Finite Group Databasebeta

← #13161200

Subgroups — #13161200

Order 1728.

1056 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 176 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×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3#11723435C211yesmaximal, characteristic
4C2×#88798C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic
8#10592885—31nomaximal
9#10591594—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×Q8×C3⋊D3—22no—
12C3×Q8×C3⋊D3—22no—
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634—21no—
16C2×C3×#634C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×#216—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22#88806—21no—
23#88803—21no—
24#88798—22no—
25C2×Q8×C3⋊D3D311yescharacteristic
26C2×C3×D3×Q8—34no—
27C2×C32×Q16—31no—
28#64155—31no—
29C2×#3756—31no—
30C2×C3×#216D314yescomplemented
31#63856—34no—
32C2×#3718—34no—
33#63662—31no—
34#63662—34no—
35#63662—64no—
36#63647—31no—
37#63643—31no—
38#63566—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8D412yes—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3—24no—
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C3×#634—24no—
52C3×#634—21no—
53C3×#634D412yes—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5228—91no—
60#5186—94no—
61C2×C32×Q8—24no—
62C2×C32×Q8—31no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65Q8×C3⋊D3—22no—
66Q8×C3⋊D3—22no—
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×#634—21no—
70C2×#634C2×D311yescharacteristic
71C3×D3×Q8—68no—
72C3×D3×Q8—68no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—64no—
75C2×C3×Dic6—34no—
76C2×C3×Dic6—64no—
77C2×C3×Dic6C2×D314yes—
78C32×Q16—62no—
79C2×C32×C8—31no—
80#3756—62no—
81#3752—61no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—34no—
84C4⋊(C3⋊Dic3)—64no—
85#3750—61no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C3×#216—28no—
90C2×C3×C3⋊C8C2×D314yes—
91#3724—64no—
92(C2×D3)⋊Dic3—64no—
93#3718—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3Q1612yescomplemented
98C33×C4—22no—
99C33×C4C2×D411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3⋊(C3⋊Dic3)—41no—
102C3×C3⋊Dic3—21no—
103C3×C3⋊Dic3—22no—
104C32×Dic3—41no—
105C22×C3×Q8—91no—
106C2×D3×Q8—34no—
107C2×C3×Q16—34no—
108#946—94no—
109C2×#216—94no—
110C2×#216—91no—
111C2×#216C3⋊D311yescharacteristic, complemented
112#924—91no—
113#835—94no—
114#835—91no—
115#835—91no—
116#835—184no—
117#830—91no—
118(C2×D3)⋊C8—94no—
119#816—94no—
120C22×C3⋊D3#6711yescharacteristic
121C22×C3×D3—34no—
122C32×Q8—21no—
123C32×Q8—28no—
124C32×Q8—24no—
125C32×Q8—44no—
126C32×Q8—61no—
127C32×Q8—61no—
128C32×Q8#6718yes—
129C32×Q8#6712yes—
130C2×C32×C4—21no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—44no—
134C2×C32×C4—61no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—61no—
139C2×C3⋊Dic3—64no—
140C2×C3⋊Dic3—124no—
141C2×C3⋊Dic3#6711yescharacteristic
142C4×C3⋊D3—24no—
143C4×C3⋊D3—21no—
144C4×C3⋊D3—21no—
145#634—24no—
146#634—21no—
147#634#6712yes—
148C2×C3×Dic3—24no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—64no—
151C3×C4×D3—616no—
152C3×C4×D3—64no—
153C3×C4×D3—64no—
154C3×Dic6—24no—
155C3×Dic6—24no—
156C3×Dic6—38no—
157C3×Dic6—616no—
158C3×Dic6—64no—
159C32×C8—61no—
160C3⋊(C3⋊C8)—61no—
161C3⋊(C3⋊C8)—64no—
162C3⋊(C3⋊C8)—124no—
163C3×C3⋊C8—24no—
164#451—271noSylow
165C2×C33#12011yescharacteristic
166C2×C33C2×Q1611yescharacteristic
167C2×C33#13711yescharacteristic
168C3×C3⋊D3—22no—
169C2×C3×Q8—24no—
170C2×C3×Q8—34no—
171C2×C3×Q8—91no—
172C2×C3×Q8—181no—
173C2×C3×Q8—182no—
174C2×C3×Q8—182no—
175C2×C3×Q8D3214yes—
176C2×C3×Q8C2×C3⋊D311yescharacteristic
177C22×C3×C4—91no—
178C22×C3×C4—181no—
179D3×Q8—68no—
180D3×Q8—68no—
181C2×C4×D3—34no—
182C2×C4×D3—64no—
183C2×Dic6—34no—
184C2×Dic6—64no—
185C2×Dic6C2×C3⋊D311yescharacteristic
186C3×Q16—68no—
187C2×C3×C8—34no—
188#217—181no—
189#216—22no—
190#216—182no—
191#216—188no—
192(C2×D3)⋊C4—184no—
193C4⋊Dic3—94no—
194C4⋊Dic3—91no—
195C4⋊Dic3—184no—
196Dic3⋊C4—181no—
197Dic3⋊C4—184no—
198C2×C3⋊C8—94no—
199C2×C3⋊C8—91no—
200C2×C3⋊C8—184no—
201C2×C3⋊C8C2×C3⋊D311yescharacteristic
202C22×C32—24no—
203C22×C32D3×D414yes—
204C22×C32C2×#6711yescharacteristic
205C2×C3⋊D3—21no—
206C2×C3⋊D3—21no—
207C2×C3⋊D3#21612yescomplemented
208C2×C3×D3—38no—
209C2×C3×D3—64no—
210C2×C3×D3—64no—
211C32×C4—22no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—28no—
215C32×C4—48no—
216C32×C4—121no—
217C32×C4D3×D414yes—
218C32×C4C2×#6714yes—
219C32×C4C2×#6711yescharacteristic
220C32×C4C2×#6711yescharacteristic
221C3⋊Dic3—21no—
222C3⋊Dic3—22no—
223C3⋊Dic3—121no—
224C3⋊Dic3—124no—
225C3⋊Dic3—244no—
226C3×Dic3—44no—
227C3×Dic3—68no—
228C3×Dic3—64no—
229C22×Q8—91no—
230C2×Q16—31no—
231C2×Q16—271no—
232C22⋊Q8—271no—
233#103—271no—
234#103—271no—
235C22⋊C8—271no—
236C33#45111yescharacteristic, complemented, Sylow
237C23×C3—91no—
238C22×D3—34no—
239C3×Q8—24no—
240C3×Q8—21no—
241C3×Q8—28no—
242C3×Q8—44no—
243C3×Q8—64no—
244C3×Q8—64no—
245C3×Q8—92no—
246C3×Q8—184no—
247C3×Q8—181no—
248C3×Q8#62518yes—
249C3×Q8#63812yes—
250C2×C3×C4—24no—
251C2×C3×C4—21no—
252C2×C3×C4—24no—
253C2×C3×C4—44no—
254C2×C3×C4—64no—
255C2×C3×C4—91no—
256C2×C3×C4—184no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—34no—
264C2×Dic3—64no—
265C2×Dic3—184no—
266C2×Dic3—181no—
267C2×Dic3—364no—
268C4×D3—616no—
269C4×D3—64no—
270C4×D3—64no—
271Dic6—21no—
272Dic6—21no—
273Dic6—38no—
274Dic6—616no—
275Dic6—64no—
276C3×C8—64no—
277C3⋊C8—21no—
278C3⋊C8—184no—
279C3⋊C8—181no—
280C3⋊C8—364no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32D3×Q1614yes—
285C2×C32#91814yes—
286C2×C32#93714yes—
287C2×C32#94211yescharacteristic
288C2×C32C2×#21611yescharacteristic
289C2×C32#95311yescharacteristic
290C3⋊D3—22no—
291C3×D3—68no—
292C2×Q8—31no—
293C2×Q8—91no—
294C2×Q8—181no—
295C2×Q8—182no—
296C2×Q8—182no—
297C2×Q8D3×C3⋊D311yescharacteristic
298C22×C4—91no—
299C22×C4—181no—
300Q16—62no—
301Q16—542no—
302C2×C8—31no—
303C2×C8—271no—
304C4⋊C4—271no—
305C4⋊C4—541no—
306C22⋊C4—541no—
307C22×C3—24no—
308C22×C3—92no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3D3×#6714yes—
312C22×C3D4×C3⋊D311yescharacteristic
313C2×D3—38no—
314C2×D3—64no—
315C2×D3—64no—
316C3×C4—22no—
317C3×C4—28no—
318C3×C4—24no—
319C3×C4—24no—
320C3×C4—48no—
321C3×C4—124no—
322C3×C4—181no—
323C3×C4—182no—
324C3×C4C2×#62514yes—
325C3×C4D3×#6714yes—
326C3×C4D4×C3⋊D311yescharacteristic
327C3×C4C2×#63811yescharacteristic
328Dic3—41no—
329Dic3—68no—
330Dic3—64no—
331Dic3—361no—
332Dic3—364no—
333Dic3—724no—
334C32—24no—
335C32#518614yescomplemented
336C32#522811yescharacteristic, complemented
337C23—91no—
338Q8—21no—
339Q8—61no—
340Q8—61no—
341Q8—92no—
342Q8—184no—
343Q8—181no—
344Q8#629012yes—
345C2×C4—21no—
346C2×C4—61no—
347C2×C4—91no—
348C2×C4—184no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4—541no—
353C2×C4C2×D3×C3⋊D311yescharacteristic
354C8—61no—
355C8—541no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—182no—
360C2×C3D3×#21614yes—
361C2×C3#6394814yes—
362C2×C3#6397614yes—
363C2×C3Q16×C3⋊D311yescharacteristic
364C2×C3#6412711yescharacteristic
365C2×C3#6414611yescharacteristic
366D3—68no—
367C22—92no—
368C22—181no—
369C22—181no—
370C22C3⋊D3×#6711yescharacteristic
371C4—22no—
372C4—121no—
373C4—181no—
374C4—182no—
375C4—1081no—
376C4C2×#629011yescharacteristic
377C4C3⋊D3×#6711yescharacteristic
378C3—24no—
379C3#1059159414yescomplemented
380C3#1059288511yescharacteristic, complemented
381C2—182no—
382C2C3⋊D3×#21611yescharacteristic
383C2#1172358611yescharacteristic
384C2#1172363411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊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×C32×#216C2semidirect
4. C2×#88798C2semidirect
5. #11722205C2semidirect
6. #11722040C2semidirect
30. C2×C3×#216D3semidirect
97. C2×C3×C3⋊D3Q16semidirect
111. C2×#216C3⋊D3semidirect
207. C2×C3⋊D3#216semidirect
236. C33#451semidirect
335. C32#5186semidirect
336. C32#5228semidirect
379. C3#10591594semidirect
380. C3#10592885semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C32×#2162
#117234352
C2×#887982
#117222052
#117220402
#117220232
#105928853
#105915943