Finite Group Databasebeta

← #13161068

Subgroups — #13161068

Order 1728.

1048 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723977C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic
3#11723501C211yesmaximal, characteristic, complemented
4C2×#88796C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7#11722041C211yesmaximal, characteristic, complemented
8#10592486—31nomaximal
9#10591434—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14#88864—21no—
15#88864—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3753—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88803—21no—
22#88796—22no—
23#64145—31no—
24#64145—34no—
25#64145—64no—
26C2×#3753D311yescharacteristic
27C2×#3744—31no—
28#64102—31no—
29C2×C3×#213—34no—
30#63898—34no—
31C2×#3716—34no—
32C32×#102—31no—
33#63641—31no—
34#63637—31no—
35C3×#832D314yescomplemented
36#63572—34no—
37#63567—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46#6308—21no—
47#6308—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)D412yes—
50C2×C32×Dic3—21no—
51Dic3×C3⋊D3—42no—
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8—21no—
54#5153—91no—
55#4787—94no—
56C2×C32×D4—24no—
57C2×C32×D4C2×D314yes—
58C2×C32×D4C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4×C3⋊D3—61no—
61C2×#634—31no—
62C2×#634—34no—
63C2×#634—64no—
64C2×C3×D12—34no—
65C2×D3×Dic3—64no—
66C2×C32×C8—31no—
67C32×C4⋊C4—31no—
68#3757—61no—
69#3757—61no—
70#3757—64no—
71#3757—64no—
72#3757—68no—
73#3757—68no—
74#3753—22no—
75#3752—61no—
76C4×C3⋊Dic3—31no—
77C4×C3⋊Dic3—34no—
78C4×C3⋊Dic3—64no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80#3744—62no—
81C3×#213—68no—
82C3×C4⋊Dic3C2×D314yes—
83C2×C3×C3⋊C8—34no—
84C2×C3×C3⋊C8C2×D314yes—
85(C2×D3)⋊Dic3—64no—
86#3716—68no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3⋊(C3⋊Dic3)—21no—
97C3⋊(C3⋊Dic3)—21no—
98C3⋊(C3⋊Dic3)—41no—
99C32×Dic3—41no—
100C2×C3×D8—91no—
101#936—94no—
102#936—91no—
103#936—184no—
104D4×Dic3—91no—
105C2×#214—91no—
106C2×#213—34no—
107C2×#204—94no—
108#893—94no—
109C3×#102—34no—
110#832C3⋊D311yescharacteristic, complemented
111#816—91no—
112Dic3⋊C8—94no—
113Dic3⋊C8—91no—
114#810—94no—
115C23×C32—28no—
116C23×C32—24no—
117C23×C32—21no—
118C22×C3⋊D3—21no—
119C22×C3×D3—64no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—24no—
123C32×D4—24no—
124C32×D4—44no—
125C32×D4—44no—
126C2×C32×C4—24no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—31no—
131C2×C3⋊Dic3—31no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—34no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—68no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—61no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)#6712yes—
141C4×C3⋊D3—122no—
142#634—61no—
143#634—61no—
144#634—64no—
145#634—64no—
146#634—124no—
147#634—124no—
148C2×C3×Dic3—24no—
149C3×D12—38no—
150C3×D12—64no—
151D3×Dic3—128no—
152C32×C8—61no—
153C3⋊(C3⋊C8)—21no—
154C3×C3⋊C8—24no—
155C3×C3⋊C8—64no—
156#463—271noSylow
157C2×C33—81no—
158C2×C33#12111yescharacteristic
159C2×C33#13511yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C3×C3⋊D3—42no—
162C2×C3×D4—24no—
163C2×C3×D4—91no—
164C2×C3×D4D3214yes—
165C2×C3×D4C2×C3⋊D311yescharacteristic
166C22×Dic3—181no—
167C2×D12—34no—
168C2×C4×D3—184no—
169C2×Dic6—94no—
170C2×Dic6—91no—
171C2×Dic6—184no—
172C3×D8—182no—
173C2×C3×C8—34no—
174C2×C3×C8—91no—
175C3×C4⋊C4—34no—
176#217—181no—
177#217—184no—
178#217—184no—
179#217—181no—
180#217—181no—
181#217—188no—
182#217—188no—
183#214—182no—
184#213—68no—
185(C2×D3)⋊C4—184no—
186C4⋊Dic3C2×C3⋊D311yescharacteristic
187C4×Dic3—94no—
188C4×Dic3—91no—
189C4×Dic3—184no—
190C2×C3⋊C8—34no—
191C2×C3⋊C8C2×C3⋊D311yescharacteristic
192#204—188no—
193C22×C32—24no—
194C22×C32—48no—
195C22×C32—48no—
196C22×C32—48no—
197C22×C32—44no—
198C22×C32—44no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—44no—
202C22×C32—41no—
203C22×C32D3×D414yes—
204C22×C32D3×D411yescharacteristic
205C2×C3⋊D3—22no—
206C2×C3⋊D3—41no—
207C2×C3⋊D3—41no—
208C2×C3×D3—68no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—121no—
214C32×C4#23514yes—
215C32×C4D3×D414yes—
216C32×C4#23711yescharacteristic
217C32×C4C2×#6711yescharacteristic
218C3⋊Dic3—61no—
219C3⋊Dic3—61no—
220C3⋊Dic3—64no—
221C3⋊Dic3—64no—
222C3⋊Dic3—124no—
223C3⋊Dic3—124no—
224C3⋊Dic3—128no—
225C3⋊Dic3—124no—
226C3⋊Dic3—121no—
227C3×Dic3—44no—
228C2×SD16—271no—
229C2×D8—91no—
230#124—271no—
231C4×D4—271no—
232C4⋊C8—271no—
233#103—271no—
234#102—31no—
235C33#46311yescharacteristic, complemented, Sylow
236C23×C3—28no—
237C23×C3—24no—
238C23×C3—21no—
239C23×C3—181no—
240C22×D3—64no—
241C3×D4—24no—
242C3×D4—24no—
243C3×D4—21no—
244C3×D4—21no—
245C3×D4—44no—
246C3×D4—44no—
247C3×D4—92no—
248C3×D4—181no—
249C2×C3×C4—24no—
250C2×C3×C4—64no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253C2×Dic3—21no—
254C2×Dic3—94no—
255C2×Dic3—94no—
256C2×Dic3—91no—
257C2×Dic3—91no—
258C2×Dic3—184no—
259C2×Dic3—184no—
260C2×Dic3—188no—
261C2×Dic3—184no—
262C2×Dic3—181no—
263C2×Dic3—362no—
264D12—38no—
265D12—64no—
266C4×D3—368no—
267Dic6—184no—
268Dic6—184no—
269Dic6—181no—
270Dic6—181no—
271Dic6—364no—
272Dic6—364no—
273C3×C8—64no—
274C3×C8—181no—
275C3⋊C8—21no—
276C3⋊C8—64no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—88no—
281C2×C32—81no—
282C2×C32—84no—
283C2×C32#88414yes—
284C2×C32#91114yes—
285C2×C32#91111yescharacteristic
286C2×C32#91211yescharacteristic
287C2×C32#91614yes—
288C2×C32#93911yescharacteristic
289C3⋊D3—42no—
290C3×D3—128no—
291C2×Q8—271no—
292C2×D4—91no—
293C2×D4D3×C3⋊D311yescharacteristic
294C22×C4—541no—
295SD16—542no—
296D8—182no—
297C2×C8—31no—
298C2×C8—91no—
299C4⋊C4—31no—
300C22⋊C4—541no—
301C22⋊C4—541no—
302C22⋊C4—541no—
303C42—271no—
304C22×C3—24no—
305C22×C3—48no—
306C22×C3—48no—
307C22×C3—48no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—44no—
314C22×C3—182no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3#381114yes—
318C22×C3D4×C3⋊D311yescharacteristic
319C2×D3—68no—
320C2×D3—124no—
321C2×D3—124no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—124no—
325C3×C4#380414yes—
326C3×C4D3×#6714yes—
327C3×C4#382811yescharacteristic
328C3×C4D4×C3⋊D311yescharacteristic
329Dic3—41no—
330Dic3—181no—
331Dic3—181no—
332Dic3—184no—
333Dic3—184no—
334Dic3—364no—
335Dic3—364no—
336Dic3—368no—
337Dic3—361no—
338Dic3—364no—
339C32—24no—
340C32#478714yescomplemented
341C32#515311yescharacteristic, complemented
342C23—21no—
343C23—181no—
344Q8—541no—
345Q8—541no—
346D4—21no—
347D4—21no—
348D4—92no—
349D4—181no—
350C2×C4—61no—
351C2×C4—271no—
352C2×C4—271no—
353C2×C4—541no—
354C2×C4—1082no—
355C2×C4C2×D3×C3⋊D311yescharacteristic
356C8—61no—
357C8—181no—
358C2×C3—24no—
359C2×C3—24no—
360C2×C3—24no—
361C2×C3—88no—
362C2×C3—84no—
363C2×C3—81no—
364C2×C3—362no—
365C2×C3#6392714yes—
366C2×C3#6393614yes—
367C2×C3#6397114yes—
368C2×C3#6409311yescharacteristic
369C2×C3#6412011yescharacteristic
370C2×C3#6412511yescharacteristic
371D3—128no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—182no—
376C22—361no—
377C22—361no—
378C22#8903911yescharacteristic
379C4—121no—
380C4—541no—
381C4—541no—
382C4—1081no—
383C4#8903111yescharacteristic
384C4C3⋊D3×#6711yescharacteristic
385C3—24no—
386C3#1059143414yescomplemented
387C3#1059248611yescharacteristic, complemented
388C2—81no—
389C2—362no—
390C2#1172354811yescharacteristic
391C2#1172356511yescharacteristic
392C2#1172362411yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723977C2semidirect
3. #11723501C2semidirect
4. C2×#88796C2semidirect
5. C32×#832C2semidirect
6. #11722050C2semidirect
7. #11722041C2semidirect
35. C3×#832D3semidirect
110. #832C3⋊D3semidirect
235. C33#463semidirect
340. C32#4787semidirect
341. C32#5153semidirect
386. C3#10591434semidirect
387. C3#10592486semidirect

Maximal subgroups

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

subgroupindex
#117239772
C2×C3×#37532
#117235012
C2×#887962
C32×#8322
#117220502
#117220412
#105924863
#105914343