Finite Group Databasebeta

← #13140952

Subgroups — #13140952

Order 1728.

432 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C2×#88549C211yesmaximal, characteristic, complemented
2#64151⋊C3C211yesmaximal, characteristic, complemented
3#11721068C211yesmaximal, characteristic
4#11721055C211yesmaximal, characteristic, complemented
5#63663⋊C3C211yesmaximal, characteristic, complemented
6#11720325C211yesmaximal, characteristic, complemented
7#11720316C211yesmaximal, characteristic, complemented
8#10591642—31nomaximal
9#10591542—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12#3828⋊C3C2211yescharacteristic
13#88549—21no—
14#88549—21no—
15C2×#6180C2211yescharacteristic
16#3755⋊C3—21no—
17#3756⋊C3—21no—
18#3748⋊C3C2211yescharacteristic
19#3748⋊C3C2211yescharacteristic
20#3748⋊C3C2211yescharacteristic
21#88437—21no—
22#88436—21no—
23#88430—21no—
24((C3⋊D3)⋊C3)⋊C8—21no—
25#64151D311yescharacteristic, complemented
26C3×#942—31no—
27C2×C3×#215—31no—
28C2×C3×#215—31no—
29C2×C3×#215—61no—
30#63830—31no—
31#63823—31no—
32#63817—31no—
33#63814—31no—
34#63663D311yescharacteristic, complemented
35C3×#836—31no—
36#63577—31no—
37#63577—31no—
38#63558—31no—
39#63558—31no—
40C22×#272—21no—
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C4⋊#272—21no—
46C4⋊#272—21no—
47#638⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
49C4×(C3⋊D3)⋊C3—21no—
50#634⋊C3D411yescharacteristic
51#6180D411yescharacteristic
52#6180D411yescharacteristic
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5203—91no—
57#5177—91no—
58#4754—91no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C2×D311yescharacteristic
62#3828C2×D311yescharacteristic
63C3×#235—31no—
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67#3756—21no—
68#3755—21no—
69#3748C2×D311yescharacteristic
70#3748C2×D311yescharacteristic
71#3748C2×D311yescharacteristic
72C3×#216—61no—
73C3×#215—61no—
74C3×#215—61no—
75C3×#215—61no—
76C3×#215—61no—
77C3×#215—61no—
78C3×#215—62no—
79C3×#215—62no—
80C3×#208—31no—
81C3×#208—31no—
82C3×#208—31no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—61no—
86#3716—61no—
87#3715—61no—
88#3714—61no—
89#3713—61no—
90#3712—61no—
91#3711—61no—
92#3710—61no—
93D3×C3⋊C8—61no—
94C22×H(3,𝔽3)#4111yescharacteristic
95C2×#272—41no—
96C2×#272—81no—
97C2×(C3⋊D3)⋊C3—21no—
98C4×H(3,𝔽3)—41no—
99C4×H(3,𝔽3)C2×D411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101(C3⋊Dic3)⋊C3—21no—
102C3×#137—91no—
103C2×C3×SD16—91no—
104#949—91no—
105#948—91no—
106#942—31no—
107#942—91no—
108C2×#215—31no—
109C2×#215—31no—
110C2×#215—61no—
111#932—91no—
112#908—91no—
113#907—91no—
114#901—91no—
115C3×#101—91no—
116#836—31no—
117#836—91no—
118#833—91no—
119#823—91no—
120#823—91no—
121#822—91no—
122#822—91no—
123#819—91no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—121no—
127C32×Q8—22no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—41no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139#638—21no—
140C4⋊(C3⋊D3)#6711yescharacteristic
141C4×C3⋊D3—21no—
142#634#6711yescharacteristic
143C3×#67—61no—
144C3×D12—31no—
145C3×D12—62no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—121no—
151C3×C4×D3—61no—
152C3×Dic6—31no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—31no—
157C3×C3⋊C8—31no—
158C3×C3⋊C8—31no—
159C3×C3⋊C8—31no—
160C3×C3⋊C8—61no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165#471—271noSylow
166C2×H(3,𝔽3)—21no—
167C2×H(3,𝔽3)#12111yescharacteristic
168#272—81no—
169(C3⋊D3)⋊C3—41no—
170C3×#41—91no—
171C2×C3×Q8—31no—
172C2×C3×Q8—31no—
173C2×C3×Q8—61no—
174C2×C3×Q8D3211yescharacteristic
175C2×C3×D4—91no—
176#235—31no—
177#235—91no—
178C2×D12—31no—
179C2×D12—31no—
180C2×D12—61no—
181C3×Q16—181no—
182C3×SD16—181no—
183C3×SD16—181no—
184C3×SD16—181no—
185C3×M16—91no—
186C3×M16—91no—
187C3×M16—91no—
188C2×C3×C8—91no—
189#216—61no—
190#216—181no—
191#215—61no—
192#215—61no—
193#215—61no—
194#215—61no—
195#215—61no—
196#215—62no—
197#215—62no—
198#215—181no—
199#215—181no—
200#214—181no—
201#213—181no—
202#213—181no—
203#208—31no—
204#208—31no—
205#208—31no—
206#208—91no—
207#208—91no—
208#208—91no—
209#208—181no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—61no—
213C2×C3⋊C8—181no—
214D24—181no—
215#204—181no—
216ΓL(1,𝔽25)—181no—
217ΓL(1,𝔽25)—181no—
218D3×C8—181no—
219D3×C8—181no—
220C22×C32—21no—
221C22×C32#23511yescharacteristic
222C22×C32#23711yescharacteristic
223C2×C3⋊D3—21no—
224C2×C3×D3—61no—
225C2×C3×D3—122no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—242no—
229C2×C3×D3—241no—
230C2×C3×D3—241no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—42no—
234C32×C4—41no—
235C32×C4—41no—
236C32×C4D3×D411yescharacteristic
237C32×C4D3×D411yescharacteristic
238C32×C4D3×D411yescharacteristic
239C32×C4C2×#6711yescharacteristic
240C3⋊Dic3—21no—
241C3×Dic3—61no—
242#137—91no—
243AGL(1,ℤ/8)—271no—
244C2×SD16—91no—
245#131—271no—
246#108—271no—
247#101—91no—
248#100—271no—
249H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
250C23×C3—181no—
251C22×D3—61no—
252C22×D3—61no—
253C22×D3—121no—
254C3×Q8—21no—
255C3×Q8—21no—
256C3×Q8—61no—
257C3×Q8—61no—
258C3×Q8—61no—
259C3×Q8—61no—
260C3×Q8—62no—
261C3×Q8—91no—
262C3×Q8—121no—
263C3×D4—91no—
264C3×D4—181no—
265C3×D4—181no—
266C3×D4—181no—
267C2×C3×C4—21no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—121no—
274C2×C3×C4—181no—
275C2×C3×C4C2×D3211yescharacteristic
276#67—61no—
277#67—181no—
278D12—31no—
279D12—61no—
280D12—61no—
281D12—61no—
282D12—61no—
283D12—62no—
284D12—91no—
285D12—121no—
286C4×D3—61no—
287C4×D3—181no—
288Dic6—31no—
289Dic6—91no—
290C3×C8—91no—
291C3×C8—91no—
292C3×C8—181no—
293C3×C8—181no—
294C3×C8—181no—
295C3⋊C8—31no—
296C3⋊C8—31no—
297C3⋊C8—31no—
298C3⋊C8—31no—
299C3⋊C8—61no—
300C3⋊C8—61no—
301C3⋊C8—61no—
302C3⋊C8—61no—
303C3⋊C8—61no—
304C3⋊C8—181no—
305C3⋊C8—181no—
306C3⋊C8—181no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32#88411yescharacteristic
312C2×C32#93911yescharacteristic
313C3⋊D3—41no—
314C3×D3—121no—
315C3×D3—242no—
316C3×D3—241no—
317C3×D3—241no—
318#41—91no—
319C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C2×D4—91no—
321Q16—181no—
322SD16—181no—
323SD16—181no—
324SD16—181no—
325SD16—541no—
326D8—541no—
327M16—91no—
328M16—91no—
329M16—91no—
330M16—541no—
331C2×C8—91no—
332C2×C8—541no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—61no—
336C22×C3—181no—
337C22×C3—361no—
338C22×C3—721no—
339C22×C3#380411yescharacteristic
340C2×D3—61no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—122no—
344C2×D3—181no—
345C2×D3—242no—
346C2×D3—241no—
347C2×D3—241no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—41no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—122no—
358C3×C4—181no—
359C3×C4D3×#6711yescharacteristic
360C3×C4#381111yescharacteristic
361Dic3—61no—
362Dic3—181no—
363C32—21no—
364C32#475411yescharacteristic, complemented
365C32#520311yescharacteristic, complemented
366C23—181no—
367Q8—21no—
368Q8—21no—
369Q8—91no—
370D4—91no—
371D4—181no—
372D4—181no—
373D4—181no—
374C2×C4—21no—
375C2×C4—181no—
376C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C8—91no—
378C8—91no—
379C8—181no—
380C8—181no—
381C8—181no—
382C2×C3—21no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—61no—
386C2×C3—61no—
387C2×C3—62no—
388C2×C3—61no—
389C2×C3—361no—
390C2×C3—721no—
391C2×C3#6397111yescharacteristic
392D3—121no—
393D3—241no—
394D3—241no—
395D3—242no—
396D3—361no—
397C22—181no—
398C22—361no—
399C22—721no—
400C22#8867111yescharacteristic
401C4—41no—
402C4—181no—
403C4#8867611yescharacteristic
404C4#8867711yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059164311yescharacteristic
409C2—21no—
410C2—361no—
411C2—721no—
412C2#1172131211yescharacteristic
subgroupG/H
centerC2#11721312
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC2×C3#63971
prime radical#11721068C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#471
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88549C2semidirect
2. #64151⋊C3C2semidirect
4. #11721055C2semidirect
5. #63663⋊C3C2semidirect
6. #11720325C2semidirect
7. #11720316C2semidirect
25. #64151D3semidirect
34. #63663D3semidirect
249. H(3,𝔽3)#471semidirect
364. C32#4754semidirect
365. C32#5203semidirect

Maximal subgroups

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

subgroupindex
C2×#885492
#64151⋊C32
#117210682
#117210552
#63663⋊C32
#117203252
#117203162
#105916423
#105915423