Finite Group Databasebeta

← #13161338

Subgroups — #13161338

Order 1728.

1128 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1128 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×D3×#634C211yesmaximal, characteristic
2C2×#88863C211yesmaximal, characteristic, complemented
3C2×C3×#3756C211yesmaximal, characteristic, complemented
4C32×#946C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
8#10592749—31nomaximal
9#10591653—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6293—21no—
15C2×D3×C3⋊Dic3—21no—
16D3×#634—22no—
17D3×#634—22no—
18#88863—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3756—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×(C2×D3)⋊C4—21no—
23C32×C4⋊Dic3C2211yescharacteristic
24C32×Dic3⋊C4—21no—
25C22×#634—31no—
26C2×D3×Dic6—34no—
27C32×C22⋊Q8—31no—
28C2×#3756—31no—
29C2×#3756—34no—
30C2×#3756—64no—
31C2×#3756D311yescharacteristic, complemented
32C3×#946D314yes—
33C2×C3×#216—34no—
34#63657—31no—
35#63637—31no—
36#63637—31no—
37#63567—34no—
38#63565—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C32×D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46#6308—21no—
47#6308D412yes—
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634D412yes—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55#6293—24no—
56D3×C3⋊Dic3—24no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5186—91no—
60#5050—94no—
61C2×C32×Q8—24no—
62C2×C32×Q8C2×D314yes—
63C2×C32×Q8C2×D311yescharacteristic
64C22×C32×C4—31no—
65C22×C3⋊Dic3—61no—
66C2×#634—31no—
67C2×#634—34no—
68C2×#634—62no—
69C2×#634—61no—
70C2×#634—62no—
71C2×#634—64no—
72C2×#634C2×D311yescharacteristic
73C2×C3×C4×D3C2×D314yes—
74C2×C3×Dic6—34no—
75C2×#627—64no—
76C2×D3×Dic3—64no—
77D3×Dic6—68no—
78D3×Dic6—68no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—61no—
81C32×C22⋊C4—61no—
82#3756—22no—
83#3756—62no—
84#3756—68no—
85#3756—616no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×#216—68no—
91C3×(C2×D3)⋊C4—24no—
92C3×C4⋊Dic3C2×D314yes—
93C3×Dic3⋊C4—24no—
94C2×C3×C3⋊C8—34no—
95C22×C33C2×D411yescharacteristic
96C2×C32×D3—21no—
97C2×C32×D3—21no—
98C2×C32×D3Q1612yescomplemented
99C33×C4—41no—
100C33×C4C2×D411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C3⋊(C3⋊Dic3)—22no—
103C3×C3⋊Dic3—22no—
104C32×Dic3—21no—
105C32×Dic3—41no—
106C2×D3×Q8—91no—
107C22×Dic6—94no—
108C2×C3×Q16—91no—
109C3×C22⋊Q8—34no—
110#946C3⋊D311yescharacteristic
111C2×#216—34no—
112C2×#216—94no—
113C2×#216—91no—
114C2×#216—184no—
115#835—91no—
116#830—94no—
117(C2×D3)⋊C8—91no—
118#816—91no—
119#810—94no—
120#810—94no—
121C23×C32—31no—
122C22×C3×D3#6714yes—
123C32×Q8—28no—
124C32×Q8—24no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—24no—
128C32×Q8—44no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—31no—
133C2×C32×C4—44no—
134C2×C32×C4—61no—
135C2×C32×C4—61no—
136C2×C32×C4—61no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—124no—
144#634—21no—
145#634—32no—
146#634—38no—
147#634—68no—
148#634—64no—
149#634—61no—
150#634—64no—
151#634—124no—
152#634#6712yes—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3#6714yes—
156C3×C4×D3—24no—
157C3×C4×D3—24no—
158C3×Dic6—38no—
159C3×Dic6—64no—
160#627—616no—
161D3×Dic3—616no—
162C3⋊(C3⋊C8)—21no—
163C3⋊(C3⋊C8)—68no—
164C3⋊(C3⋊C8)—64no—
165C3⋊(C3⋊C8)—61no—
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—91no—
174C2×C3×Q8D3214yes—
175C2×C3×Q8C2×C3⋊D311yescharacteristic
176C22×C3×C4—34no—
177C22×Dic3—184no—
178D3×Q8—182no—
179D3×Q8—182no—
180C2×C4×D3—181no—
181C2×C4×D3C2×C3⋊D311yescharacteristic
182C2×Dic6—34no—
183C2×Dic6—94no—
184C2×Dic6—91no—
185C2×Dic6—184no—
186C2×Dic6—188no—
187C2×Dic6—188no—
188C2×Dic6—184no—
189C2×Dic6—181no—
190C3×Q16—182no—
191C2×C3×C8—91no—
192C3×C4⋊C4—34no—
193C3×C4⋊C4—64no—
194C3×C22⋊C4—64no—
195#216—68no—
196#216—188no—
197#216—182no—
198#216—1816no—
199(C2×D3)⋊C4—21no—
200C4⋊Dic3C2×C3⋊D311yescharacteristic
201Dic3⋊C4—21no—
202C2×C3⋊C8—34no—
203C2×C3⋊C8—94no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—184no—
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—24no—
216C32×C4—24no—
217C32×C4—48no—
218C32×C4—44no—
219C32×C4—41no—
220C32×C4—61no—
221C32×C4—121no—
222C32×C4D3×D414yes—
223C32×C4D3×D414yes—
224C32×C4D3×D411yescharacteristic
225C32×C4C2×#6711yescharacteristic
226C3⋊Dic3—22no—
227C3⋊Dic3—62no—
228C3⋊Dic3—68no—
229C3⋊Dic3—128no—
230C3×Dic3—24no—
231C3×Dic3—44no—
232C3×Dic3—68no—
233C22×Q8—271no—
234C2×Q16—91no—
235C2×Q16—271no—
236C22⋊Q8—31no—
237#103—271no—
238#103—271no—
239C22⋊C8—271no—
240C33#45111yescharacteristic, complemented, Sylow
241C23×C3—34no—
242C22×D3#63811yescharacteristic
243C3×Q8—28no—
244C3×Q8—24no—
245C3×Q8—24no—
246C3×Q8—21no—
247C3×Q8—21no—
248C3×Q8—44no—
249C3×Q8—92no—
250C3×Q8—181no—
251C2×C3×C4—24no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—34no—
255C2×C3×C4—44no—
256C2×C3×C4—64no—
257C2×C3×C4—64no—
258C2×C3×C4—64no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—64no—
264C2×Dic3—1816no—
265C2×Dic3—184no—
266C2×Dic3—181no—
267C2×Dic3—364no—
268C2×Dic3#63811yescharacteristic
269C4×D3—21no—
270C4×D3—21no—
271C4×D3—184no—
272Dic6—38no—
273Dic6—64no—
274Dic6—98no—
275Dic6—92no—
276Dic6—188no—
277Dic6—184no—
278Dic6—1816no—
279Dic6—184no—
280Dic6—181no—
281Dic6—364no—
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×C32D3×Q1611yescharacteristic
292C2×C32#91811yescharacteristic
293C2×C32#93714yes—
294C2×C32#93711yescharacteristic
295C2×C32C2×#21614yes—
296C2×C32#95114yes—
297C3×D3—28no—
298C2×Q8—91no—
299C2×Q8—271no—
300C2×Q8—541no—
301C2×Q8—542no—
302C2×Q8—542no—
303C2×Q8D3×C3⋊D311yescharacteristic
304C22×C4—31no—
305C22×C4—541no—
306Q16—182no—
307Q16—542no—
308C2×C8—91no—
309C2×C8—271no—
310C4⋊C4—31no—
311C4⋊C4—61no—
312C22⋊C4—61no—
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—24no—
323C3×C4—24no—
324C3×C4—48no—
325C3×C4—41no—
326C3×C4—44no—
327C3×C4—64no—
328C3×C4—124no—
329C3×C4—182no—
330C3×C4D3×#6714yes—
331C3×C4#381114yes—
332C3×C4D4×C3⋊D311yescharacteristic
333C3×C4D4×C3⋊D311yescharacteristic
334Dic3—21no—
335Dic3—41no—
336Dic3—68no—
337Dic3—182no—
338Dic3—188no—
339Dic3—368no—
340C32—24no—
341C32#505014yescomplemented
342C32#518611yescharacteristic, complemented
343C23—31no—
344Q8—21no—
345Q8—21no—
346Q8—92no—
347Q8—181no—
348Q8—272no—
349Q8—544no—
350Q8—541no—
351C2×C4—21no—
352C2×C4—31no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—181no—
357C2×C4—544no—
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#6394314yes—
368C2×C3#6398314yes—
369C2×C3#6414611yescharacteristic
370C2×C3C2×#375611yescharacteristic
371C2×C3#6416011yescharacteristic
372D3—22no—
373C22—32no—
374C22—61no—
375C22—61no—
376C22D3×#63811yescharacteristic
377C4—41no—
378C4—61no—
379C4—121no—
380C4—182no—
381C4—542no—
382C4C3⋊D3×#6711yescharacteristic
383C4#8903911yescharacteristic
384C3—24no—
385C3#1059165314yescomplemented
386C3#1059274911yescharacteristic, complemented
387C2—62no—
388C2D3×#375611yescharacteristic
389C2#1172357411yescharacteristic
390C2#1172364311yescharacteristic
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
3. C2×C3×#3756C2semidirect
5. #11722041C2semidirect
6. #11722039C2semidirect
31. C2×#3756D3semidirect
98. C2×C32×D3Q16semidirect
214. C2×C3×D3#216semidirect
240. C33#451semidirect
321. C2×D3#3756semidirect
341. C32#5050semidirect
342. C32#5186semidirect
385. C3#10591653semidirect
386. C3#10592749semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
C2×#888632
C2×C3×#37562
C32×#9462
#117220412
#117220392
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927493
#105916533