Finite Group Databasebeta

← #13160080

Subgroups — #13160080

Order 1728.

1252 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1252 conjugacy classes of subgroups — too many to draw. Only its 170 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×#89015C211yesmaximal, characteristic
2C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3#11723413C211yesmaximal, characteristic, complemented
4C3×#63643C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592471—31nomaximal
9#10590956—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6291—21no—
14C2×D3×C3⋊Dic3—21no—
15#89015—22no—
16#89015C414yescomplemented
17C2×C33×C8C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C8×C3⋊(C3⋊D3)—22no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22#88804—21no—
23Dic3×C3⋊Dic3—21no—
24C2×#3830—31no—
25C2×#3796—34no—
26C2×C8×C3⋊D3—31no—
27C2×C8×C3⋊D3—34no—
28C2×C8×C3⋊D3—64no—
29#64101—31no—
30#63849—34no—
31C32×#102—31no—
32#63659—31no—
33#63643D311yescharacteristic, complemented
34#63637—31no—
35C3×#821D314yescomplemented
36C3×#816—34no—
37#63565—34no—
38#63562—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C32×D3—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4×C3⋊(C3⋊D3)C2×C412yes—
44C4×C3⋊(C3⋊D3)D412yes—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634C2×C412yes—
49C2×C32×Dic3—21no—
50C32×D12—21no—
51C32×D12C2×C412yes—
52#6291—24no—
53D3×C3⋊Dic3—24no—
54C33×C8—21no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56#4804—91no—
57#4772—94no—
58C2×C32×D4—31no—
59C2×#638—61no—
60C22×C3⋊Dic3—61no—
61#3830—34no—
62#3830—62no—
63C2×C4×C3⋊D3—31no—
64C2×C4×C3⋊D3—34no—
65C2×C4×C3⋊D3—64no—
66C2×#634C2×D311yescharacteristic
67C2×C3×D12C2×D314yes—
68C2×C3×Dic6—34no—
69C2×#626—64no—
70C2×D3×Dic3—64no—
71#3796—316no—
72#3796—68no—
73C2×C32×C8—24no—
74C2×C32×C8C2×D314yes—
75C2×C32×C8C2×D311yescharacteristic
76C32×C4⋊C4—31no—
77#3752—61no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4×C3⋊Dic3—61no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—64no—
83C8×C3⋊D3—62no—
84C8×C3⋊D3—68no—
85C8×C3⋊D3—616no—
86C3×C4⋊Dic3—34no—
87C3×C4⋊Dic3C2×D314yes—
88#3722—64no—
89Dic32—64no—
90C22×C33C2×D411yescharacteristic
91C2×C3⋊(C3⋊D3)—21no—
92C2×C3⋊(C3⋊D3)—21no—
93C2×C3⋊(C3⋊D3)C22⋊C412yes—
94C2×C32×D3—22no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3⋊(C3⋊Dic3)C22⋊C412yes—
100C3×C3⋊Dic3—22no—
101C3×C3⋊Dic3—41no—
102C32×Dic3—41no—
103C2×#239—91no—
104C2×#237—94no—
105C2×D3×C8—94no—
106C2×D3×C8—91no—
107C2×D3×C8—184no—
108#892—94no—
109#892—91no—
110C3×#103—91no—
111C3×#102—34no—
112#835—91no—
113#832—94no—
114#821C3⋊D311yescharacteristic, complemented
115#816—34no—
116#810—94no—
117#809—91no—
118C23×C32—61no—
119C22×C3⋊D3—31no—
120C22×C3⋊D3—34no—
121C22×C3⋊D3—64no—
122C22×C3×D3—24no—
123C32×D4—32no—
124C32×D4—61no—
125C2×C32×C4—24no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129#638—64no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—34no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—64no—
136C4×C3⋊D3—32no—
137C4×C3⋊D3—32no—
138C4×C3⋊D3—38no—
139C4×C3⋊D3—38no—
140C4×C3⋊D3—68no—
141C4×C3⋊D3—68no—
142#634—21no—
143#634C4×D312yes—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—64no—
147C3×D12—24no—
148C3×D12C4×D318yes—
149C3×Dic6—38no—
150C3×Dic6—64no—
151#626—616no—
152D3×Dic3—616no—
153C32×C8—28no—
154C32×C8—24no—
155C32×C8—21no—
156C3⋊(C3⋊C8)—68no—
157C3⋊(C3⋊C8)—64no—
158C3⋊(C3⋊C8)—61no—
159#416—271noSylow
160C2×C33C2×C22⋊C411yescharacteristic
161C2×C33#13511yescharacteristic
162C2×C33#13511yescharacteristic
163C3⋊(C3⋊D3)—22no—
164C32×D3—42no—
165C2×C3×Q8—91no—
166C2×C3×D4—34no—
167C2×#67—184no—
168C22×Dic3—184no—
169#239—94no—
170#239—182no—
171#237—916no—
172#237—188no—
173C2×D12—181no—
174C2×D12C2×C3⋊D311yescharacteristic
175C2×C4×D3—94no—
176C2×C4×D3—91no—
177C2×C4×D3—184no—
178C2×C4×D3—181no—
179C2×Dic6—34no—
180C2×C3×C8—24no—
181C2×C3×C8D3214yes—
182C2×C3×C8C2×C3⋊D311yescharacteristic
183C3×C4⋊C4—34no—
184C3×C4⋊C4—91no—
185(C2×D3)⋊C4—181no—
186(C2×D3)⋊C4—184no—
187C4⋊Dic3—34no—
188C4⋊Dic3C2×C3⋊D311yescharacteristic
189C4×Dic3—181no—
190C4×Dic3—184no—
191C2×C3⋊C8—94no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—184no—
194D3×C8—188no—
195D3×C8—182no—
196D3×C8—1816no—
197C22×C32—24no—
198C22×C32—62no—
199C22×C32—121no—
200C22×C32—121no—
201C22×C32D3×D414yes—
202C22×C32D3×D411yescharacteristic
203C2×C3⋊D3—32no—
204C2×C3⋊D3—38no—
205C2×C3⋊D3—68no—
206C2×C3⋊D3—68no—
207C2×C3⋊D3—68no—
208C2×C3⋊D3—64no—
209C2×C3⋊D3—64no—
210C2×C3⋊D3—61no—
211C2×C3⋊D3—61no—
212C2×C3×D3—28no—
213C2×C3×D3—44no—
214C2×C3×D3—44no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—121no—
218C32×C4C2×C4×D314yes—
219C32×C4C2×C4×D311yescharacteristic
220C32×C4D3×D414yes—
221C32×C4D3×D411yescharacteristic
222C3⋊Dic3—22no—
223C3⋊Dic3—32no—
224C3⋊Dic3—38no—
225C3⋊Dic3—41no—
226C3⋊Dic3—68no—
227C3×Dic3—44no—
228C3×Dic3—68no—
229C3×Dic3—124no—
230C2×#41—271no—
231C22×C8—271no—
232#117—271no—
233#103—91no—
234#103—271no—
235#102—31no—
236#102—271no—
237C33#41611yescharacteristic, complemented, Sylow
238C23×C3—64no—
239C22×D3—21no—
240C22×D3—94no—
241C22×D3—91no—
242C22×D3—184no—
243C3×Q8—92no—
244C3×Q8—181no—
245C3×D4—38no—
246C3×D4—64no—
247C2×C3×C4—24no—
248C2×C3×C4—64no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—1816no—
254C2×Dic3—21no—
255C2×Dic3—64no—
256C2×Dic3—64no—
257C2×Dic3—94no—
258C2×Dic3—91no—
259C2×Dic3—184no—
260C2×Dic3—1816no—
261D12—21no—
262D12—184no—
263D12C4×C3⋊D312yes—
264C4×D3—98no—
265C4×D3—98no—
266C4×D3—92no—
267C4×D3—92no—
268C4×D3—188no—
269C4×D3—188no—
270C4×D3—184no—
271Dic6—38no—
272Dic6—64no—
273C3×C8—28no—
274C3×C8—24no—
275C3×C8—21no—
276C3⋊C8—188no—
277C3⋊C8—184no—
278C3⋊C8—181no—
279C2×C32—24no—
280C2×C32—24no—
281C2×C32—24no—
282C2×C32—122no—
283C2×C32D3×C22⋊C414yes—
284C2×C32D3×C22⋊C411yescharacteristic
285C2×C32#91214yes—
286C2×C32#91614yes—
287C2×C32#91611yescharacteristic
288C2×C32#91911yescharacteristic
289C3⋊D3—616no—
290C3⋊D3—68no—
291C3⋊D3—62no—
292C3×D3—48no—
293#41—274no—
294#41—542no—
295C2×Q8—91no—
296C2×D4—31no—
297C2×D4—541no—
298C22×C4—271no—
299C22×C4—541no—
300C2×C8—271no—
301C2×C8—542no—
302C2×C8D3×C3⋊D311yescharacteristic
303C4⋊C4—31no—
304C4⋊C4—91no—
305C22⋊C4—541no—
306C42—541no—
307C22×C3—24no—
308C22×C3—68no—
309C22×C3—124no—
310C22×C3—124no—
311C22×C3#380214yes—
312C22×C3D4×C3⋊D311yescharacteristic
313C2×D3—22no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—92no—
317C2×D3—98no—
318C2×D3—188no—
319C2×D3—188no—
320C2×D3—188no—
321C2×D3—184no—
322C2×D3—184no—
323C2×D3—181no—
324C2×D3—181no—
325C3×C4—24no—
326C3×C4—24no—
327C3×C4—124no—
328C3×C4—182no—
329C3×C4—361no—
330C3×C4C4×D3214yes—
331C3×C4#381114yes—
332C3×C4C2×C4×C3⋊D311yescharacteristic
333C3×C4D4×C3⋊D311yescharacteristic
334Dic3—41no—
335Dic3—68no—
336Dic3—92no—
337Dic3—98no—
338Dic3—124no—
339Dic3—188no—
340C32—24no—
341C32#477214yescomplemented
342C32#480411yescharacteristic, complemented
343C23—61no—
344C23—271no—
345Q8—92no—
346Q8—181no—
347D4—32no—
348D4—61no—
349D4—544no—
350C2×C4—61no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—271no—
354C2×C4—272no—
355C2×C4—272no—
356C2×C4—544no—
357C2×C4C2×D3×C3⋊D311yescharacteristic
358C8—21no—
359C8—541no—
360C2×C3—24no—
361C2×C3—24no—
362C2×C3—24no—
363C2×C3—128no—
364C2×C3#6380914yes—
365C2×C3#6381014yes—
366C2×C3#6392114yes—
367C2×C3C22⋊C4×C3⋊D311yescharacteristic
368C2×C3#6412111yescharacteristic
369C2×C3#6412511yescharacteristic
370D3—42no—
371D3—1816no—
372D3—182no—
373D3—188no—
374C22—62no—
375C22—121no—
376C22—121no—
377C22—272no—
378C22—541no—
379C22—541no—
380C22#8902611yescharacteristic
381C4—121no—
382C4—182no—
383C4—272no—
384C4—361no—
385C4C4×D3×C3⋊D311yescharacteristic
386C4#8903911yescharacteristic
387C3—24no—
388C3#1059095614yescomplemented
389C3#1059247111yescharacteristic, complemented
390C2—122no—
391C2—542no—
392C2#1172334211yescharacteristic
393C2#1172334311yescharacteristic
394C2#1172353911yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89015C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C8×C3⋊(C3⋊D3)C2semidirect
3. #11723413C2semidirect
4. C3×#63643C2semidirect
5. C32×#821C2semidirect
6. #11722039C2semidirect
7. #11722031C2semidirect
16. #89015C4semidirect
33. #63643D3semidirect
35. C3×#821D3semidirect
114. #821C3⋊D3semidirect
237. C33#416semidirect
341. C32#4772semidirect
342. C32#4804semidirect
388. C3#10590956semidirect
389. C3#10592471semidirect

Maximal subgroups

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

subgroupindex
C2×#890152
C2×C8×C3⋊(C3⋊D3)2
#117234132
C3×#636432
C32×#8212
#117220392
#117220312
#105924713
#105909563