Finite Group Databasebeta

← #13140928

Subgroups — #13140928

Order 1728.

402 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 63 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#64151⋊C3C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3#11721061C211yesmaximal, characteristic
4#11721054C211yesmaximal, characteristic, complemented
5#11720584C211yesmaximal, characteristic, complemented
6#11720320C211yesmaximal, characteristic, complemented
7#11720319C211yesmaximal, characteristic, complemented
8#10591643—31nomaximal
9#10591542—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13#88538C2211yescharacteristic
14#88538C2211yescharacteristic
15#3755⋊C3—21no—
16#3755⋊C3—21no—
17#3755⋊C3—21no—
18#3756⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88432—21no—
22#88431—21no—
23#88430—21no—
24((C3⋊D3)⋊C3)⋊C8—21no—
25#64151D311yescharacteristic, complemented
26C2×#3755D311yescharacteristic, complemented
27C3×#942—31no—
28C2×C3×#215—31no—
29#63830—31no—
30#63825—31no—
31#63817—31no—
32#63815—31no—
33C3×#836—31no—
34C3×#836—31no—
35C3×#836—61no—
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#638⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
49C4×(C3⋊D3)⋊C3—21no—
50#634⋊C3D411yescharacteristic
51#6180—21no—
52#6180—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
62#3828C2×D311yescharacteristic
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C3×#235—31no—
65C2×C3×D12—31no—
66#3756—21no—
67#3755—21no—
68#3755—21no—
69#3755—21no—
70#3748C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#216—61no—
73C3×#215—61no—
74C3×#215—61no—
75C3×#215—61no—
76C3×#208—31no—
77C3×#208—31no—
78C3×#208—31no—
79C3×#208—31no—
80C3×#208—31no—
81C3×#208—61no—
82C3×#208—61no—
83C2×C3×C3⋊C8—31no—
84#3717—61no—
85#3716—61no—
86#3714—61no—
87#3714—61no—
88#3712—61no—
89#3711—61no—
90#3710—61no—
91D3×C3⋊C8—61no—
92C22×H(3,𝔽3)#4111yescharacteristic
93C2×(C3⋊D3)⋊C3—21no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—81no—
96C4×H(3,𝔽3)—41no—
97C4×H(3,𝔽3)C2×D411yescharacteristic
98C4×H(3,𝔽3)C2×D411yescharacteristic
99(C3⋊Dic3)⋊C3—21no—
100C3×#137—91no—
101C2×C3×SD16—91no—
102#949—91no—
103#948—91no—
104#942—31no—
105#942—91no—
106C2×#215—31no—
107C2×#215—91no—
108#932—91no—
109#908—91no—
110#907—91no—
111#901—91no—
112C3×#101—91no—
113#836—31no—
114#836—31no—
115#836—61no—
116#833—91no—
117#823—91no—
118#823—91no—
119#822—91no—
120#822—91no—
121#819—91no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—41no—
129C32×Q8—41no—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136#638—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)#6711yescharacteristic
140C4×C3⋊D3—21no—
141#634#6711yescharacteristic
142C3×#67—61no—
143C3×D12—31no—
144C3×D12—61no—
145C3×D12—61no—
146C3×C4×D3—61no—
147C3×Dic6—31no—
148C3⋊(C3⋊C8)—21no—
149C3⋊(C3⋊C8)#6711yescharacteristic
150C3⋊(C3⋊C8)#6711yescharacteristic
151C3×C3⋊C8—31no—
152C3×C3⋊C8—31no—
153C3×C3⋊C8—61no—
154C3×C3⋊C8—61no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—121no—
159C3×C3⋊C8—121no—
160#471—271noSylow
161C2×H(3,𝔽3)—21no—
162C2×H(3,𝔽3)#12111yescharacteristic
163(C3⋊D3)⋊C3—41no—
164(C3⋊D3)⋊C3—81no—
165C3×#41—91no—
166C2×C3×Q8—31no—
167C2×C3×Q8—31no—
168C2×C3×Q8—61no—
169C2×C3×Q8D3211yescharacteristic
170C2×C3×D4—91no—
171#235—31no—
172#235—91no—
173C2×D12—31no—
174C2×D12—91no—
175C3×Q16—181no—
176C3×SD16—181no—
177C3×SD16—181no—
178C3×SD16—181no—
179C3×M16—91no—
180C3×M16—91no—
181C3×M16—91no—
182C2×C3×C8—91no—
183#216—61no—
184#216—181no—
185#215—61no—
186#215—61no—
187#215—61no—
188#215—181no—
189#215—181no—
190#215—181no—
191#215—181no—
192#214—181no—
193#213—181no—
194#213—181no—
195#208—31no—
196#208—31no—
197#208—31no—
198#208—31no—
199#208—31no—
200#208—61no—
201#208—61no—
202#208—91no—
203#208—181no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—181no—
207D24—181no—
208#204—181no—
209ΓL(1,𝔽25)—181no—
210ΓL(1,𝔽25)—181no—
211D3×C8—181no—
212D3×C8—181no—
213C22×C32—21no—
214C22×C32#23711yescharacteristic
215C22×C32#23711yescharacteristic
216C2×C3⋊D3—21no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—81no—
219C2×C3×D3—61no—
220C2×C3×D3—121no—
221C2×C3×D3—241no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—42no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4D3×D411yescharacteristic
228C32×C4D3×D411yescharacteristic
229C32×C4C2×#6711yescharacteristic
230C32×C4C2×#6711yescharacteristic
231C3⋊Dic3—21no—
232C3×Dic3—61no—
233#137—91no—
234AGL(1,ℤ/8)—271no—
235C2×SD16—91no—
236#131—271no—
237#108—271no—
238#101—91no—
239#100—271no—
240H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
241C23×C3—181no—
242C22×D3—61no—
243C22×D3—181no—
244C3×Q8—21no—
245C3×Q8—21no—
246C3×Q8—61no—
247C3×Q8—61no—
248C3×Q8—61no—
249C3×Q8—61no—
250C3×Q8—91no—
251C3×Q8—121no—
252C3×Q8—121no—
253C3×D4—91no—
254C3×D4—181no—
255C3×D4—181no—
256C3×D4—181no—
257C2×C3×C4—21no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—62no—
263C2×C3×C4—61no—
264C2×C3×C4—181no—
265C2×C3×C4C2×D3211yescharacteristic
266#67—61no—
267#67—181no—
268D12—31no—
269D12—61no—
270D12—61no—
271D12—91no—
272D12—181no—
273D12—181no—
274C4×D3—61no—
275C4×D3—181no—
276Dic6—31no—
277Dic6—91no—
278C3×C8—91no—
279C3×C8—91no—
280C3×C8—181no—
281C3×C8—181no—
282C3×C8—181no—
283C3⋊C8—31no—
284C3⋊C8—31no—
285C3⋊C8—61no—
286C3⋊C8—61no—
287C3⋊C8—61no—
288C3⋊C8—61no—
289C3⋊C8—61no—
290C3⋊C8—91no—
291C3⋊C8—91no—
292C3⋊C8—121no—
293C3⋊C8—121no—
294C3⋊C8—181no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—41no—
299C2×C32#93811yescharacteristic
300C2×C32#93911yescharacteristic
301C3⋊D3—41no—
302C3⋊D3—81no—
303C3×D3—121no—
304C3×D3—241no—
305#41—91no—
306C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
307C2×D4—91no—
308Q16—181no—
309SD16—181no—
310SD16—181no—
311SD16—181no—
312SD16—541no—
313D8—541no—
314M16—91no—
315M16—91no—
316M16—91no—
317M16—541no—
318C2×C8—91no—
319C2×C8—541no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—61no—
323C22×C3—181no—
324C22×C3—361no—
325C22×C3—721no—
326C22×C3#380511yescharacteristic
327C2×D3—61no—
328C2×D3—121no—
329C2×D3—181no—
330C2×D3—241no—
331C2×D3—361no—
332C2×D3—721no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—41no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—121no—
341C3×C4—121no—
342C3×C4—122no—
343C3×C4—181no—
344C3×C4D3×#6711yescharacteristic
345C3×C4D3×#6711yescharacteristic
346Dic3—61no—
347Dic3—181no—
348C32—21no—
349C32#517711yescharacteristic, complemented
350C32#520311yescharacteristic, complemented
351C23—181no—
352Q8—21no—
353Q8—21no—
354Q8—91no—
355D4—91no—
356D4—181no—
357D4—181no—
358D4—181no—
359C2×C4—21no—
360C2×C4—181no—
361C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C8—91no—
363C8—91no—
364C8—181no—
365C8—181no—
366C8—181no—
367C2×C3—21no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—121no—
374C2×C3—361no—
375C2×C3—721no—
376C2×C3#6397011yescharacteristic
377D3—121no—
378D3—241no—
379D3—361no—
380D3—721no—
381C22—181no—
382C22—361no—
383C22—721no—
384C22#8867211yescharacteristic
385C4—41no—
386C4—181no—
387C4#8867611yescharacteristic
388C4#8867611yescharacteristic
389C3—31no—
390C3—31no—
391C3—61no—
392C3#1059164211yescharacteristic
393C2—21no—
394C2—361no—
395C2—721no—
396C2#1172131311yescharacteristic
subgroupG/H
centerC2#11721313
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC2×C3#63970
prime radical#11721061C2
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. #64151⋊C3C2semidirect
2. C2×#3755⋊C3C2semidirect
4. #11721054C2semidirect
5. #11720584C2semidirect
6. #11720320C2semidirect
7. #11720319C2semidirect
25. #64151D3semidirect
26. C2×#3755D3semidirect
240. H(3,𝔽3)#471semidirect
349. C32#5177semidirect
350. C32#5203semidirect

Maximal subgroups

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

subgroupindex
#64151⋊C32
C2×#3755⋊C32
#117210612
#117210542
#117205842
#117203202
#117203192
#105916433
#105915423