Finite Group Databasebeta

← #13138694

Subgroups — #13138694

Order 1728.

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

normal subgroups only

This group has 492 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×D9×Dic6C211yesmaximal, characteristic
2C2×C3×#3663C211yesmaximal, characteristic
3#11720907C211yesmaximal, characteristic
4#11720539C211yesmaximal, characteristic, complemented
5C9×#816C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10590948—31nomaximal
9#10588482—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×Dic3×D9—21no—
14C2×#6189—21no—
15D9×Dic6—22no—
16D9×Dic6C414yescomplemented
17C2×C3×C8×C9C2211yescharacteristic
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C9×C4⋊Dic3C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C3×#3663—22no—
22(C2×D9)⋊Dic3—21no—
23Dic3×Dic9—21no—
24C2×D3×Dic6—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26#63841—31no—
27C2×Q8×D9—31no—
28#63643—31no—
29C3×#816D311yescharacteristic
30#63567—31no—
31#63566—31no—
32C2×#3663D311yescharacteristic
33#63454—31no—
34C9×#103—31no—
35#63395—31no—
36#63376—31no—
37#63376—61no—
38#63368—31no—
39C22×C3×D9D411yescharacteristic
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9—21no—
43#6226—21no—
44#6226C2×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—
52Dic3×D9—24no—
53#6189—24no—
54C3×C8×C9—21no—
55C3×C9⋊C8—21no—
56#5120—91no—
57#4802—91no—
58C2×#634—31no—
59C2×C3×C4×D3—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61C2×#627—61no—
62C2×D3×Dic3—61no—
63D3×Dic6—34no—
64D3×Dic6—62no—
65C2×C32×C8C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)—31no—
67C3×C4⋊Dic3C2×D311yescharacteristic
68C2×C3×C3⋊C8—31no—
69C3×ΓL(1,𝔽25)—62no—
70(C2×D3)⋊Dic3—61no—
71Dic32—61no—
72C2×Q8×C9—31no—
73Q8×D9—34no—
74Q8×D9—62no—
75C2×C4×D9—61no—
76C2×C4×D9C2×D311yescharacteristic
77C2×Dic18—31no—
78C2×Dic18—61no—
79C2×Dic18—61no—
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
88#3663—22no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C2×C3×D9C22⋊C412yes—
93C3×C4×C9C22×C411yescharacteristic
94C3×C4×C9C2×D411yescharacteristic
95C3⋊Dic9—22no—
96C3⋊Dic9—41no—
97C9×Dic3—22no—
98C9×Dic3—41no—
99C3×Dic9C22⋊C412yes—
100C2×D3×Q8—91no—
101C22×Dic6—91no—
102C2×C3×M16—91no—
103#926—91no—
104C2×ΓL(1,𝔽25)—31no—
105#892—91no—
106C3×#103—31no—
107#835—91no—
108#816—91no—
109#816—91no—
110#816—91no—
111#816—91no—
112#816—181no—
113#816D911yescharacteristic, complemented
114#810—91no—
115C22×C3×D3—31no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—61no—
119#634—32no—
120#634—61no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—31no—
124C3×C4×D3—32no—
125C3×C4×D3—32no—
126C3×Dic6—21no—
127C3×Dic6C4×D312yes—
128#627—64no—
129D3×Dic3—64no—
130C22×D9D1211yescharacteristic
131C32×C8—21no—
132C3×C3⋊C8—61no—
133Q8×C9—32no—
134Q8×C9—61no—
135C2×C4×C9—21no—
136C2×C4×C9—61no—
137C2×C4×C9—61no—
138C2×C4×C9C22×D311yescharacteristic
139C2×Dic9—61no—
140C2×Dic9—61no—
141C2×Dic9—121no—
142C2×Dic9—121no—
143C2×Dic9D1211yescharacteristic
144C4×D9—64no—
145C4×D9C4×D312yes—
146C4×D9#6712yes—
147Dic18—32no—
148Dic18—62no—
149Dic18—61no—
150Dic18—64no—
151Dic18—121no—
152C8×C9—21no—
153C8×C9—41no—
154C9⋊C8—21no—
155#419—271noSylow
156C2×C3×C9C2×C22⋊C411yescharacteristic
157C2×C3×C9#13711yescharacteristic
158C2×C3×C9#13711yescharacteristic
159C3×D9—22no—
160C2×C3×Q8—31no—
161C22×C3×C4—91no—
162C22×Dic3—181no—
163D3×Q8—94no—
164D3×Q8—182no—
165C2×C4×D3—31no—
166C2×C4×D3—181no—
167C2×Dic6—91no—
168C2×Dic6—91no—
169C2×Dic6—94no—
170C2×Dic6—181no—
171C2×Dic6—182no—
172C2×Dic6—181no—
173C2×Dic6—181no—
174C2×Dic6C2×D911yescharacteristic
175C3×M16—182no—
176C2×C3×C8—21no—
177C2×C3×C8—91no—
178C2×C3×C8C2×D911yescharacteristic
179C2×C3×C8D3211yescharacteristic
180C3×C4⋊C4—31no—
181#217—181no—
182(C2×D3)⋊C4—181no—
183C4⋊Dic3—91no—
184C4⋊Dic3—91no—
185C4⋊Dic3—181no—
186C4⋊Dic3C2×D911yescharacteristic
187C4×Dic3—181no—
188C4×Dic3—181no—
189C2×C3⋊C8—31no—
190ΓL(1,𝔽25)—62no—
191C22×C32D3×D411yescharacteristic
192C2×C3×D3—32no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C32×C4C2×C4×D311yescharacteristic
196C32×C4D3×D411yescharacteristic
197C3⋊Dic3—62no—
198C3⋊Dic3—121no—
199C3×Dic3—22no—
200C3×Dic3—32no—
201C3×Dic3—41no—
202C22×C9—21no—
203C22×C9C2×D1211yescharacteristic
204C2×D9—21no—
205C2×D9—21no—
206C2×D9(C2×D3)⋊C412yes—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—62no—
210C4×C9—121no—
211C4×C9C2×C4×D311yescharacteristic
212C4×C9C2×#6711yescharacteristic
213Dic9—62no—
214Dic9—122no—
215Dic9—121no—
216Dic9—241no—
217Dic9(C2×D3)⋊C412yes—
218C22×Q8—271no—
219C2×M16—91no—
220#117—271no—
221#103—31no—
222#103—271no—
223#103—271no—
224#103—271no—
225C3×C9#41911yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C22×D3—31no—
228C3×Q8—32no—
229C3×Q8—61no—
230C2×C3×C4—21no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—91no—
234C2×C3×C4—92no—
235C2×C3×C4—92no—
236C2×C3×C4C22×D911yescharacteristic
237C2×C3×C4C2×D3211yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—31no—
241C2×Dic3—184no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244C2×Dic3—181no—
245C2×Dic3—181no—
246C2×Dic3—361no—
247C2×Dic3—361no—
248C4×D3—32no—
249C4×D3—32no—
250C4×D3—184no—
251Dic6—21no—
252Dic6—92no—
253Dic6—92no—
254Dic6—182no—
255Dic6—184no—
256Dic6—184no—
257Dic6—181no—
258Dic6—181no—
259Dic6—361no—
260Dic6C4×D912yes—
261C3×C8—21no—
262C3×C8—21no—
263C3×C8—41no—
264C3×C8—181no—
265C3⋊C8—61no—
266C2×C32D3×C22⋊C411yescharacteristic
267C2×C32#91511yescharacteristic
268C2×C32#91811yescharacteristic
269C3×D3—62no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9#90911yescharacteristic
274C2×C9#90911yescharacteristic
275C2×C9C2×(C2×D3)⋊C411yescharacteristic
276D9—22no—
277C2×Q8—31no—
278C2×Q8—274no—
279C2×Q8—271no—
280C2×Q8—541no—
281C2×Q8—542no—
282C22×C4—91no—
283C22×C4—541no—
284M16—182no—
285C2×C8—91no—
286C2×C8D3×D911yescharacteristic
287C4⋊C4—31no—
288C4⋊C4—271no—
289C22⋊C4—541no—
290C42—541no—
291C22×C3—21no—
292C22×C3—92no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3D4×D911yescharacteristic
296C22×C3D3×D1211yescharacteristic
297C2×D3—32no—
298C2×D3—61no—
299C2×D3—61no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—62no—
303C3×C4—92no—
304C3×C4—121no—
305C3×C4C2×C4×D911yescharacteristic
306C3×C4D4×D911yescharacteristic
307C3×C4C4×D3211yescharacteristic
308C3×C4D3×#6711yescharacteristic
309Dic3—22no—
310Dic3—32no—
311Dic3—41no—
312Dic3—182no—
313Dic3—182no—
314Dic3—362no—
315Dic3—361no—
316Dic3—361no—
317Dic3—721no—
318C32#480211yescharacteristic
319C9—21no—
320C9#512011yescharacteristic, complemented
321C23—91no—
322Q8—32no—
323Q8—61no—
324Q8—272no—
325Q8—544no—
326Q8—541no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—91no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—544no—
333C2×C4—541no—
334C2×C4—541no—
335C2×C4C2×D3×D911yescharacteristic
336C8—21no—
337C8—181no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—182no—
342C2×C3C22⋊C4×D911yescharacteristic
343C2×C3#6347711yescharacteristic
344C2×C3#6348011yescharacteristic
345C2×C3#6380011yescharacteristic
346C2×C3#6380111yescharacteristic
347C2×C3D3×(C2×D3)⋊C411yescharacteristic
348D3—62no—
349C22—92no—
350C22—181no—
351C22—181no—
352C22D9×D1211yescharacteristic
353C4—62no—
354C4—92no—
355C4—121no—
356C4—542no—
357C4—1081no—
358C4C4×D3×D911yescharacteristic
359C4D9×#6711yescharacteristic
360C3—21no—
361C3#1058848211yescharacteristic, complemented
362C3#1059094811yescharacteristic
363C2—182no—
364C2#1172082711yescharacteristic
365C2#1172082811yescharacteristic
366C2D9×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22D9×D12
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#419
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
4. #11720539C2semidirect
5. C9×#816C2semidirect
6. #11720298C2semidirect
7. #11720269C2semidirect
16. D9×Dic6C4semidirect
113. #816D9semidirect
225. C3×C9#419semidirect
320. C9#5120semidirect
361. C3#10588482semidirect

Maximal subgroups

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

subgroupindex
C2×D9×Dic62
C2×C3×#36632
#117209072
#117205392
C9×#8162
#117202982
#117202692
#105909483
#105884823