Finite Group Databasebeta

← #13159969

Subgroups — #13159969

Order 1728.

920 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 156 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×C32×#204C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic
3#11723397C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722184C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592862—31nomaximal
9#10590876—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6290—21no—
14C2×#6290—21no—
15C2×C33×C8C2211yescharacteristic
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C32×#204—22no—
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22#88793—22no—
23C2×C32×SD16—31no—
24Q8×C3⋊Dic3—31no—
25C2×#3755—31no—
26#64149—31no—
27C2×C3×#204D314yescomplemented
28#63914—34no—
29#63843—34no—
30C2×#3715—34no—
31#63659—31no—
32#63648D311yescharacteristic, complemented
33#63641—31no—
34#63641—34no—
35#63641—64no—
36C3×#821—34no—
37#63562—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C2×C3×C3⋊Dic3—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C2×C32×Dic3—21no—
47C32×D12—21no—
48C32×D12—21no—
49C32×Dic6—21no—
50C32×Dic6D412yescomplemented
51#6290—42no—
52#6290—42no—
53C33×C8—21no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55#5163—94no—
56#4762—91no—
57C2×C32×Q8—31no—
58C2×C32×D4—31no—
59C2×#638—61no—
60C2×#638—61no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×D12—34no—
63C2×C3×D12C2×D314yes—
64C2×C3×Dic6C2×D314yes—
65C2×#625—64no—
66C2×#625—64no—
67C32×SD16—62no—
68C2×C32×C8—24no—
69C2×C32×C8C2×D314yes—
70C2×C32×C8C2×D311yescharacteristic
71#3755—62no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C4×C3⋊Dic3—31no—
75C4×C3⋊Dic3—34no—
76C4×C3⋊Dic3—61no—
77C4×C3⋊Dic3—64no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—34no—
80C2×C3⋊(C3⋊C8)—64no—
81C3×C4⋊Dic3—34no—
82C3×#204—28no—
83Dic3⋊Dic3—64no—
84Dic32—64no—
85#3715—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—21no—
96C3⋊(C3⋊Dic3)SD1612yescomplemented
97C3×C3⋊Dic3—41no—
98C32×Dic3—22no—
99C2×C3×SD16—34no—
100Q8×Dic3—94no—
101C2×#215—94no—
102#940—94no—
103#940—91no—
104C2×#214—91no—
105C2×#204C3⋊D311yescharacteristic, complemented
106#887—91no—
107C3×#102—91no—
108#832—94no—
109#821—34no—
110Dic3⋊C8—94no—
111Dic3⋊C8—91no—
112Dic3⋊C8—184no—
113#809—91no—
114C23×C32—61no—
115C22×C3⋊D3—21no—
116C22×C3×D3—24no—
117C22×C3×D3—64no—
118C32×Q8—32no—
119C32×Q8—61no—
120C32×D4—61no—
121C32×D4—61no—
122C2×C32×C4—24no—
123C2×C32×C4—61no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126#638—122no—
127#638—122no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—31no—
130C2×C3⋊Dic3—31no—
131C2×C3⋊Dic3—34no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—64no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—64no—
139C3×D12—24no—
140C3×D12—24no—
141C3×D12—64no—
142C3×D12—64no—
143C3×Dic6—24no—
144C3×Dic6#6718yescomplemented
145#625—128no—
146#625—128no—
147C32×C8—21no—
148C32×C8—24no—
149C32×C8—44no—
150C3⋊(C3⋊C8)—61no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—124no—
153#460—271noSylow
154C2×C33#12111yescharacteristic
155C2×C33C2×SD1611yescharacteristic
156C2×C33AGL(1,ℤ/8)11yescharacteristic
157C3×C3⋊D3—81no—
158C32×D3—81no—
159C2×C3×Q8—34no—
160C2×C3×D4—34no—
161C2×C3×D4—91no—
162C2×#67—181no—
163C2×#67—181no—
164C2×#67—184no—
165C2×#67—184no—
166C2×D12—34no—
167C2×D12C2×C3⋊D311yescharacteristic
168C2×Dic6C2×C3⋊D311yescharacteristic
169C3×SD16—68no—
170C2×C3×C8—24no—
171C2×C3×C8D3214yes—
172C2×C3×C8C2×C3⋊D311yescharacteristic
173C3×C4⋊C4—91no—
174#215—188no—
175#214—182no—
176C4⋊Dic3—34no—
177C4⋊Dic3—184no—
178Dic3⋊C4—181no—
179C4×Dic3—94no—
180C4×Dic3—91no—
181C4×Dic3—184no—
182C4×Dic3—181no—
183C4×Dic3—184no—
184C2×C3⋊C8—94no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—184no—
187#204—22no—
188C22×C32—24no—
189C22×C32—121no—
190C22×C32—121no—
191C22×C32—121no—
192C22×C32D3×D414yes—
193C22×C32D3×D411yescharacteristic
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—41no—
197C2×C3×D3—44no—
198C2×C3×D3—44no—
199C2×C3×D3—44no—
200C2×C3×D3—124no—
201C2×C3×D3—124no—
202C2×C3×D3—124no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—62no—
206C32×C4#23511yescharacteristic
207C32×C4D3×D411yescharacteristic
208C32×C4#23714yes—
209C32×C4C2×#6714yes—
210C3⋊Dic3—32no—
211C3⋊Dic3—38no—
212C3⋊Dic3—41no—
213C3⋊Dic3—68no—
214C3⋊Dic3—61no—
215C3⋊Dic3—64no—
216C3⋊Dic3—124no—
217C3×Dic3—28no—
218C3×Dic3—124no—
219C2×SD16—31no—
220C2×SD16—271no—
221#127—271no—
222C4×Q8—271no—
223C4⋊C8—271no—
224#102—91no—
225#102—271no—
226C33#46011yescharacteristic, complemented, Sylow
227C23×C3—64no—
228C23×C3—181no—
229C22×D3—21no—
230C22×D3—64no—
231C3×Q8—38no—
232C3×Q8—64no—
233C3×D4—64no—
234C3×D4—64no—
235C3×D4—181no—
236C3×D4—181no—
237C2×C3×C4—24no—
238C2×C3×C4—64no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3214yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242#67—362no—
243#67—368no—
244#67—362no—
245#67—368no—
246C2×Dic3—21no—
247C2×Dic3—64no—
248C2×Dic3—94no—
249C2×Dic3—91no—
250C2×Dic3—94no—
251C2×Dic3—91no—
252C2×Dic3—184no—
253C2×Dic3—184no—
254D12—21no—
255D12—21no—
256D12—64no—
257D12—64no—
258Dic6—21no—
259Dic6#63812yescomplemented
260C3×C8—24no—
261C3×C8—21no—
262C3×C8—44no—
263C3⋊C8—184no—
264C3⋊C8—181no—
265C3⋊C8—364no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—24no—
269C2×C32—241no—
270C2×C32#88411yescharacteristic
271C2×C32#91111yescharacteristic
272C2×C32D3×SD1614yes—
273C2×C32D3×SD1611yescharacteristic
274C2×C32#91414yes—
275C2×C32#93914yes—
276C3⋊D3—81no—
277C3×D3—84no—
278C3×D3—244no—
279C2×Q8—31no—
280C2×D4—31no—
281C2×D4—91no—
282C2×D4—541no—
283C2×D4—541no—
284SD16—62no—
285SD16—542no—
286C2×C8—271no—
287C2×C8D3×C3⋊D311yescharacteristic
288C4⋊C4—91no—
289C4⋊C4—541no—
290C42—271no—
291C42—541no—
292C22×C3—24no—
293C22×C3—124no—
294C22×C3—124no—
295C22×C3—124no—
296C22×C3—361no—
297C22×C3—361no—
298C22×C3—361no—
299C22×C3#380214yes—
300C22×C3D4×C3⋊D311yescharacteristic
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—124no—
305C2×D3—124no—
306C2×D3—124no—
307C3×C4—24no—
308C3×C4—24no—
309C3×C4—68no—
310C3×C4—361no—
311C3×C4#379514yes—
312C3×C4D3×#6714yes—
313C3×C4#383011yescharacteristic
314C3×C4C2×#63811yescharacteristic
315Dic3—22no—
316Dic3—98no—
317Dic3—92no—
318Dic3—124no—
319Dic3—188no—
320Dic3—184no—
321Dic3—181no—
322Dic3—364no—
323C32—24no—
324C32#476211yescharacteristic, complemented
325C32#516314yescomplemented
326C23—61no—
327C23—181no—
328Q8—32no—
329Q8—61no—
330D4—61no—
331D4—61no—
332D4—181no—
333D4—181no—
334D4—1082no—
335D4—1082no—
336C2×C4—61no—
337C2×C4—181no—
338C2×C4—271no—
339C2×C4—271no—
340C2×C4C2×D3×C3⋊D311yescharacteristic
341C8—21no—
342C8—541no—
343C2×C3—24no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—244no—
347C2×C3—721no—
348C2×C3#6379614yes—
349C2×C3#6379814yes—
350C2×C3#6388514yes—
351C2×C3SD16×C3⋊D311yescharacteristic
352C2×C3#6412311yescharacteristic
353C2×C3#6414811yescharacteristic
354D3—81no—
355D3—244no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—361no—
360C22—361no—
361C22—361no—
362C22#8902611yescharacteristic
363C4—62no—
364C4—272no—
365C4—361no—
366C4—541no—
367C4#8901411yescharacteristic
368C4D3×#63811yescharacteristic
369C3—24no—
370C3#1059087614yescomplemented
371C3#1059286211yescharacteristic, complemented
372C2—241no—
373C2—721no—
374C2#1172332111yescharacteristic
375C2#1172332311yescharacteristic
376C2#1172347411yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723529C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#460
LCSG ▷ C2×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
1. C2×C32×#204C2semidirect
3. #11723397C2semidirect
4. C2×#88793C2semidirect
5. #11722184C2semidirect
6. C3×#63648C2semidirect
7. #11722031C2semidirect
27. C2×C3×#204D3semidirect
32. #63648D3semidirect
50. C32×Dic6D4semidirect
96. C3⋊(C3⋊Dic3)SD16semidirect
105. C2×#204C3⋊D3semidirect
144. C3×Dic6#67semidirect
226. C33#460semidirect
259. Dic6#638semidirect
324. C32#4762semidirect
325. C32#5163semidirect
370. C3#10590876semidirect
371. C3#10592862semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2042
#117235292
#117233972
C2×#887932
#117221842
C3×#636482
#117220312
#105928623
#105908763