Finite Group Databasebeta

← #13161346

Subgroups — #13161346

Order 1728.

1216 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 134 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#11724581C211yesmaximal, characteristic
2#11723988C211yesmaximal, characteristic, complemented
3C3×#64151C211yesmaximal, characteristic, complemented
4C32×#942C211yesmaximal, characteristic, complemented
5#11722045C211yesmaximal, characteristic, complemented
6#11722043C211yesmaximal, characteristic, complemented
7#11722027C211yesmaximal, characteristic, complemented
8#10592901—31nomaximal
9#10591658—34nomaximal
10C2×C33×Q8C2211yescharacteristic, complemented
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C32×#235C2211yescharacteristic
14#89039—22no—
15C3⋊D3×D12—21no—
16D3×C4⋊(C3⋊D3)—21no—
17#89022—21no—
18#89018—21no—
19#88859—22no—
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
21C3×#3756—21no—
22C3×#3755—21no—
23C3×#3748C2211yescharacteristic, complemented
24C32×#216—21no—
25C32×#215—21no—
26C32×#208C2211yescharacteristic, complemented
27#64366—31no—
28#64308—34no—
29C32×#137—31no—
30#64156—31no—
31#64156—34no—
32#64156—64no—
33#64151D311yescharacteristic, complemented
34C3×#942—34no—
35C3×#942D314yescomplemented
36#63664—31no—
37#63652—31no—
38#63650—31no—
39#63568—34no—
40#63568—34no—
41#63559—34no—
42C22×C3⋊(C3⋊D3)D412yes—
43C2×D3×C3⋊D3—22no—
44C33×Q8—21no—
45C33×Q8—21no—
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))—21no—
51C3×#638—21no—
52C3×C4⋊(C3⋊D3)D411yescharacteristic
53C3×C4×C3⋊D3—21no—
54C3×#634D411yescharacteristic
55C32×#67—21no—
56C32×D12D411yescharacteristic
57C32×C4×D3—21no—
58C32×Dic6D411yescharacteristic
59#6292—22no—
60#6291—22no—
61#6289—22no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5202—94no—
65#5202—91no—
66C32×#41—31no—
67C2×C32×Q8—24no—
68C2×C32×Q8C2×D314yescomplemented
69C2×C32×Q8C2×D311yescharacteristic, complemented
70#3832—61no—
71D4×C3⋊D3—61no—
72D4×C3⋊D3—62no—
73#3828—61no—
74#3828C2×D311yescharacteristic
75C2×C4⋊(C3⋊D3)—31no—
76C2×C4⋊(C3⋊D3)—34no—
77C2×C4⋊(C3⋊D3)—61no—
78C2×C4⋊(C3⋊D3)—64no—
79C3×#235—34no—
80C3×#235C2×D314yes—
81#3811—68no—
82D3×D12—64no—
83D3×D12—64no—
84#3799—64no—
85#3799—64no—
86C32×Q16—61no—
87C32×SD16—61no—
88C32×M16—31no—
89#3756—21no—
90#3755—21no—
91#3752—62no—
92#3752—68no—
93#3752—616no—
94C4×C3⋊Dic3—31no—
95C4×C3⋊Dic3—34no—
96C4×C3⋊Dic3—64no—
97#3748C2×D311yescharacteristic, complemented
98C3×#216—24no—
99C3×#216—64no—
100C3×#215—24no—
101C3×#215—64no—
102C3×#208—34no—
103C3×#208C2×D314yescomplemented
104C22×C33C2×D411yescharacteristic
105C2×C3⋊(C3⋊D3)—22no—
106C2×C3⋊(C3⋊D3)—42no—
107C2×C3×C3⋊D3—21no—
108C2×C32×D3—21no—
109D3×C3⋊D3—82no—
110C33×C4—41no—
111C33×C4C2×D411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C3×C3⋊Dic3—21no—
115C32×Dic3—21no—
116#1009—94no—
117#1009—91no—
118C3×#137—34no—
119C3×#137—91no—
120#947—94no—
121#947—91no—
122#947—184no—
123#942—34no—
124#942C3⋊D311yescharacteristic, complemented
125#837—94no—
126#837—91no—
127#825—94no—
128#825—91no—
129#823—94no—
130#823—91no—
131C22×C3⋊D3—32no—
132C22×C3⋊D3—38no—
133C22×C3⋊D3—68no—
134C22×C3⋊D3—62no—
135C2×D32—68no—
136C32×Q8—21no—
137C32×Q8—21no—
138C32×Q8—24no—
139C32×Q8—24no—
140C32×Q8—31no—
141C32×Q8—44no—
142C32×Q8—44no—
143C32×D4—31no—
144C32×D4—61no—
145C2×C32×C4—28no—
146C2×C32×C4—24no—
147C2×C32×C4—21no—
148C2×C32×C4—24no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D314yes—
151C2×C32×C4C22×D311yescharacteristic
152#638—21no—
153#638—62no—
154C2×C3⋊Dic3—68no—
155C2×C3⋊Dic3—64no—
156C2×C3⋊Dic3—61no—
157C4⋊(C3⋊D3)—62no—
158C4⋊(C3⋊D3)—61no—
159C4⋊(C3⋊D3)—61no—
160C4⋊(C3⋊D3)—64no—
161C4⋊(C3⋊D3)—64no—
162C4⋊(C3⋊D3)—124no—
163C4⋊(C3⋊D3)—124no—
164C4⋊(C3⋊D3)#6711yescharacteristic
165C4×C3⋊D3—21no—
166C4×C3⋊D3—62no—
167#634#6711yescharacteristic
168C3×#67—24no—
169C3×#67—64no—
170C3×D12—34no—
171C3×D12#6714yes—
172C3×C4×D3—24no—
173C3×C4×D3—64no—
174C3×Dic6—34no—
175C3×Dic6#6714yes—
176#626—68no—
177#626—68no—
178#624—68no—
179C32×C8—61no—
180C3⋊(C3⋊C8)—21no—
181C3×C3⋊C8—24no—
182C3×C3⋊C8—64no—
183#455—271noSylow
184C2×C33—21no—
185C2×C33#12011yescharacteristic
186C3⋊(C3⋊D3)—42no—
187C3×C3⋊D3—41no—
188C32×D3—41no—
189C3×#41—34no—
190C3×#41—91no—
191C2×C3×Q8—24no—
192C2×C3×Q8D3214yescomplemented
193C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
194#239—181no—
195#239—184no—
196D3×D4—181no—
197D3×D4—184no—
198D3×D4—182no—
199D3×D4—188no—
200#235—34no—
201#235—184no—
202#235—181no—
203#235C2×C3⋊D311yescharacteristic
204C2×D12—94no—
205C2×D12—91no—
206C2×D12—184no—
207C2×D12—184no—
208C2×D12—181no—
209C3×Q16—64no—
210C3×Q16—181no—
211C3×SD16—64no—
212C3×SD16—181no—
213C3×M16—34no—
214C3×M16—91no—
215#216—21no—
216#216—64no—
217#215—21no—
218#215—64no—
219(C2×D3)⋊C4—188no—
220(C2×D3)⋊C4—182no—
221(C2×D3)⋊C4—1816no—
222C4×Dic3—94no—
223C4×Dic3—91no—
224C4×Dic3—184no—
225#208—34no—
226#208C2×C3⋊D311yescharacteristic, complemented
227C22×C32—24no—
228C22×C32—61no—
229C22×C32D3×D414yes—
230C22×C32D3×D411yescharacteristic
231C2×C3⋊D3—21no—
232C2×C3⋊D3—62no—
233C2×C3⋊D3—68no—
234C2×C3⋊D3—128no—
235C2×C3⋊D3—122no—
236C2×C3⋊D3—128no—
237C2×C3⋊D3—1216no—
238C2×C3⋊D3—242no—
239C2×C3×D3—24no—
240C2×C3×D3—64no—
241D32—248no—
242C32×C4—24no—
243C32×C4—24no—
244C32×C4—48no—
245C32×C4—44no—
246C32×C4—41no—
247C32×C4—61no—
248C32×C4D3×D414yes—
249C32×C4D3×D411yescharacteristic
250C32×C4C2×#6714yes—
251C32×C4C2×#6711yescharacteristic
252C3⋊Dic3—21no—
253C3⋊Dic3—128no—
254C3⋊Dic3—124no—
255C3⋊Dic3—121no—
256C3×Dic3—24no—
257C3×Dic3—64no—
258H(5,𝔽2)—271no—
259#137—31no—
260#137—91no—
261#124—271no—
262#104—271no—
263#104—271no—
264#100—271no—
265C33#45511yescharacteristic, complemented, Sylow
266C22×D3—98no—
267C22×D3—92no—
268C22×D3—188no—
269C22×D3—182no—
270C22×D3—188no—
271C3×Q8—24no—
272C3×Q8—24no—
273C3×Q8—21no—
274C3×Q8—21no—
275C3×Q8—34no—
276C3×Q8—44no—
277C3×Q8—44no—
278C3×Q8—91no—
279C3×D4—34no—
280C3×D4—64no—
281C3×D4—91no—
282C3×D4—181no—
283C2×C3×C4—28no—
284C2×C3×C4—24no—
285C2×C3×C4—21no—
286C2×C3×C4—24no—
287C2×C3×C4—64no—
288C2×C3×C4—181no—
289C2×C3×C4C2×D3214yes—
290C2×C3×C4C22×C3⋊D311yescharacteristic
291#67—21no—
292#67—64no—
293#67—182no—
294#67—188no—
295C2×Dic3—188no—
296C2×Dic3—184no—
297C2×Dic3—181no—
298D12—34no—
299D12—188no—
300D12—182no—
301D12—184no—
302D12—184no—
303D12—181no—
304D12—181no—
305D12—364no—
306D12—364no—
307D12#63811yescharacteristic
308C4×D3—21no—
309C4×D3—64no—
310C4×D3—182no—
311C4×D3—188no—
312Dic6—34no—
313Dic6#63811yescharacteristic
314C3×C8—64no—
315C3×C8—181no—
316C3⋊C8—21no—
317C3⋊C8—64no—
318C2×C32—28no—
319C2×C32—24no—
320C2×C32—21no—
321C2×C32—24no—
322C2×C32—121no—
323C2×C32#93714yes—
324C2×C32#93711yescharacteristic
325C3⋊D3—41no—
326C3⋊D3—122no—
327C3⋊D3—128no—
328C3⋊D3—248no—
329C3×D3—44no—
330C3×D3—124no—
331#41—31no—
332#41—91no—
333#41—541no—
334#41—541no—
335C2×Q8D3×C3⋊D311yescharacteristic, complemented
336C2×D4—271no—
337C2×D4—541no—
338C2×D4—541no—
339C2×D4—542no—
340Q16—61no—
341Q16—181no—
342SD16—61no—
343SD16—181no—
344M16—31no—
345M16—91no—
346C22⋊C4—542no—
347C42—271no—
348C22×C3—24no—
349C22×C3—64no—
350C22×C3—181no—
351C22×C3#381114yes—
352C22×C3D4×C3⋊D311yescharacteristic
353C2×D3—21no—
354C2×D3—64no—
355C2×D3—182no—
356C2×D3—188no—
357C2×D3—368no—
358C2×D3—362no—
359C2×D3—368no—
360C2×D3—3616no—
361C2×D3—722no—
362C2×D3—728no—
363C3×C4—24no—
364C3×C4—24no—
365C3×C4—48no—
366C3×C4—41no—
367C3×C4—44no—
368C3×C4—64no—
369C3×C4—181no—
370C3×C4D3×#6714yes—
371C3×C4D3×#6714yes—
372C3×C4D4×C3⋊D311yescharacteristic
373C3×C4C2×#63811yescharacteristic
374Dic3—21no—
375Dic3—64no—
376Dic3—368no—
377Dic3—361no—
378Dic3—364no—
379C32—24no—
380C32#520214yescomplemented
381C32#520211yescharacteristic, complemented
382C23—272no—
383C23—542no—
384Q8—21no—
385Q8—21no—
386Q8—31no—
387Q8—91no—
388D4—31no—
389D4—61no—
390D4—91no—
391D4—181no—
392D4—542no—
393D4—542no—
394D4—541no—
395D4—541no—
396C2×C4—21no—
397C2×C4—61no—
398C2×C4—181no—
399C2×C4—542no—
400C2×C4—541no—
401C2×C4C2×D3×C3⋊D311yescharacteristic
402C8—61no—
403C8—181no—
404C2×C3—28no—
405C2×C3—21no—
406C2×C3—24no—
407C2×C3—24no—
408C2×C3—124no—
409C2×C3—361no—
410C2×C3#6398314yes—
411C2×C3#6414611yescharacteristic
412D3—41no—
413D3—124no—
414D3—362no—
415D3—368no—
416D3—728no—
417C22—61no—
418C22—181no—
419C22—542no—
420C22—1082no—
421C22—2162no—
422C22#8903911yescharacteristic
423C4—41no—
424C4—61no—
425C4—181no—
426C4—1081no—
427C4D3×#63811yescharacteristic
428C4C3⋊D3×#6711yescharacteristic
429C3—24no—
430C3#1059165814yescomplemented
431C3#1059290111yescharacteristic, complemented
432C2—21no—
433C2—121no—
434C2—361no—
435C2—1082no—
436C2#1172364311yescharacteristic
subgroupG/H
centerC2#11723643
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#120
prime radical#11724581C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11723988C2semidirect
3. C3×#64151C2semidirect
4. C32×#942C2semidirect
5. #11722045C2semidirect
6. #11722043C2semidirect
7. #11722027C2semidirect
10. C2×C33×Q8C22semidirect
20. C4×C3⋊(C3⋊Dic3)C22semidirect
23. C3×#3748C22semidirect
26. C32×#208C22semidirect
33. #64151D3semidirect
35. C3×#942D3semidirect
68. C2×C32×Q8C2×D3semidirect
69. C2×C32×Q8C2×D3semidirect
97. #3748C2×D3semidirect
103. C3×#208C2×D3semidirect
124. #942C3⋊D3semidirect
192. C2×C3×Q8D32semidirect
193. C2×C3×Q8C2×C3⋊D3semidirect
226. #208C2×C3⋊D3semidirect
265. C33#455semidirect
335. C2×Q8D3×C3⋊D3semidirect
380. C32#5202semidirect
381. C32#5202semidirect
430. C3#10591658semidirect
431. C3#10592901semidirect

Maximal subgroups

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

subgroupindex
#117245812
#117239882
C3×#641512
C32×#9422
#117220452
#117220432
#117220272
#105929013
#105916583