Finite Group Databasebeta

← #13159367

Subgroups — #13159367

Order 1728.

1156 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1156 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
1#11723936C211yesmaximal, characteristic, complemented
2#11723532C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic, complemented
4C2×#88788C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#808C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592564—31nomaximal
9#10590508—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×#6292—21no—
15C33×C4⋊C4C2211yescharacteristic
16#88859—21no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88803—21no—
21#88791—22no—
22#88788—22no—
23C2×#3745—31no—
24C2×#3743—31no—
25#64104—31no—
26#64104—31no—
27#64104—34no—
28#64104—64no—
29#63916—34no—
30C2×#3714—34no—
31C2×#3712—34no—
32C32×#106—31no—
33#63636—31no—
34#63636D311yescharacteristic, complemented
35C3×#809—34no—
36C3×#808D314yescomplemented
37#63564—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C3×C3⋊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—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C2×C32×Dic3—21no—
52#6292—42no—
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8D412yescomplemented
55#4865—94no—
56#4782—91no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—61no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×C4⋊(C3⋊D3)—64no—
62C2×C4⋊(C3⋊D3)—68no—
63C2×C4⋊(C3⋊D3)—61no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—34no—
67C2×C4×C3⋊D3—64no—
68C2×C3×D12—34no—
69C2×#626—64no—
70C2×C32×C8—31no—
71C32×C4⋊C4—24no—
72C32×C4⋊C4—31no—
73C32×C4⋊C4C2×D314yes—
74C32×C4⋊C4C2×D311yescharacteristic
75#3752—61no—
76#3752—64no—
77#3752—61no—
78#3752—68no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80#3745—62no—
81#3743—62no—
82C3×C4⋊Dic3C2×D314yes—
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×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C33×C4—41no—
99C33×C4C2×D411yescharacteristic
100C33×C4#4111yescharacteristic
101C3⋊(C3⋊Dic3)—21no—
102C32×Dic3—41no—
103#938—91no—
104C2×#213—91no—
105C2×D24—94no—
106C2×ΓL(1,𝔽25)—94no—
107C2×ΓL(1,𝔽25)—91no—
108#895—94no—
109#895—94no—
110#895—91no—
111#895—184no—
112C3×#106—34no—
113C3×#102—91no—
114#821—91no—
115#809—34no—
116#809—94no—
117#808C3⋊D311yescharacteristic, complemented
118C22×C3⋊D3—21no—
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—64no—
126C2×C32×C4—28no—
127C2×C32×C4—24no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—61no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—31no—
134C2×C3⋊Dic3—34no—
135C2×C3⋊Dic3—64no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—124no—
143C4⋊(C3⋊D3)—124no—
144C4⋊(C3⋊D3)—1216no—
145C4⋊(C3⋊D3)—122no—
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—24no—
155C3×D12—64no—
156C3×D12—64no—
157#626—128no—
158C32×C8—32no—
159C3⋊(C3⋊C8)—21no—
160C3×C3⋊C8—64no—
161C3×C3⋊C8D1218yescomplemented
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—
168C3×C3⋊D3—81no—
169C2×C3×D4—91no—
170C2×#67—181no—
171C2×D12—34no—
172C2×D12—94no—
173C2×D12—91no—
174C2×D12—184no—
175C2×D12—184no—
176C2×D12—181no—
177C2×D12—188no—
178C2×D12—184no—
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—34no—
186C3×C4⋊C4D3214yes—
187C3×C4⋊C4C2×C3⋊D311yescharacteristic
188#217—181no—
189#213—182no—
190(C2×D3)⋊C4—184no—
191(C2×D3)⋊C4—181no—
192(C2×D3)⋊C4—184no—
193(C2×D3)⋊C4—188no—
194C4⋊Dic3C2×C3⋊D311yescharacteristic
195C2×C3⋊C8—34no—
196C2×C3⋊C8C2×C3⋊D311yescharacteristic
197D24—188no—
198ΓL(1,𝔽25)—188no—
199ΓL(1,𝔽25)—182no—
200C22×C32—24no—
201C22×C32D3×D414yes—
202C22×C32D3×D411yescharacteristic
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—68no—
207C2×C3⋊D3—61no—
208C2×C3⋊D3—61no—
209C2×C3⋊D3—64no—
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—121no—
219C2×C3⋊D3—124no—
220C2×C3⋊D3—121no—
221C2×C3⋊D3—124no—
222C2×C3⋊D3—124no—
223C2×C3⋊D3—121no—
224C2×C3×D3—124no—
225C2×C3×D3—124no—
226C2×C3×D3—124no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—48no—
230C32×C4—41no—
231C32×C4—44no—
232C32×C4—121no—
233C32×C4C2×D1214yes—
234C32×C4#23511yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4#23914yes—
237C3⋊Dic3—61no—
238C3⋊Dic3—64no—
239C3⋊Dic3—124no—
240C3×Dic3—44no—
241C2×D8—271no—
242C2×M16—271no—
243#121—271no—
244#121—271no—
245#106—31no—
246#102—91no—
247#102—271no—
248C33#46911yescharacteristic, complemented, Sylow
249C23×C3—181no—
250C22×D3—64no—
251C22×D3—94no—
252C22×D3—91no—
253C22×D3—184no—
254C22×D3—188no—
255C22×D3—184no—
256C22×D3—181no—
257C3×D4—181no—
258C3×D4—181no—
259C2×C3×C4—28no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—64no—
264C2×C3×C4C2×D3214yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266#67—362no—
267C2×Dic3—21no—
268C2×Dic3—94no—
269C2×Dic3—91no—
270C2×Dic3—184no—
271D12—64no—
272D12—64no—
273D12—184no—
274D12—184no—
275D12—181no—
276D12—181no—
277D12—364no—
278D12—364no—
279D12—3616no—
280D12—362no—
281D12—368no—
282D12—368no—
283C4×D3—184no—
284C4×D3—184no—
285C4×D3—181no—
286C4×D3—181no—
287C4×D3—364no—
288C4×D3—364no—
289C3×C8—38no—
290C3×C8—181no—
291C3⋊C8—64no—
292C3⋊C8C4⋊(C3⋊D3)12yescomplemented
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32#88411yescharacteristic
297C2×C32#89514yes—
298C2×C32#91111yescharacteristic
299C2×C32#91414yes—
300C2×C32#91414yes—
301C2×C32#91411yescharacteristic
302C3⋊D3—81no—
303C3⋊D3—128no—
304C3⋊D3—124no—
305C3⋊D3—121no—
306C3⋊D3—248no—
307C3⋊D3—244no—
308C3⋊D3—241no—
309C3×D3—244no—
310C2×D4—91no—
311C2×D4—271no—
312C2×D4—541no—
313C2×D4—541no—
314C22×C4—271no—
315D8—542no—
316M16—542no—
317C2×C8—31no—
318C2×C8—91no—
319C4⋊C4—31no—
320C4⋊C4D3×C3⋊D311yescharacteristic
321C22⋊C4—541no—
322C22⋊C4—541no—
323C22×C3—24no—
324C22×C3—361no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3#381114yes—
328C22×C3D4×C3⋊D311yescharacteristic
329C2×D3—124no—
330C2×D3—124no—
331C2×D3—124no—
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—124no—
356C3×C4#379914yes—
357C3×C4D3×D1214yes—
358C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
359C3×C4#383211yescharacteristic
360Dic3—41no—
361Dic3—181no—
362Dic3—184no—
363Dic3—364no—
364C32—24no—
365C32#478211yescharacteristic, complemented
366C32#486514yescomplemented
367C23—181no—
368C23—271no—
369C23—541no—
370D4—181no—
371D4—181no—
372D4—541no—
373D4—541no—
374D4—1082no—
375D4—1082no—
376C2×C4—21no—
377C2×C4—61no—
378C2×C4—271no—
379C2×C4—541no—
380C2×C4—541no—
381C2×C4C2×D3×C3⋊D311yescharacteristic
382C8—32no—
383C8—181no—
384C2×C3—24no—
385C2×C3—24no—
386C2×C3—24no—
387C2×C3—721no—
388C2×C3#6391014yes—
389C2×C3#6393714yes—
390C2×C3#6394114yes—
391C2×C3#6410411yescharacteristic
392C2×C3#6412311yescharacteristic
393C2×C3#6412311yescharacteristic
394D3—244no—
395D3—368no—
396D3—364no—
397D3—361no—
398D3—728no—
399D3—724no—
400D3—721no—
401C22—361no—
402C22—361no—
403C22—361no—
404C22—541no—
405C22—541no—
406C22—541no—
407C22—1081no—
408C22—1081no—
409C22—1081no—
410C22#8903911yescharacteristic
411C4—41no—
412C4—121no—
413C4—541no—
414C4#8901811yescharacteristic
415C4D3×C4⋊(C3⋊D3)11yescharacteristic
416C3—24no—
417C3#1059050814yescomplemented
418C3#1059256411yescharacteristic, complemented
419C2—721no—
420C2—1081no—
421C2—2161no—
422C2#1172348611yescharacteristic
423C2#1172356611yescharacteristic
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. #11723532C2semidirect
3. C2×#88791C2semidirect
4. C2×#88788C2semidirect
5. C3×#63636C2semidirect
6. C32×#808C2semidirect
7. #11722035C2semidirect
34. #63636D3semidirect
36. C3×#808D3semidirect
54. C32×C3⋊C8D4semidirect
117. #808C3⋊D3semidirect
161. C3×C3⋊C8D12semidirect
248. C33#469semidirect
292. C3⋊C8C4⋊(C3⋊D3)semidirect
365. C32#4782semidirect
366. C32#4865semidirect
417. C3#10590508semidirect
418. C3#10592564semidirect

Maximal subgroups

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

subgroupindex
#117239362
#117235322
C2×#887912
C2×#887882
C3×#636362
C32×#8082
#117220352
#105925643
#105905083