Finite Group Databasebeta

← #13138704

Subgroups — #13138704

Order 1728.

524 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 109 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×#88635C211yesmaximal, characteristic
2C2×C3×C8×D9C211yesmaximal, characteristic, complemented
3#11720907C211yesmaximal, characteristic, complemented
4#11720543C211yesmaximal, characteristic, complemented
5C9×#816C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10590690—31nomaximal
9#10588484—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×#6192—21no—
14C2×(C3⋊D9)⋊C4—21no—
15#88635—22no—
16#88635C414yescomplemented
17C2×C3×C8×C9C2211yescharacteristic
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C9×C4⋊Dic3C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C3×C8×D9—22no—
22(C2×D9)⋊Dic3—21no—
23Dic3×Dic9—21no—
24C2×#3799—31no—
25C2×C3×D3×C8—31no—
26#63841—31no—
27C2×#3701—31no—
28#63648—31no—
29C3×#816D311yescharacteristic
30#63566—31no—
31#63564—31no—
32C2×C8×D9D311yescharacteristic, complemented
33#63454—31no—
34C9×#103—31no—
35#63395—31no—
36#63380—31no—
37#63380—61no—
38#63369—31no—
39C22×C3⋊D9—21no—
40C22×C3×D9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)C2×C412yes—
45C2×C9×Dic3—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9D411yescharacteristic
48C3×C4×D9C2×C412yes—
49C3×C4×D9D412yes—
50C9×Dic6—21no—
51C9×Dic6C2×C412yes—
52#6192—24no—
53(C3⋊D9)⋊C4—24no—
54C3×C8×C9—21no—
55C3×C9⋊C8—21no—
56#5113—91no—
57#4804—91no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C3×C4×D3—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61C2×#626—61no—
62C2×#624—61no—
63#3799—34no—
64#3799—62no—
65C2×C32×C8C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)—31no—
67C3×C4⋊Dic3C2×D311yescharacteristic
68C2×C3×C3⋊C8—31no—
69C3×D3×C8—62no—
70(C2×D3)⋊Dic3—61no—
71Dic32—61no—
72C2×Q8×C9—31no—
73#3701—34no—
74#3701—62no—
75C2×D36—31no—
76C2×D36—61no—
77C2×D36—61no—
78C2×C4×D9—61no—
79C2×C4×D9C2×D311yescharacteristic
80C2×C8×C9—21no—
81C2×C8×C9C2×D311yescharacteristic
82C9×C4⋊C4—31no—
83(C2×D9)⋊C4—61no—
84C4⋊Dic9—31no—
85C4⋊Dic9—61no—
86C4×Dic9—61no—
87C2×C9⋊C8C2×D311yescharacteristic
88C8×D9—22no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3⋊D9—22no—
91C2×C3⋊D9—41no—
92C2×C3⋊D9—41no—
93C2×C3×D9—21no—
94C2×C3×D9—21no—
95C2×C3×D9C22⋊C412yes—
96C3×C4×C9C22×C411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3⋊Dic9—41no—
99C9×Dic3—22no—
100C9×Dic3—41no—
101C3×Dic9C22⋊C412yes—
102C2×#239—91no—
103C2×#235—91no—
104C22×C3×C8—91no—
105#926—91no—
106C2×D3×C8—31no—
107#892—91no—
108C3×#103—31no—
109#835—91no—
110#821—91no—
111#821—91no—
112#821—91no—
113#821—181no—
114#816—91no—
115#816D911yescharacteristic, complemented
116#809—91no—
117C22×C3⋊D3—61no—
118C22×C3×D3—31no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—61no—
121C4⋊(C3⋊D3)—32no—
122C4⋊(C3⋊D3)—61no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—31no—
126C3×C4×D3—32no—
127C3×C4×D3—32no—
128C3×Dic6—21no—
129C3×Dic6C4×D312yes—
130#626—64no—
131#624—64no—
132C22×D9—61no—
133C22×D9—121no—
134C22×D9D1211yescharacteristic
135C32×C8—21no—
136C3×C3⋊C8—61no—
137Q8×C9—32no—
138Q8×C9—61no—
139C2×C4×C9—21no—
140C2×C4×C9—61no—
141C2×C4×C9—61no—
142C2×C4×C9C22×D311yescharacteristic
143C2×Dic9—61no—
144C2×Dic9—121no—
145C2×Dic9D1211yescharacteristic
146D36—32no—
147D36—62no—
148D36—64no—
149D36—61no—
150D36—121no—
151C4×D9—64no—
152C4×D9C4×D312yes—
153C4×D9#6712yes—
154C8×C9—21no—
155C8×C9—41no—
156C9⋊C8—21no—
157#416—271noSylow
158C2×C3×C9C2×C22⋊C411yescharacteristic
159C2×C3×C9#13511yescharacteristic
160C2×C3×C9#13511yescharacteristic
161C3⋊D9—42no—
162C3×D9—22no—
163C2×C3×Q8—31no—
164C22×C3×C4—91no—
165C2×#67—181no—
166#239—94no—
167#239—182no—
168#235—94no—
169#235—182no—
170C2×D12—91no—
171C2×D12—91no—
172C2×D12—181no—
173C2×D12—181no—
174C2×C4×D3—31no—
175C2×C4×D3—181no—
176C2×C4×D3—181no—
177C2×Dic6C2×D911yescharacteristic
178C2×C3×C8—21no—
179C2×C3×C8—91no—
180C2×C3×C8—182no—
181C2×C3×C8C2×D911yescharacteristic
182C2×C3×C8D3211yescharacteristic
183C3×C4⋊C4—31no—
184#217—181no—
185(C2×D3)⋊C4—181no—
186C4⋊Dic3—91no—
187C4⋊Dic3—91no—
188C4⋊Dic3—181no—
189C4⋊Dic3C2×D911yescharacteristic
190C4×Dic3—181no—
191C4×Dic3—181no—
192C2×C3⋊C8—31no—
193D3×C8—62no—
194C22×C32D3×D411yescharacteristic
195C2×C3⋊D3—62no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—121no—
198C2×C3×D3—32no—
199C2×C3×D3—61no—
200C2×C3×D3—61no—
201C32×C4C2×C4×D311yescharacteristic
202C32×C4D3×D411yescharacteristic
203C3⋊Dic3—121no—
204C3×Dic3—22no—
205C3×Dic3—32no—
206C3×Dic3—41no—
207C22×C9—21no—
208C22×C9C2×D1211yescharacteristic
209C2×D9—21no—
210C2×D9—21no—
211C2×D9—62no—
212C2×D9—122no—
213C2×D9—121no—
214C2×D9—121no—
215C2×D9—241no—
216C2×D9—241no—
217C2×D9(C2×D3)⋊C412yes—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—62no—
221C4×C9—121no—
222C4×C9C2×C4×D311yescharacteristic
223C4×C9C2×#6711yescharacteristic
224Dic9—121no—
225Dic9—241no—
226Dic9(C2×D3)⋊C412yes—
227C2×#41—271no—
228C22×C8—91no—
229#117—271no—
230#103—31no—
231#103—271no—
232#102—271no—
233#102—271no—
234C3×C9#41611yescharacteristic, complemented, Sylow
235C23×C3—91no—
236C22×D3—31no—
237C22×D3—181no—
238C22×D3—181no—
239C22×D3—361no—
240C3×Q8—32no—
241C3×Q8—61no—
242C2×C3×C4—21no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—91no—
246C2×C3×C4—92no—
247C2×C3×C4—92no—
248C2×C3×C4C22×D911yescharacteristic
249C2×C3×C4C2×D3211yescharacteristic
250#67—184no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—31no—
254C2×Dic3—181no—
255C2×Dic3—181no—
256C2×Dic3—361no—
257D12—92no—
258D12—92no—
259D12—182no—
260D12—184no—
261D12—181no—
262D12—181no—
263D12—361no—
264C4×D3—32no—
265C4×D3—32no—
266C4×D3—184no—
267C4×D3—184no—
268Dic6—21no—
269Dic6C4×D912yes—
270C3×C8—21no—
271C3×C8—21no—
272C3×C8—41no—
273C3×C8—181no—
274C3⋊C8—61no—
275C2×C32D3×C22⋊C411yescharacteristic
276C2×C32#91611yescharacteristic
277C2×C32#91911yescharacteristic
278C3⋊D3—122no—
279C3×D3—62no—
280C2×C9—21no—
281C2×C9—21no—
282C2×C9—21no—
283C2×C9#90411yescharacteristic
284C2×C9#90411yescharacteristic
285C2×C9C2×(C2×D3)⋊C411yescharacteristic
286D9—22no—
287D9—122no—
288D9—242no—
289#41—274no—
290#41—542no—
291C2×Q8—31no—
292C2×D4—271no—
293C2×D4—541no—
294C22×C4—91no—
295C22×C4—541no—
296C2×C8—91no—
297C2×C8—182no—
298C2×C8D3×D911yescharacteristic
299C4⋊C4—31no—
300C4⋊C4—271no—
301C22⋊C4—541no—
302C42—541no—
303C22×C3—21no—
304C22×C3—92no—
305C22×C3—181no—
306C22×C3—181no—
307C22×C3D4×D911yescharacteristic
308C22×C3D3×D1211yescharacteristic
309C2×D3—32no—
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—182no—
313C2×D3—182no—
314C2×D3—362no—
315C2×D3—361no—
316C2×D3—361no—
317C2×D3—361no—
318C2×D3—361no—
319C2×D3—721no—
320C2×D3—721no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—62no—
324C3×C4—92no—
325C3×C4—121no—
326C3×C4C2×C4×D911yescharacteristic
327C3×C4D4×D911yescharacteristic
328C3×C4C4×D3211yescharacteristic
329C3×C4D3×#6711yescharacteristic
330Dic3—22no—
331Dic3—32no—
332Dic3—41no—
333Dic3—361no—
334Dic3—361no—
335Dic3—721no—
336C32#480411yescharacteristic
337C9—21no—
338C9#511311yescharacteristic, complemented
339C23—91no—
340C23—541no—
341Q8—32no—
342Q8—61no—
343D4—272no—
344D4—544no—
345D4—541no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—91no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—544no—
352C2×C4—541no—
353C2×C4C2×D3×D911yescharacteristic
354C8—21no—
355C8—181no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—182no—
360C2×C3C22⋊C4×D911yescharacteristic
361C2×C3#6347811yescharacteristic
362C2×C3#6348111yescharacteristic
363C2×C3#6380611yescharacteristic
364C2×C3#6380811yescharacteristic
365C2×C3D3×(C2×D3)⋊C411yescharacteristic
366D3—62no—
367D3—362no—
368D3—362no—
369D3—722no—
370C22—92no—
371C22—181no—
372C22—181no—
373C22—542no—
374C22—1081no—
375C22—1081no—
376C22D9×D1211yescharacteristic
377C4—62no—
378C4—92no—
379C4—121no—
380C4—1081no—
381C4C4×D3×D911yescharacteristic
382C4D9×#6711yescharacteristic
383C3—21no—
384C3#1058848411yescharacteristic, complemented
385C3#1059069011yescharacteristic
386C2—182no—
387C2—1082no—
388C2#1172082111yescharacteristic
389C2#1172083011yescharacteristic
390C2D9×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22D9×D12
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88635C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#416
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×D9C2semidirect
3. #11720907C2semidirect
4. #11720543C2semidirect
5. C9×#816C2semidirect
6. #11720298C2semidirect
7. #11720267C2semidirect
16. #88635C4semidirect
32. C2×C8×D9D3semidirect
115. #816D9semidirect
234. C3×C9#416semidirect
338. C9#5113semidirect
384. C3#10588484semidirect

Maximal subgroups

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

subgroupindex
C2×#886352
C2×C3×C8×D92
#117209072
#117205432
C9×#8162
#117202982
#117202672
#105906903
#105884843