Finite Group Databasebeta

← #13149813

Subgroups — #13149813

Order 1728.

1007 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 139 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#11722491C212yesmaximal, complemented
2C2×C8×#272C211yesmaximal, characteristic, complemented
3#64113⋊C3C212yesmaximal, complemented
4#11721060C212yesmaximal, complemented
5#11720844C212yesmaximal, complemented
6#11720843C212yesmaximal, complemented
7#11720838C211yesmaximal, characteristic, complemented
8#11720837C212yesmaximal, complemented
9#11720833C211yesmaximal, characteristic, complemented
10D3×#904—32nomaximal
11C2×C4×#272C2211yescharacteristic
12#3828⋊C3C2214yescomplemented
13#88677—22no—
14#88672—22no—
15#88655C2212yes—
16C4×((C3⋊D3)⋊C3)⋊C2—22no—
17#88651C2212yes—
18#88650—22no—
19#88649C2214yescomplemented
20C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
21C2×#6180C2211yescharacteristic, complemented
22C8×#272C2211yescharacteristic
23C8×#272C2211yescharacteristic
24C8×#272C2211yescharacteristic, complemented
25C8×#272C2211yescharacteristic, complemented
26#3745⋊C3C2212yescomplemented
27#3744⋊C3C2214yescomplemented
28#3746⋊C3C2212yescomplemented
29#88433C2212yescomplemented
30#88432C2212yescomplemented
31#88431C2214yescomplemented
32D3×#235—34no—
33#64113D312yescomplemented
34C3×#904—32no—
35C2×C3×D3×C8—32no—
36C2×C3×D3×C8—61no—
37#63829—34no—
38#63808—32no—
39#63807—32no—
40#63806—34no—
41D3×Dic12—32no—
42D3×D24—32no—
43D3×#204—34no—
44C22×#272D411yescharacteristic
45C2×((C3⋊D3)⋊C3)⋊C2—22no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×#1084D411yescharacteristic
48C4×#272D411yescharacteristic
49C4×#272D411yescharacteristic
50C4×#272C2311yescharacteristic
51C4×#272C2311yescharacteristic
52#638⋊C3—24no—
53C4⋊((C3⋊D3)⋊C3)C2314yes—
54C4×(C3⋊D3)⋊C3—24no—
55#634⋊C3C2314yes—
56#6200—24no—
57#6199—22no—
58#6198—22no—
59((C3⋊D3)⋊C3)⋊C4—24no—
60#6196—22no—
61C8×H(3,𝔽3)C2311yescharacteristic
62C8×H(3,𝔽3)C2311yescharacteristic
63#6180C2311yescharacteristic
64#6180C2311yescharacteristic
65D3×#135—92no—
66C2×#904—91no—
67#3828C2×D314yescomplemented
68C3×#235—34no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—61no—
71D3×#67—64no—
72#3804—64no—
73D3×D12—34no—
74C4×D32—64no—
75#3799—34no—
76#3798—64no—
77#3795—34no—
78D3×Dic6—34no—
79C2×C32×C8—21no—
80C2×C32×C8C2×D312yescomplemented
81#3746C2×D312yescomplemented
82#3745C2×D312yescomplemented
83#3744C2×D314yescomplemented
84C2×C3×C3⋊C8—32no—
85C2×C3×C3⋊C8—61no—
86C3×Dic12—32no—
87C3×D24—32no—
88C3×#204—34no—
89C3×D3×C8—32no—
90C3×D3×C8—32no—
91C3×D3×C8—32no—
92C3×D3×C8—32no—
93C3×D3×C8—61no—
94C3×D3×C8—61no—
95C3×D3×C8—61no—
96C3×D3×C8—61no—
97#3719—34no—
98#3717—34no—
99#3716—34no—
100#3714—34no—
101C22×H(3,𝔽3)C2×D411yescharacteristic
102C2×#272—22no—
103C2×#272C2×D411yescharacteristic
104C2×#272C2×D411yescharacteristic
105C2×(C3⋊D3)⋊C3—24no—
106((C3⋊D3)⋊C3)⋊C2—44no—
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2411yescharacteristic
109#1084C2×D411yescharacteristic
110#1084C2×D411yescharacteristic
111(C3⋊Dic3)⋊C3—24no—
112D3×#41—94no—
113C2×#235—92no—
114C3×#135—92no—
115C22×C3×C8—91no—
116#950—94no—
117#919—92no—
118D3×Q16—92no—
119#916—94no—
120D3×SD16—94no—
121#912—92no—
122D3×D8—92no—
123C2×Dic12—91no—
124#904—32no—
125#904—92no—
126#904—92no—
127#904—92no—
128#904—92no—
129C2×D24—91no—
130C2×#204—92no—
131C2×D3×C8—32no—
132C2×D3×C8—61no—
133C22×C3×D3—32no—
134C22×C3×D3—61no—
135C2×D32—64no—
136C2×C32×C4—21no—
137C2×C32×C4C22×D312yes—
138#638—24no—
139C4⋊(C3⋊D3)C22×D314yes—
140C4×C3⋊D3—24no—
141#634C22×D314yes—
142C3×#67—64no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—61no—
145C3×D12—34no—
146C3×C4×D3—32no—
147C3×C4×D3—32no—
148C3×C4×D3—32no—
149C3×C4×D3—32no—
150C3×C4×D3—61no—
151C3×C4×D3—61no—
152C3×C4×D3—61no—
153C3×C4×D3—61no—
154C3×C4×D3—64no—
155C3×Dic6—34no—
156#627—64no—
157#626—64no—
158#626—64no—
159#625—64no—
160#624—64no—
161D3×Dic3—64no—
162D3×Dic3—64no—
163C32×C8—21no—
164C32×C8—21no—
165C32×C8C22×D312yes—
166C32×C8C22×D312yes—
167C3×C3⋊C8—32no—
168C3×C3⋊C8—32no—
169C3×C3⋊C8—61no—
170C3×C3⋊C8—61no—
171C2×#135—271noSylow
172C2×H(3,𝔽3)—21no—
173C2×H(3,𝔽3)C22×D411yescharacteristic
174#272—21no—
175#272#13512yescomplemented
176(C3⋊D3)⋊C3—44no—
177C3×#41—94no—
178C22×C3×C4—91no—
179C2×#67—182no—
180#239—94no—
181D3×Q8—94no—
182#237—94no—
183#237—184no—
184D3×D4—94no—
185D3×D4—184no—
186#235—34no—
187#235—94no—
188#235—94no—
189#235—184no—
190#235—182no—
191#235—182no—
192C2×D12—92no—
193C2×C4×D3—32no—
194C2×C4×D3—61no—
195C2×C4×D3—184no—
196C2×C4×D3—182no—
197C2×Dic6—92no—
198C3×Q16—92no—
199C3×SD16—94no—
200C3×D8—92no—
201C2×C3×C8—32no—
202C2×C3×C8—61no—
203C2×C3×C8—91no—
204C2×C3×C8—91no—
205C2×C3×C8—91no—
206C2×C3×C8—91no—
207C2×C3×C8—91no—
208C2×C3×C8D3211yescharacteristic
209#216—94no—
210#215—94no—
211#214—94no—
212#213—94no—
213C2×C3⋊C8—32no—
214C2×C3⋊C8—61no—
215Dic12—32no—
216Dic12—92no—
217Dic12—92no—
218D24—32no—
219D24—92no—
220D24—92no—
221#204—34no—
222#204—94no—
223#204—94no—
224D3×C8—32no—
225D3×C8—32no—
226D3×C8—32no—
227D3×C8—32no—
228D3×C8—61no—
229D3×C8—61no—
230D3×C8—61no—
231D3×C8—61no—
232C22×C32—21no—
233C22×C32D3×D412yes—
234C2×C3⋊D3—24no—
235C2×C3×D3—32no—
236C2×C3×D3—32no—
237C2×C3×D3—61no—
238C2×C3×D3—61no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C2×C3×D3—64no—
242D32—128no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4D3×D412yes—
246C32×C4C23×D312yes—
247C3⋊Dic3—24no—
248C3×Dic3—32no—
249C3×Dic3—32no—
250C3×Dic3—61no—
251C3×Dic3—61no—
252C3×Dic3—64no—
253C2×#41—272no—
254#135—92no—
255#135—272no—
256#135—272no—
257#135—272no—
258C2×Q16—271no—
259C2×SD16—272no—
260C2×D8—271no—
261C22×C8—91no—
262H(3,𝔽3)C2×#13511yescharacteristic, complemented, Sylow
263C23×C3—91no—
264C22×D3—32no—
265C22×D3—61no—
266C22×D3—184no—
267C22×D3—182no—
268C3×Q8—94no—
269C3×D4—94no—
270C3×D4—184no—
271C2×C3×C4—32no—
272C2×C3×C4—61no—
273C2×C3×C4—91no—
274C2×C3×C4—91no—
275C2×C3×C4—91no—
276C2×C3×C4—91no—
277C2×C3×C4—91no—
278C2×C3×C4—184no—
279C2×C3×C4C2×D3211yescharacteristic
280#67—64no—
281#67—184no—
282#67—184no—
283#67—184no—
284#67—184no—
285C2×Dic3—32no—
286C2×Dic3—61no—
287C2×Dic3—184no—
288C2×Dic3—182no—
289D12—34no—
290D12—94no—
291D12—182no—
292D12—184no—
293C4×D3—32no—
294C4×D3—32no—
295C4×D3—32no—
296C4×D3—32no—
297C4×D3—61no—
298C4×D3—61no—
299C4×D3—61no—
300C4×D3—61no—
301C4×D3—64no—
302C4×D3—184no—
303C4×D3—184no—
304C4×D3—184no—
305C4×D3—184no—
306Dic6—34no—
307Dic6—94no—
308Dic6—184no—
309Dic6—182no—
310C3×C8—32no—
311C3×C8—32no—
312C3×C8—61no—
313C3×C8—61no—
314C3×C8—91no—
315C3×C8—91no—
316C3×C8C2×D3211yescharacteristic
317C3×C8C2×D3211yescharacteristic
318C3⋊C8—32no—
319C3⋊C8—32no—
320C3⋊C8—61no—
321C3⋊C8—61no—
322C2×C32—22no—
323C2×C32—22no—
324C2×C32—21no—
325C2×C32C2×D3×D412yes—
326C3⋊D3—44no—
327C3×D3—34no—
328C3×D3—62no—
329C3×D3—62no—
330C3×D3—62no—
331C3×D3—124no—
332#41—94no—
333#41—274no—
334#41—542no—
335#41—542no—
336C2×Q8—272no—
337C2×D4—272no—
338C2×D4—542no—
339C22×C4—91no—
340C22×C4—542no—
341Q16—92no—
342Q16—272no—
343SD16—94no—
344SD16—274no—
345D8—92no—
346D8—272no—
347C2×C8—91no—
348C2×C8—91no—
349C2×C8—91no—
350C2×C8—91no—
351C2×C8—91no—
352C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
353C22×C3—32no—
354C22×C3—61no—
355C22×C3—91no—
356C22×C3—91no—
357C22×C3—182no—
358C22×C3—184no—
359C22×C3#380211yescharacteristic
360C2×D3—32no—
361C2×D3—32no—
362C2×D3—61no—
363C2×D3—61no—
364C2×D3—64no—
365C2×D3—64no—
366C2×D3—64no—
367C2×D3—184no—
368C2×D3—368no—
369C2×D3—364no—
370C3×C4—32no—
371C3×C4—32no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—91no—
375C3×C4—91no—
376C3×C4—184no—
377C3×C4#380211yescharacteristic
378C3×C4C22×D3211yescharacteristic
379Dic3—32no—
380Dic3—32no—
381Dic3—61no—
382Dic3—61no—
383Dic3—64no—
384Dic3—184no—
385C32—21no—
386C32D3×#13512yescomplemented
387C23—91no—
388C23—542no—
389Q8—94no—
390Q8—542no—
391D4—94no—
392D4—184no—
393D4—542no—
394D4—544no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—91no—
399C2×C4—91no—
400C2×C4—184no—
401C2×C4—544no—
402C2×C4—542no—
403C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
404C8—91no—
405C8—91no—
406C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
407C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
408C2×C3—21no—
409C2×C3—32no—
410C2×C3—62no—
411C2×C3—62no—
412C2×C3—61no—
413C2×C3—92no—
414C2×C3—181no—
415C2×C3—364no—
416C2×C3C2×#380211yescharacteristic
417D3—34no—
418D3—62no—
419D3—62no—
420D3—62no—
421D3—124no—
422D3—364no—
423C22—91no—
424C22—91no—
425C22—182no—
426C22—184no—
427C22—1084no—
428C22#8865511yescharacteristic
429C4—91no—
430C4—91no—
431C4—184no—
432C4#8865511yescharacteristic
433C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
434C3—32no—
435C3—61no—
436C3#1059468411yescharacteristic
437C2—21no—
438C2—92no—
439C2—181no—
440C2—364no—
441C2C2×#8865511yescharacteristic
subgroupG/H
centerC4#88655
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×#3802
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#135
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11722491C2semidirect
2. C2×C8×#272C2semidirect
3. #64113⋊C3C2semidirect
4. #11721060C2semidirect
5. #11720844C2semidirect
6. #11720843C2semidirect
7. #11720838C2semidirect
8. #11720837C2semidirect
9. #11720833C2semidirect
12. #3828⋊C3C22semidirect
19. #88649C22semidirect
20. C2×C8×H(3,𝔽3)C22semidirect
21. C2×#6180C22semidirect
24. C8×#272C22semidirect
25. C8×#272C22semidirect
26. #3745⋊C3C22semidirect
27. #3744⋊C3C22semidirect
28. #3746⋊C3C22semidirect
29. #88433C22semidirect
30. #88432C22semidirect
31. #88431C22semidirect
33. #64113D3semidirect
67. #3828C2×D3semidirect
80. C2×C32×C8C2×D3semidirect
81. #3746C2×D3semidirect
82. #3745C2×D3semidirect
83. #3744C2×D3semidirect
175. #272#135semidirect
262. H(3,𝔽3)C2×#135semidirect
352. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
386. C32D3×#135semidirect

Maximal subgroups

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

subgroupindex
#117224912
C2×C8×#2722
#64113⋊C32
#117210602
#117208442
#117208432
#117208382
#117208372
#117208332
D3×#9043