Finite Group Databasebeta

← #13140990

Subgroups — #13140990

Order 1728.

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

normal subgroups only

This group has 424 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
1#11721859C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3#11721071C211yesmaximal, characteristic
4#11721064C211yesmaximal, characteristic, complemented
5#63663⋊C3C211yesmaximal, characteristic, complemented
6#11720320C211yesmaximal, characteristic, complemented
7#11720319C211yesmaximal, characteristic, complemented
8#10591643—31nomaximal
9#10591642—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88740C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13#88550—21no—
14#88549—21no—
15#88538C2211yescharacteristic
16#3755⋊C3—21no—
17#3755⋊C3—21no—
18#3748⋊C3C2211yescharacteristic
19#3748⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88438—21no—
22#88436—21no—
23#88435—21no—
24#88434—21no—
25C2×#3755D311yescharacteristic, complemented
26C3×#942—31no—
27C3×#942—31no—
28C3×#942—61no—
29C2×C3×#215—31no—
30#63825—31no—
31#63823—31no—
32#63815—31no—
33#63814—31no—
34#63663D311yescharacteristic, complemented
35C3×#836—31no—
36#63578—31no—
37#63577—31no—
38#63559—31no—
39#63558—31no—
40C22×(C3⋊D3)⋊C3—21no—
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#6235—21no—
46C4⋊#272D411yescharacteristic
47C4×#272—21no—
48#6229D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
56#5203—91no—
57#5177—91no—
58#4754—91no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C2×D311yescharacteristic
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C3×#235—31no—
64C3×#235—31no—
65C3×#235—61no—
66C2×C3×D12—31no—
67#3755—21no—
68#3755—21no—
69#3748C2×D311yescharacteristic
70#3748C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#216—61no—
73C3×#216—61no—
74C3×#216—62no—
75C3×#215—61no—
76C3×#215—61no—
77C3×#215—61no—
78C3×#215—61no—
79C3×#215—62no—
80C3×#208—31no—
81C3×#208—31no—
82C3×#208—31no—
83C3×#208—31no—
84C3×#208—61no—
85C2×C3×C3⋊C8—31no—
86#3717—61no—
87#3715—61no—
88#3714—61no—
89#3713—61no—
90#3711—61no—
91#3711—61no—
92D3×C3⋊C8—61no—
93D3×C3⋊C8—61no—
94C22×H(3,𝔽3)#4111yescharacteristic
95C2×#272—21no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—81no—
98C4×H(3,𝔽3)—41no—
99C4×H(3,𝔽3)C2×D411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101#1084—21no—
102C3×#137—91no—
103C2×C3×SD16—91no—
104#949—91no—
105#948—91no—
106#942—31no—
107#942—31no—
108#942—61no—
109C2×#215—31no—
110C2×#215—91no—
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—21no—
125C22×C3×D3—61no—
126C32×Q8—22no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—41no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—41no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C3×#67—61no—
141C3×#67—61no—
142C3×#67—121no—
143C3×D12—31no—
144C3×D12—31no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×C4×D3—61no—
149C3×C4×D3—61no—
150C3×C4×D3—121no—
151C3×Dic6—31no—
152C3×Dic6—31no—
153C3×Dic6—61no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)#6711yescharacteristic
157C3⋊(C3⋊C8)#6711yescharacteristic
158C3×C3⋊C8—31no—
159C3×C3⋊C8—31no—
160C3×C3⋊C8—62no—
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—41no—
169(C3⋊D3)⋊C3—81no—
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—31no—
178#235—61no—
179C2×D12—31no—
180C2×D12—91no—
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—61no—
191#216—62no—
192#215—61no—
193#215—61no—
194#215—61no—
195#215—61no—
196#215—62no—
197#215—181no—
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—31no—
207#208—61no—
208#208—91no—
209#208—91no—
210#208—181no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—91no—
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—41no—
224C2×C3⋊D3—81no—
225C2×C3×D3—61no—
226C2×C3×D3—61no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—241no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—42no—
233C32×C4—41no—
234C32×C4—41no—
235C32×C4D3×D411yescharacteristic
236C32×C4D3×D411yescharacteristic
237C32×C4D3×D411yescharacteristic
238C32×C4C2×#6711yescharacteristic
239C3×Dic3—61no—
240C3×Dic3—61no—
241C3×Dic3—121no—
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—181no—
253C3×Q8—21no—
254C3×Q8—21no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—61no—
258C3×Q8—61no—
259C3×Q8—62no—
260C3×Q8—91no—
261C3×Q8—121no—
262C3×D4—91no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—181no—
266C2×C3×C4—21no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—121no—
273C2×C3×C4—181no—
274C2×C3×C4C2×D3211yescharacteristic
275#67—61no—
276#67—61no—
277#67—121no—
278D12—31no—
279D12—31no—
280D12—61no—
281D12—61no—
282D12—61no—
283D12—181no—
284D12—181no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—121no—
288Dic6—31no—
289Dic6—31no—
290Dic6—61no—
291C3×C8—91no—
292C3×C8—91no—
293C3×C8—181no—
294C3×C8—181no—
295C3×C8—181no—
296C3⋊C8—31no—
297C3⋊C8—31no—
298C3⋊C8—61no—
299C3⋊C8—61no—
300C3⋊C8—62no—
301C3⋊C8—61no—
302C3⋊C8—61no—
303C3⋊C8—91no—
304C3⋊C8—91no—
305C3⋊C8—181no—
306C3⋊C8—181no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—41no—
311C2×C32#88411yescharacteristic
312C2×C32#93811yescharacteristic
313C3⋊D3—81no—
314C3×D3—121no—
315C3×D3—121no—
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—61no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—241no—
345C2×D3—361no—
346C2×D3—721no—
347C3×C4—31no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—41no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—121no—
355C3×C4—121no—
356C3×C4—122no—
357C3×C4—181no—
358C3×C4D3×#6711yescharacteristic
359C3×C4#381111yescharacteristic
360Dic3—61no—
361Dic3—61no—
362Dic3—121no—
363C32—21no—
364C32#475411yescharacteristic, complemented
365C32#517711yescharacteristic, 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—61no—
388C2×C3—121no—
389C2×C3—361no—
390C2×C3—721no—
391C2×C3#6395811yescharacteristic
392D3—121no—
393D3—121no—
394D3—241no—
395D3—241no—
396D3—721no—
397C22—181no—
398C22—361no—
399C22—721no—
400C22#8867111yescharacteristic
401C4—41no—
402C4—181no—
403C4#8867611yescharacteristic
404C4#8867711yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059154211yescharacteristic
409C2—21no—
410C2—361no—
411C2—721no—
412C2#1172130311yescharacteristic
subgroupG/H
centerC2#11721303
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC2×C3#63958
prime radical#11721071C2
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. #11721859C2semidirect
2. C2×#3755⋊C3C2semidirect
4. #11721064C2semidirect
5. #63663⋊C3C2semidirect
6. #11720320C2semidirect
7. #11720319C2semidirect
25. C2×#3755D3semidirect
34. #63663D3semidirect
249. H(3,𝔽3)#471semidirect
364. C32#4754semidirect
365. C32#5177semidirect

Maximal subgroups

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

subgroupindex
#117218592
C2×#3755⋊C32
#117210712
#117210642
#63663⋊C32
#117203202
#117203192
#105916433
#105916423