊(C3⋊D3)—61no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×C4⋊(C3⋊D3)—64no—
64C2×C4⋊(C3⋊D3)—68no—
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—34no—
67C2×C4×C3⋊D3—64no—
68C2×C3×D12C2×D314yes—
69C2×#626—64no—
70C2×C32×C8—31no—
71C32×C4⋊C4—24no—
72C32×C4⋊C4C2×D314yes—
73C32×C4⋊C4C2×D311yescharacteristic
74#3757—61no—
75#3753—62no—
76#3752—61no—
77#3752—64no—
78#3752—68no—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81#3743—62no—
82C3×C4⋊Dic3—34no—
83C2×C3×C3⋊C8—34no—
84C2×C3×C3⋊C8C2×D314yes—
85(C2×D3)⋊Dic3—64no—
86#3714—68no—
87#3712—68no—
88C22×C33C2×D411yescharacteristic
89C2×C3⋊(C3⋊D3)—21no—
90C2×C3⋊(C3⋊D3)—21no—
91C2×C3⋊(C3⋊D3)—21no—
92C2×C3⋊(C3⋊D3)—41no—
93C2×C3⋊(C3⋊D3)—41no—
94C2×C3⋊(C3⋊D3)—41no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C2×C32×D3—41no—
98C33×C4—41no—
99C33×C4C2×D411yescharacteristic
100C33×C4#4111yescharacteristic
101C3⋊(C3⋊Dic3)—21no—
102C3×C3⋊Dic3—41no—
103#938—94no—
104C2×#213—94no—
105C2×D24—91no—
106C2×ΓL(1,𝔽25)—94no—
107C2×ΓL(1,𝔽25)—91no—
108#895—91no—
109#895—94no—
110#895—91no—
111#895—184no—
112C3×#106—91no—
113C3×#102—34no—
114#821—94no—
115#809—91no—
116#809C3⋊D311yescharacteristic, complemented
117#808—34no—
118C23×C32—61no—
119C22×C3⋊D3—31no—
120C22×C3⋊D3—34no—
121C22×C3⋊D3—64no—
122C22×C3⋊D3—68no—
123C22×C3⋊D3—64no—
124C22×C3⋊D3—61no—
125C22×C3×D3—24no—
126C32×D4—61no—
127C32×D4—61no—
128C2×C32×C4—28no—
129C2×C32×C4—24no—
130C2×C32×C4—21no—
131C2×C32×C4—24no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—122no—
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—31no—
137C2×C3⋊Dic3—34no—
138C2×C3⋊Dic3—64no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—64no—
143C4⋊(C3⋊D3)—124no—
144C4⋊(C3⋊D3)—124no—
145C4⋊(C3⋊D3)—1216no—
146C4⋊(C3⋊D3)—128no—
147C4⋊(C3⋊D3)—122no—
148C4×C3⋊D3—61no—
149C4×C3⋊D3—61no—
150C4×C3⋊D3—64no—
151C4×C3⋊D3—64no—
152C4×C3⋊D3—124no—
153C4×C3⋊D3—124no—
154C2×C3×Dic3—64no—
155C3×D12—24no—
156C3×D12—24no—
157#626—128no—
158C32×C8—61no—
159C3⋊(C3⋊C8)D1212yescomplemented
160C3×C3⋊C8—24no—
161C3×C3⋊C8—38no—
162#469—271noSylow
163C2×C33#12111yescharacteristic
164C2×C33AGL(1,ℤ/8)11yescharacteristic
165C2×C33AGL(1,ℤ/8)11yescharacteristic
166C3⋊(C3⋊D3)—41no—
167C3⋊(C3⋊D3)—81no—
168C32×D3—81no—
169C2×C3×D4—34no—
170C2×#67—184no—
171C2×D12—94no—
172C2×D12—91no—
173C2×D12—184no—
174C2×D12—184no—
175C2×D12—181no—
176C2×D12—188no—
177C2×D12—181no—
178C2×D12C2×C3⋊D311yescharacteristic
179C2×C4×D3—94no—
180C2×C4×D3—91no—
181C2×C4×D3—184no—
182C2×C3×C8—34no—
183C2×C3×C8—91no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—91no—
186C3×C4⋊C4D3214yes—
187C3×C4⋊C4C2×C3⋊D311yescharacteristic
188#217—184no—
189#213—188no—
190(C2×D3)⋊C4—184no—
191(C2×D3)⋊C4—181no—
192(C2×D3)⋊C4—181no—
193(C2×D3)⋊C4—188no—
194C4⋊Dic3—34no—
195C2×C3⋊C8—34no—
196C2×C3⋊C8C2×C3⋊D311yescharacteristic
197D24—182no—
198ΓL(1,𝔽25)—182no—
199ΓL(1,𝔽25)—188no—
200C22×C32—24no—
201C22×C32—121no—
202C22×C32—121no—
203C22×C32—121no—
204C22×C32D3×D414yes—
205C22×C32D3×D411yescharacteristic
206C2×C3⋊D3—68no—
207C2×C3⋊D3—61no—
208C2×C3⋊D3—64no—
209C2×C3⋊D3—61no—
210C2×C3⋊D3—61no—
211C2×C3⋊D3—64no—
212C2×C3⋊D3—64no—
213C2×C3⋊D3—124no—
214C2×C3⋊D3—124no—
215C2×C3⋊D3—128no—
216C2×C3⋊D3—128no—
217C2×C3⋊D3—128no—
218C2×C3⋊D3—124no—
219C2×C3⋊D3—121no—
220C2×C3⋊D3—124no—
221C2×C3⋊D3—121no—
222C2×C3⋊D3—124no—
223C2×C3⋊D3—121no—
224C2×C3×D3—44no—
225C2×C3×D3—44no—
226C2×C3×D3—44no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—48no—
230C32×C4—44no—
231C32×C4—41no—
232C32×C4C2×D1211yescharacteristic
233C32×C4#23514yes—
234C32×C4D3×D414yes—
235C32×C4#23911yescharacteristic
236C3⋊Dic3—41no—
237C3⋊Dic3—61no—
238C3⋊Dic3—64no—
239C3⋊Dic3—124no—
240C3×Dic3—124no—
241C2×D8—271no—
242C2×M16—271no—
243#121—271no—
244#121—271no—
245#106—91no—
246#102—31no—
247#102—271no—
248C33#46911yescharacteristic, complemented, Sylow
249C23×C3—64no—
250C22×D3—21no—
251C22×D3—94no—
252C22×D3—91no—
253C22×D3—184no—
254C22×D3—188no—
255C22×D3—184no—
256C22×D3—181no—
257C3×D4—64no—
258C3×D4—64no—
259C2×C3×C4—28no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—181no—
264C2×C3×C4C2×D3214yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266#67—368no—
267C2×Dic3—64no—
268C2×Dic3—94no—
269C2×Dic3—91no—
270C2×Dic3—184no—
271D12—21no—
272D12—21no—
273D12—184no—
274D12—184no—
275D12—181no—
276D12—181no—
277D12—364no—
278D12—364no—
279D12—3616no—
280D12—362no—
281D12—368no—
282D12—362no—
283C4×D3—184no—
284C4×D3—184no—
285C4×D3—181no—
286C4×D3—181no—
287C4×D3—364no—
288C4×D3—364no—
289C3×C8—64no—
290C3×C8—92no—
291C3⋊C8—21no—
292C3⋊C8—38no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—241no—
297C2×C32#88414yes—
298C2×C32#89511yescharacteristic
299C2×C32#91114yes—
300C2×C32#91414yes—
301C2×C32#91411yescharacteristic
302C2×C32#91411yescharacteristic
303C3⋊D3—128no—
304C3⋊D3—121no—
305C3⋊D3—124no—
306C3⋊D3—248no—
307C3⋊D3—241no—
308C3⋊D3—244no—
309C3×D3—84no—
310C2×D4—31no—
311C2×D4—271no—
312C2×D4—541no—
313C2×D4—541no—
314C22×C4—271no—
315D8—542no—
316M16—542no—
317C2×C8—31no—
318C2×C8—91no—
319C4⋊C4—91no—
320C4⋊C4D3×C3⋊D311yescharacteristic
321C22⋊C4—541no—
322C22⋊C4—541no—
323C22×C3—24no—
324C22×C3—124no—
325C22×C3—124no—
326C22×C3—124no—
327C22×C3#381114yes—
328C22×C3D4×C3⋊D311yescharacteristic
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—188no—
333C2×D3—181no—
334C2×D3—181no—
335C2×D3—181no—
336C2×D3—184no—
337C2×D3—184no—
338C2×D3—184no—
339C2×D3—364no—
340C2×D3—364no—
341C2×D3—368no—
342C2×D3—368no—
343C2×D3—368no—
344C2×D3—361no—
345C2×D3—361no—
346C2×D3—361no—
347C2×D3—364no—
348C2×D3—364no—
349C2×D3—364no—
350C3×C4—24no—
351C3×C4—24no—
352C3×C4—48no—
353C3×C4—41no—
354C3×C4—44no—
355C3×C4—361no—
356C3×C4#379914yes—
357C3×C4D3×D1214yes—
358C3×C4#382811yescharacteristic
359C3×C4D4×C3⋊D311yescharacteristic
360Dic3—124no—
361Dic3—181no—
362Dic3—184no—
363Dic3—364no—
364C32—24no—
365C32#478214yescomplemented
366C32#486511yescharacteristic, complemented
367C23—61no—
368C23—271no—
369C23—541no—
370D4—61no—
371D4—61no—
372D4—541no—
373D4—541no—
374D4—1082no—
375D4—1082no—
376C2×C4—21no—
377C2×C4—181no—
378C2×C4—271no—
379C2×C4—541no—
380C2×C4—541no—
381C2×C4C2×D3×C3⋊D311yescharacteristic
382C8—61no—
383C8—92no—
384C2×C3—24no—
385C2×C3—24no—
386C2×C3—24no—
387C2×C3—244no—
388C2×C3#6391014yes—
389C2×C3#6393714yes—
390C2×C3#6394114yes—
391C2×C3#6409311yescharacteristic
392C2×C3#6412011yescharacteristic
393C2×C3#6412311yescharacteristic
394D3—81no—
395D3—368no—
396D3—364no—
397D3—361no—
398D3—728no—
399D3—724no—
400D3—721no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—541no—
405C22—541no—
406C22—541no—
407C22—1081no—
408C22—1081no—
409C22—1081no—
410C22#8903911yescharacteristic
411C4—41no—
412C4—361no—
413C4—541no—
414C4#8902211yescharacteristic
415C4C3⋊D3×D1211yescharacteristic
416C3—24no—
417C3#1059050814yescomplemented
418C3#1059248111yescharacteristic, complemented
419C2—241no—
420C2—1081no—
421C2—2161no—
422C2#1172352511yescharacteristic
423C2#1172356411yescharacteristic
424C2#1172357211yescharacteristic
subgroupG/H
centerC22#89039
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. #11723936C2semidirect
2. #11723531C2semidirect
3. C2×#88790C2semidirect
4. C2×#88788C2semidirect
5. C3×#63635C2semidirect
6. C32×#809C2semidirect
7. #11722036C2semidirect
34. #63635D3semidirect
35. C3×#809D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
116. #809C3⋊D3semidirect
159. C3⋊(C3⋊C8)D12semidirect
248. C33#469semidirect
365. C32#4782semidirect
366. C32#4865semidirect
417. C3#10590508semidirect
418. C3#10592481semidirect

Maximal subgroups

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

subgroupindex
#117239362
#117235312
C2×#887902
C2×#887882
C3×#636352
C32×#8092
#117220362
#105924813
#105905083