Finite Group Databasebeta

← #13161049

Subgroups — #13161049

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#11723979C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic, complemented
3#11723531C211yesmaximal, characteristic, complemented
4C2×#88788C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6#11722052C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592858—31nomaximal
9#10591454—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6291—21no—
15#88864—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#213—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88788—22no—
23C2×C32×D8—31no—
24#64147—31no—
25#64147—34no—
26#64147—31no—
27#64147—64no—
28C2×#3743—31no—
29C2×C3×#213D314yescomplemented
30#63916—34no—
31C2×#3712—34no—
32#63659—31no—
33#63659D311yescharacteristic, complemented
34#63644—31no—
35C3×#832—34no—
36#63573—34no—
37#63563—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C32×D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44#6312—42no—
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
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)—21no—
54C32×C3⋊C8D412yescomplemented
55#5159—94no—
56#4782—91no—
57C2×C32×D4—24no—
58C2×C32×D4—31no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—61no—
62C2×#638—61no—
63C2×#638—64no—
64C2×#638—68no—
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—34no—
67C2×C4×C3⋊D3—64no—
68C2×C3×D12C2×D314yes—
69C2×#626—64no—
70C32×D8—62no—
71C2×C32×C8—31no—
72#3757—61no—
73#3757—61no—
74#3757—64no—
75#3757—68no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—34no—
78C4⋊(C3⋊Dic3)—64no—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81#3743—62no—
82C3×#213—28no—
83C3×C4⋊Dic3—34no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8C2×D314yes—
86(C2×D3)⋊Dic3—64no—
87#3712—68no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33C2×D411yescharacteristic
92C2×C3⋊(C3⋊D3)—21no—
93C2×C3⋊(C3⋊D3)—21no—
94C2×C3⋊(C3⋊D3)—21no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C2×C32×D3—41no—
98C33×C4C2×D411yescharacteristic
99C33×C4#4111yescharacteristic
100C3⋊(C3⋊Dic3)—21no—
101C3⋊(C3⋊Dic3)—41no—
102C3×C3⋊Dic3—41no—
103C2×C3×D8—34no—
104#938—94no—
105#938—94no—
106#938—91no—
107#938—184no—
108C2×#213C3⋊D311yescharacteristic, complemented
109C2×ΓL(1,𝔽25)—94no—
110C2×ΓL(1,𝔽25)—91no—
111#895—91no—
112C3×#102—91no—
113#832—34no—
114#832—94no—
115#821—91no—
116#817—94no—
117#808—91no—
118C23×C32—28no—
119C23×C32—24no—
120C23×C32—21no—
121C23×C32—61no—
122C22×C3⋊D3—31no—
123C22×C3⋊D3—34no—
124C22×C3⋊D3—64no—
125C22×C3×D3—24no—
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—24no—
129C32×D4—24no—
130C32×D4—44no—
131C32×D4—44no—
132C32×D4—61no—
133C32×D4—61no—
134C2×C32×C4—24no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137#638—122no—
138#638—1216no—
139#638—128no—
140#638—122no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—31no—
143C2×C3⋊Dic3—34no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3—68no—
146C2×C3⋊Dic3—64no—
147C2×C3⋊Dic3—61no—
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—32no—
159C3⋊(C3⋊C8)—21no—
160C3×C3⋊C8—64no—
161C3×C3⋊C8#6718yescomplemented
162#469—271noSylow
163C2×C33—81no—
164C2×C33#12111yescharacteristic
165C2×C33AGL(1,ℤ/8)11yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C3⋊(C3⋊D3)—41no—
168C32×D3—81no—
169C2×C3×D4—24no—
170C2×C3×D4—34no—
171C2×C3×D4D3214yes—
172C2×C3×D4C2×C3⋊D311yescharacteristic
173C2×#67—184no—
174C2×#67—184no—
175C2×#67—181no—
176C2×#67—188no—
177C2×D12—181no—
178C2×D12C2×C3⋊D311yescharacteristic
179C2×C4×D3—94no—
180C2×C4×D3—91no—
181C2×C4×D3—184no—
182C3×D8—68no—
183C2×C3×C8—34no—
184C2×C3×C8—91no—
185C3×C4⋊C4—91no—
186#217—184no—
187#217—184no—
188#217—181no—
189#217—188no—
190#213—22no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—34no—
193C4⋊Dic3—94no—
194C4⋊Dic3—91no—
195C4⋊Dic3—184no—
196C2×C3⋊C8—34no—
197C2×C3⋊C8C2×C3⋊D311yescharacteristic
198ΓL(1,𝔽25)—188no—
199ΓL(1,𝔽25)—182no—
200C22×C32—24no—
201C22×C32—48no—
202C22×C32—48no—
203C22×C32—48no—
204C22×C32—41no—
205C22×C32—44no—
206C22×C32—44no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—44no—
210C22×C32—121no—
211C22×C32—121no—
212C22×C32—121no—
213C22×C32D3×D414yes—
214C22×C32D3×D411yescharacteristic
215C2×C3⋊D3—68no—
216C2×C3⋊D3—61no—
217C2×C3⋊D3—64no—
218C2×C3⋊D3—61no—
219C2×C3⋊D3—61no—
220C2×C3⋊D3—64no—
221C2×C3⋊D3—64no—
222C2×C3⋊D3—124no—
223C2×C3⋊D3—124no—
224C2×C3×D3—44no—
225C2×C3×D3—44no—
226C2×C3×D3—44no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4#23511yescharacteristic
230C32×C4D3×D411yescharacteristic
231C32×C4#23714yes—
232C32×C4C2×#6714yes—
233C3⋊Dic3—41no—
234C3⋊Dic3—61no—
235C3⋊Dic3—64no—
236C3⋊Dic3—124no—
237C3⋊Dic3—128no—
238C3⋊Dic3—124no—
239C3⋊Dic3—121no—
240C3×Dic3—124no—
241C2×D8—31no—
242C2×M16—271no—
243#121—271no—
244#121—271no—
245#106—271no—
246#102—91no—
247#102—271no—
248C33#46911yescharacteristic, complemented, Sylow
249C23×C3—28no—
250C23×C3—24no—
251C23×C3—21no—
252C23×C3—64no—
253C22×D3—21no—
254C22×D3—94no—
255C22×D3—91no—
256C22×D3—184no—
257C3×D4—24no—
258C3×D4—24no—
259C3×D4—21no—
260C3×D4—21no—
261C3×D4—44no—
262C3×D4—44no—
263C3×D4—64no—
264C3×D4—64no—
265C2×C3×C4—24no—
266C2×C3×C4—181no—
267C2×C3×C4C2×D3214yes—
268C2×C3×C4C22×C3⋊D311yescharacteristic
269#67—368no—
270#67—3616no—
271#67—362no—
272#67—368no—
273C2×Dic3—64no—
274C2×Dic3—94no—
275C2×Dic3—91no—
276C2×Dic3—184no—
277C2×Dic3—188no—
278C2×Dic3—184no—
279C2×Dic3—181no—
280D12—21no—
281D12—21no—
282D12—362no—
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⋊C8#63812yescomplemented
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—88no—
297C2×C32—84no—
298C2×C32—81no—
299C2×C32—241no—
300C2×C32#88411yescharacteristic
301C2×C32#91114yes—
302C2×C32#91114yes—
303C2×C32#91111yescharacteristic
304C2×C32#91411yescharacteristic
305C2×C32#93814yes—
306C3⋊D3—128no—
307C3⋊D3—124no—
308C3⋊D3—121no—
309C3×D3—84no—
310C2×D4—31no—
311C2×D4—541no—
312C2×D4—541no—
313C2×D4D3×C3⋊D311yescharacteristic
314C22×C4—271no—
315D8—62no—
316M16—542no—
317C2×C8—31no—
318C2×C8—91no—
319C4⋊C4—91no—
320C4⋊C4—271no—
321C22⋊C4—541no—
322C22⋊C4—541no—
323C22×C3—24no—
324C22×C3—48no—
325C22×C3—48no—
326C22×C3—48no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—44no—
331C22×C3—44no—
332C22×C3—44no—
333C22×C3—124no—
334C22×C3—124no—
335C22×C3—124no—
336C22×C3#381114yes—
337C22×C3D4×C3⋊D311yescharacteristic
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—188no—
342C2×D3—181no—
343C2×D3—181no—
344C2×D3—181no—
345C2×D3—184no—
346C2×D3—184no—
347C2×D3—184no—
348C2×D3—364no—
349C2×D3—364no—
350C3×C4—24no—
351C3×C4—24no—
352C3×C4—361no—
353C3×C4#380414yes—
354C3×C4D3×#6714yes—
355C3×C4#383011yescharacteristic
356C3×C4C2×#63811yescharacteristic
357Dic3—124no—
358Dic3—181no—
359Dic3—184no—
360Dic3—364no—
361Dic3—368no—
362Dic3—361no—
363Dic3—364no—
364C32—24no—
365C32#478211yescharacteristic, complemented
366C32#515914yescomplemented
367C23—21no—
368C23—61no—
369C23—271no—
370D4—21no—
371D4—21no—
372D4—61no—
373D4—61no—
374D4—1082no—
375D4—1082no—
376C2×C4—181no—
377C2×C4—271no—
378C2×C4—541no—
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—88no—
388C2×C3—81no—
389C2×C3—84no—
390C2×C3—244no—
391C2×C3#6392714yes—
392C2×C3#6393714yes—
393C2×C3#6395814yes—
394C2×C3#6412011yescharacteristic
395C2×C3#6412011yescharacteristic
396C2×C3#6414711yescharacteristic
397D3—81no—
398D3—368no—
399D3—361no—
400D3—364no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—541no—
408C22—541no—
409C22—541no—
410C22#8903911yescharacteristic
411C4—361no—
412C4—541no—
413C4—1081no—
414C4#8902811yescharacteristic
415C4D3×#63811yescharacteristic
416C3—24no—
417C3#1059145414yescomplemented
418C3#1059285811yescharacteristic, complemented
419C2—81no—
420C2—241no—
421C2—1081no—
422C2#1172354811yescharacteristic
423C2#1172356411yescharacteristic
424C2#1172359911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#469
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723979C2semidirect
2. C2×C32×#213C2semidirect
3. #11723531C2semidirect
4. C2×#88788C2semidirect
5. C3×#63659C2semidirect
6. #11722052C2semidirect
7. #11722032C2semidirect
29. C2×C3×#213D3semidirect
33. #63659D3semidirect
54. C32×C3⋊C8D4semidirect
108. C2×#213C3⋊D3semidirect
161. C3×C3⋊C8#67semidirect
248. C33#469semidirect
292. C3⋊C8#638semidirect
365. C32#4782semidirect
366. C32#5159semidirect
417. C3#10591454semidirect
418. C3#10592858semidirect

Maximal subgroups

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

subgroupindex
#117239792
C2×C32×#2132
#117235312
C2×#887882
C3×#636592
#117220522
#117220322
#105928583
#105914543