Finite Group Databasebeta

← #13161210

Subgroups — #13161210

Order 1728.

1128 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 200 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×D3×Q8C211yesmaximal, characteristic
2C2×#88863C211yesmaximal, characteristic, complemented
3#11723426C211yesmaximal, characteristic
4C2×#88799C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
8#10592927—31nomaximal
9#10591595—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C32×D3×Q8—22no—
13C32×D3×Q8—22no—
14C2×C32×C4×D3—21no—
15C2×C32×C4×D3C2211yescharacteristic
16C2×C32×Dic6—21no—
17C2×C32×Dic6C2211yescharacteristic
18#88863—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88807—21no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88799—22no—
25C22×C32×Q8—31no—
26C2×C3×D3×Q8D314yes—
27C2×#3756—31no—
28C2×#3756—34no—
29C2×#3756—31no—
30C2×#3756—64no—
31#64133—31no—
32#63857—34no—
33C2×#3719—34no—
34#63662—31no—
35#63662D311yescharacteristic, complemented
36#63657—31no—
37C3×#835—34no—
38#63565—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C32×D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8D412yes—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46#6308—21no—
47#6308—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3D411yescharacteristic
51C32×C4×D3—24no—
52C32×C4×D3—21no—
53C32×C4×D3—21no—
54C32×Dic6—24no—
55C32×Dic6—21no—
56C32×Dic6D412yes—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5228—94no—
60#4809—91no—
61C2×C32×Q8—24no—
62C2×C32×Q8—31no—
63C2×C32×Q8—61no—
64C2×C32×Q8—62no—
65C2×C32×Q8—62no—
66C2×C32×Q8C2×D311yescharacteristic
67C2×C32×Q8C2×D314yes—
68C22×C32×C4—31no—
69C22×C32×C4—61no—
70C2×#634—31no—
71C2×#634—34no—
72C2×#634—64no—
73C3×D3×Q8—28no—
74C3×D3×Q8—28no—
75C2×C3×C4×D3—24no—
76C2×C3×C4×D3C2×D314yes—
77C2×C3×Dic6—24no—
78C2×C3×Dic6C2×D314yes—
79#3757—61no—
80#3756—62no—
81#3756—616no—
82#3756—62no—
83#3756—68no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85#3750—61no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×C4⋊Dic3—34no—
91C2×C3×C3⋊C8—34no—
92Dic3⋊Dic3—64no—
93(C2×D3)⋊Dic3—64no—
94#3719—68no—
95C22×C33C2×D411yescharacteristic
96C2×C32×D3—21no—
97C2×C32×D3—21no—
98C2×C32×D3Q1612yescomplemented
99C33×C4—22no—
100C33×C4C2×D411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C3⋊(C3⋊Dic3)—41no—
103C3×C3⋊Dic3—41no—
104C32×Dic3—21no—
105C32×Dic3—22no—
106C22×C3×Q8—34no—
107C2×D3×Q8C3⋊D311yescharacteristic
108C2×#216—94no—
109C2×#216—94no—
110C2×#216—91no—
111C2×#216—184no—
112#924—94no—
113C2×Dic12—91no—
114#897—91no—
115C3×#103—91no—
116#835—34no—
117#835—94no—
118#830—94no—
119(C2×D3)⋊C8—91no—
120#810—91no—
121C23×C32—31no—
122C22×C3×D3#6714yes—
123C32×Q8—21no—
124C32×Q8—28no—
125C32×Q8—24no—
126C32×Q8—32no—
127C32×Q8—44no—
128C32×Q8—64no—
129C32×Q8—61no—
130C32×Q8D1212yes—
131C32×Q8#6718yes—
132C2×C32×C4—21no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—31no—
136C2×C32×C4—44no—
137C2×C32×C4—64no—
138C2×C32×C4—61no—
139C2×C32×C4—61no—
140C2×C32×C4—61no—
141C2×C32×C4C22×D314yes—
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—64no—
146C2×C3⋊Dic3—124no—
147#634—68no—
148#634—61no—
149#634—61no—
150#634—64no—
151#634—64no—
152#634—124no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3#6714yes—
156C3×C4×D3—216no—
157C3×C4×D3—24no—
158C3×C4×D3—24no—
159C3×Dic6—216no—
160C3×Dic6—24no—
161C3×Dic6#6718yes—
162C3⋊(C3⋊C8)—21no—
163C3⋊(C3⋊C8)—68no—
164C3⋊(C3⋊C8)—61no—
165C3⋊(C3⋊C8)—64no—
166C3×C3⋊C8—64no—
167#451—271noSylow
168C2×C33#12011yescharacteristic
169C2×C33C2×Q1611yescharacteristic
170C2×C33#13711yescharacteristic
171C32×D3—22no—
172C2×C3×Q8—24no—
173C2×C3×Q8—34no—
174C2×C3×Q8—64no—
175C2×C3×Q8—68no—
176C2×C3×Q8—68no—
177C2×C3×Q8D3214yes—
178C2×C3×Q8C2×C3⋊D311yescharacteristic
179C22×C3×C4—34no—
180C22×C3×C4—64no—
181D3×Q8—22no—
182D3×Q8—22no—
183C2×C4×D3—21no—
184C2×C4×D3C2×C3⋊D311yescharacteristic
185C2×Dic6—21no—
186C2×Dic6—94no—
187C2×Dic6—91no—
188C2×Dic6—184no—
189C2×Dic6C2×C3⋊D311yescharacteristic
190C2×C3×C8—91no—
191C3×C4⋊C4—91no—
192#217—184no—
193#216—188no—
194#216—1816no—
195#216—188no—
196#216—182no—
197(C2×D3)⋊C4—181no—
198C4⋊Dic3—34no—
199C4⋊Dic3—181no—
200Dic3⋊C4—184no—
201C2×C3⋊C8—34no—
202C2×C3⋊C8—94no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—184no—
205Dic12—182no—
206C22×C32—24no—
207C22×C32—32no—
208C22×C32—61no—
209C22×C32—61no—
210C22×C32D3×D411yescharacteristic
211C22×C32C2×#6714yes—
212C2×C3×D3—24no—
213C2×C3×D3—24no—
214C2×C3×D3#21618yescomplemented
215C32×C4—22no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—28no—
219C32×C4—48no—
220C32×C4—61no—
221C32×C4—62no—
222C32×C4C2×D1211yescharacteristic
223C32×C4D3×D411yescharacteristic
224C32×C4C2×#6714yes—
225C32×C4C2×#6714yes—
226C3⋊Dic3—41no—
227C3⋊Dic3—128no—
228C3⋊Dic3—121no—
229C3⋊Dic3—124no—
230C3×Dic3—24no—
231C3×Dic3—28no—
232C3×Dic3—124no—
233C22×Q8—31no—
234C2×Q16—271no—
235C2×Q16—271no—
236C22⋊Q8—271no—
237#103—91no—
238#103—271no—
239C22⋊C8—271no—
240C33#45111yescharacteristic, complemented, Sylow
241C23×C3—34no—
242C22×D3#63811yescharacteristic
243C3×Q8—24no—
244C3×Q8—21no—
245C3×Q8—28no—
246C3×Q8—38no—
247C3×Q8—44no—
248C3×Q8—616no—
249C3×Q8—64no—
250C3×Q8#62618yes—
251C3×Q8#63812yes—
252C2×C3×C4—24no—
253C2×C3×C4—21no—
254C2×C3×C4—24no—
255C2×C3×C4—34no—
256C2×C3×C4—44no—
257C2×C3×C4—616no—
258C2×C3×C4—64no—
259C2×C3×C4—64no—
260C2×C3×C4—64no—
261C2×C3×C4—181no—
262C2×C3×C4C2×D3214yes—
263C2×C3×C4C22×C3⋊D311yescharacteristic
264C2×Dic3—21no—
265C2×Dic3—64no—
266C2×Dic3—184no—
267C2×Dic3—181no—
268C2×Dic3—364no—
269C2×Dic3#63811yescharacteristic
270C4×D3—24no—
271C4×D3—21no—
272C4×D3—21no—
273Dic6—24no—
274Dic6—21no—
275Dic6—188no—
276Dic6—184no—
277Dic6—184no—
278Dic6—181no—
279Dic6—181no—
280Dic6—364no—
281Dic6#63812yes—
282C3×C8—181no—
283C3⋊C8—64no—
284C3⋊C8—188no—
285C3⋊C8—184no—
286C3⋊C8—181no—
287C2×C32—24no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—62no—
291C2×C32#88211yescharacteristic
292C2×C32#91511yescharacteristic
293C2×C32D3×Q1611yescharacteristic
294C2×C32#94214yes—
295C2×C32C2×#21614yes—
296C2×C32#95314yes—
297C3×D3—28no—
298C2×Q8—31no—
299C2×Q8—61no—
300C2×Q8—62no—
301C2×Q8—62no—
302C2×Q8—271no—
303C2×Q8D3×C3⋊D311yescharacteristic
304C22×C4—31no—
305C22×C4—61no—
306Q16—542no—
307Q16—542no—
308C2×C8—91no—
309C2×C8—271no—
310C4⋊C4—91no—
311C4⋊C4—541no—
312C22⋊C4—541no—
313C22×C3—24no—
314C22×C3—38no—
315C22×C3—64no—
316C22×C3—64no—
317C22×C3D3×#6714yes—
318C22×C3C2×#63811yescharacteristic
319C2×D3—21no—
320C2×D3—21no—
321C2×D3#375612yescomplemented
322C3×C4—22no—
323C3×C4—28no—
324C3×C4—24no—
325C3×C4—24no—
326C3×C4—48no—
327C3×C4—64no—
328C3×C4—68no—
329C3×C4—361no—
330C3×C4C2×#62614yes—
331C3×C4D3×#6714yes—
332C3×C4C2×#63811yescharacteristic
333C3×C4C2×#63811yescharacteristic
334Dic3—21no—
335Dic3—22no—
336Dic3—124no—
337Dic3—368no—
338Dic3—361no—
339Dic3—364no—
340C32—24no—
341C32#480911yescharacteristic, complemented
342C32#522814yescomplemented
343C23—31no—
344Q8—21no—
345Q8—32no—
346Q8—64no—
347Q8—61no—
348Q8—541no—
349Q8—541no—
350Q8#629112yes—
351C2×C4—21no—
352C2×C4—31no—
353C2×C4—64no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—181no—
358C2×C4—541no—
359C2×C4C2×D3×C3⋊D311yescharacteristic
360C8—181no—
361C8—541no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—68no—
366C2×C3D3×#21614yes—
367C2×C3#6394614yes—
368C2×C3#6397714yes—
369C2×C3#6415111yescharacteristic
370C2×C3C2×#375611yescharacteristic
371C2×C3#6416211yescharacteristic
372D3—22no—
373C22—32no—
374C22—61no—
375C22—61no—
376C22D3×#63811yescharacteristic
377C4—22no—
378C4—61no—
379C4—62no—
380C4—361no—
381C4—1081no—
382C4C2×#629111yescharacteristic
383C4D3×#63811yescharacteristic
384C3—24no—
385C3#1059159514yescomplemented
386C3#1059292711yescharacteristic, complemented
387C2—62no—
388C2D3×#375611yescharacteristic
389C2#1172357911yescharacteristic
390C2#1172363011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#451
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88863C2semidirect
4. C2×#88799C2semidirect
5. C3×#63662C2semidirect
6. #11722037C2semidirect
35. #63662D3semidirect
98. C2×C32×D3Q16semidirect
214. C2×C3×D3#216semidirect
240. C33#451semidirect
321. C2×D3#3756semidirect
341. C32#4809semidirect
342. C32#5228semidirect
385. C3#10591595semidirect
386. C3#10592927semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×Q82
C2×#888632
#117234262
C2×#887992
C3×#636622
#117220372
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105929273
#105915953