Finite Group Databasebeta

← #13159426

Subgroups — #13159426

Order 1728.

1068 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 210 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
2C32×#892C211yesmaximal, characteristic
3C2×#88786C211yesmaximal, characteristic
4#11722180C211yesmaximal, characteristic, complemented
5C3×#63637C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592703—31nomaximal
9#10590600—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6293—21no—
14C2×D3×C3⋊Dic3—21no—
15D3×#634—22no—
16D3×#634C414yescomplemented
17C33×C4⋊C4C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×(C2×D3)⋊C4—21no—
21C32×C4⋊Dic3C2211yescharacteristic
22C32×C4×Dic3—21no—
23#88786—22no—
24C22×#634—31no—
25C2×D3×Dic6—34no—
26C32×#117—31no—
27C2×#3748—31no—
28C3×#892D314yes—
29C2×#3711—34no—
30#63637—31no—
31#63637—31no—
32#63637—31no—
33#63637—34no—
34#63637—64no—
35#63637D311yescharacteristic, complemented
36C3×#810—34no—
37#63567—34no—
38#63565—34no—
39C22×C32×D3D411yescharacteristic
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308C2×C412yes—
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634C2×C412yes—
48C2×C32×Dic3—21no—
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3C2×C412yes—
51C32×C4×D3D412yes—
52#6293—24no—
53D3×C3⋊Dic3—24no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5004—94no—
57#4802—91no—
58C22×C32×C4—31no—
59C22×C3⋊Dic3—61no—
60C2×#634—31no—
61C2×#634—34no—
62C2×#634—34no—
63C2×#634—62no—
64C2×#634—61no—
65C2×#634—64no—
66C2×#634C2×D311yescharacteristic
67C2×C3×C4×D3C2×D314yes—
68C2×C3×Dic6—34no—
69C2×#627—64no—
70C2×D3×Dic3—64no—
71D3×Dic6—316no—
72D3×Dic6—68no—
73C32×C4⋊C4—24no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4C2×D314yes—
76C32×C4⋊C4C2×D311yescharacteristic
77C32×C22⋊C4—61no—
78C32×C42—61no—
79#3748—62no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—64no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×(C2×D3)⋊C4—24no—
85C3×C4⋊Dic3C2×D314yes—
86C3×C4×Dic3—24no—
87C2×C3×C3⋊C8—34no—
88#3711—68no—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3C22⋊C412yes—
93C33×C4—41no—
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3⋊(C3⋊Dic3)—22no—
97C3×C3⋊Dic3—22no—
98C32×Dic3—41no—
99C32×Dic3C22⋊C412yes—
100C2×D3×Q8—91no—
101C22×Dic6—94no—
102C3×#117—34no—
103C2×#208—94no—
104C2×ΓL(1,𝔽25)—91no—
105#892C3⋊D311yescharacteristic
106C3×#103—91no—
107#835—91no—
108#816—91no—
109#810—34no—
110#810—94no—
111#810—94no—
112#810—94no—
113#810—91no—
114#810—184no—
115C23×C32—31no—
116C22×C3×D3#6714yes—
117C2×C32×C4—21no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—31no—
121C2×C32×C4—32no—
122C2×C32×C4—32no—
123C2×C32×C4—44no—
124C2×C32×C4—61no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—124no—
132#634—21no—
133#634—32no—
134#634—38no—
135#634—68no—
136#634—64no—
137#634—61no—
138#634—64no—
139#634—124no—
140#634C4×D312yes—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3#6714yes—
144C3×C4×D3C4×D318yes—
145C3×C4×D3D1218yes—
146C3×Dic6—38no—
147C3×Dic6—64no—
148#627—616no—
149D3×Dic3—616no—
150C3⋊(C3⋊C8)—21no—
151C3⋊(C3⋊C8)—61no—
152C3⋊(C3⋊C8)—64no—
153C3⋊(C3⋊C8)—124no—
154C3×C3⋊C8—64no—
155#419—271noSylow
156C2×C33C2×C22⋊C411yescharacteristic
157C2×C33#13711yescharacteristic
158C2×C33#13711yescharacteristic
159C32×D3—22no—
160C2×C3×Q8—91no—
161C22×C3×C4—34no—
162C22×Dic3—184no—
163D3×Q8—94no—
164D3×Q8—182no—
165C2×C4×D3—181no—
166C2×C4×D3C2×C3⋊D311yescharacteristic
167C2×Dic6—34no—
168C2×Dic6—916no—
169C2×Dic6—94no—
170C2×Dic6—91no—
171C2×Dic6—184no—
172C2×Dic6—188no—
173C2×Dic6—184no—
174C2×Dic6—181no—
175C2×C3×C8—91no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—34no—
178C3×C4⋊C4D3214yes—
179C3×C4⋊C4C2×C3⋊D311yescharacteristic
180C3×C22⋊C4—64no—
181C3×C42—64no—
182(C2×D3)⋊C4—21no—
183C4⋊Dic3C2×C3⋊D311yescharacteristic
184C4×Dic3—21no—
185#208—188no—
186C2×C3⋊C8—34no—
187C2×C3⋊C8—94no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—184no—
190ΓL(1,𝔽25)—182no—
191C22×C32—24no—
192C22×C32—32no—
193C22×C32—61no—
194C22×C32—61no—
195C22×C32D3×D411yescharacteristic
196C22×C32C2×#6714yes—
197C2×C3×D3—24no—
198C2×C3×D3—24no—
199C2×C3×D3(C2×D3)⋊C418yes—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—32no—
203C32×C4—41no—
204C32×C4—44no—
205C32×C4—84no—
206C32×C4—121no—
207C32×C4C2×C4×D314yes—
208C32×C4C2×C4×D311yescharacteristic
209C32×C4C2×D1214yes—
210C32×C4D3×D411yescharacteristic
211C3⋊Dic3—22no—
212C3⋊Dic3—62no—
213C3⋊Dic3—68no—
214C3⋊Dic3—128no—
215C3×Dic3—44no—
216C3×Dic3—68no—
217C3×Dic3(C2×D3)⋊C418yes—
218C22×Q8—271no—
219C2×M16—271no—
220#117—31no—
221#103—91no—
222#103—271no—
223#103—271no—
224#103—271no—
225C33#41911yescharacteristic, complemented, Sylow
226C23×C3—34no—
227C22×D3#63811yescharacteristic
228C3×Q8—92no—
229C3×Q8—181no—
230C2×C3×C4—24no—
231C2×C3×C4—21no—
232C2×C3×C4—24no—
233C2×C3×C4—34no—
234C2×C3×C4—38no—
235C2×C3×C4—38no—
236C2×C3×C4—44no—
237C2×C3×C4—64no—
238C2×C3×C4—181no—
239C2×C3×C4C2×D3214yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241C2×Dic3—21no—
242C2×Dic3—64no—
243C2×Dic3—1816no—
244C2×Dic3—184no—
245C2×Dic3—181no—
246C2×Dic3—364no—
247C2×Dic3#63811yescharacteristic
248C4×D3—184no—
249C4×D3C4×C3⋊D312yes—
250C4×D3C4⋊(C3⋊D3)12yes—
251Dic6—38no—
252Dic6—64no—
253Dic6—98no—
254Dic6—92no—
255Dic6—188no—
256Dic6—1816no—
257Dic6—184no—
258Dic6—181no—
259Dic6—184no—
260Dic6—364no—
261C3×C8—181no—
262C3⋊C8—64no—
263C3⋊C8—184no—
264C3⋊C8—181no—
265C3⋊C8—364no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—24no—
269C2×C32—62no—
270C2×C32D3×C22⋊C411yescharacteristic
271C2×C32#91511yescharacteristic
272C2×C32#91811yescharacteristic
273C2×C32C2×(C2×D3)⋊C414yes—
274C2×C32#95114yes—
275C2×C32#95114yes—
276C3×D3—28no—
277C2×Q8—91no—
278C2×Q8—274no—
279C2×Q8—271no—
280C2×Q8—541no—
281C2×Q8—542no—
282C22×C4—31no—
283C22×C4—541no—
284M16—542no—
285C2×C8—91no—
286C2×C8—271no—
287C4⋊C4—31no—
288C4⋊C4D3×C3⋊D311yescharacteristic
289C22⋊C4—61no—
290C42—61no—
291C22×C3—24no—
292C22×C3—38no—
293C22×C3—64no—
294C22×C3—64no—
295C22×C3D3×#6714yes—
296C22×C3C2×#63811yescharacteristic
297C2×D3—21no—
298C2×D3—21no—
299C2×D3#375212yes—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—38no—
303C3×C4—41no—
304C3×C4—44no—
305C3×C4—84no—
306C3×C4—124no—
307C3×C4—182no—
308C3×C4C4×D3214yes—
309C3×C4D3×D1214yes—
310C3×C4C2×C4×C3⋊D311yescharacteristic
311C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
312Dic3—41no—
313Dic3—68no—
314Dic3—182no—
315Dic3—188no—
316Dic3—368no—
317Dic3#375212yes—
318C32—24no—
319C32#480211yescharacteristic, complemented
320C32#500414yescomplemented
321C23—31no—
322Q8—92no—
323Q8—181no—
324Q8—272no—
325Q8—544no—
326Q8—541no—
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—61no—
332C2×C4—181no—
333C2×C4—544no—
334C2×C4—541no—
335C2×C4C2×D3×C3⋊D311yescharacteristic
336C8—181no—
337C8—541no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—68no—
342C2×C3D3×(C2×D3)⋊C414yes—
343C2×C3#6392914yes—
344C2×C3#6394314yes—
345C2×C3C2×#375211yescharacteristic
346C2×C3#6416011yescharacteristic
347C2×C3#6416011yescharacteristic
348D3—22no—
349C22—32no—
350C22—61no—
351C22—61no—
352C22D3×#63811yescharacteristic
353C4—32no—
354C4—41no—
355C4—121no—
356C4—182no—
357C4—542no—
358C4C4×D3×C3⋊D311yescharacteristic
359C4D3×C4⋊(C3⋊D3)11yescharacteristic
360C3—24no—
361C3#1059060014yescomplemented
362C3#1059270311yescharacteristic, complemented
363C2—62no—
364C2D3×#375211yescharacteristic
365C2#1172355011yescharacteristic
366C2#1172357411yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11722180C2semidirect
5. C3×#63637C2semidirect
6. #11722041C2semidirect
7. #11722039C2semidirect
16. D3×#634C4semidirect
35. #63637D3semidirect
225. C33#419semidirect
319. C32#4802semidirect
320. C32#5004semidirect
361. C3#10590600semidirect
362. C3#10592703semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
C32×#8922
C2×#887862
#117221802
C3×#636372
#117220412
#117220392
#105927033
#105906003