Finite Group Databasebeta

← #13139126

Subgroups — #13139126

Order 1728.

496 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88642C211yesmaximal, characteristic
2D3×C9×M16C211yesmaximal, characteristic, complemented
3D3×#3667C211yesmaximal, characteristic, complemented
4#11720557C211yesmaximal, characteristic, complemented
5C3×#63382C211yesmaximal, characteristic, complemented
6#11720289C211yesmaximal, characteristic, complemented
7#11720288C211yesmaximal, characteristic, complemented
8#10591007—31nomaximal
9#10588814—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×D3×C9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×#6195C412yes—
14C2×(C3⋊D9)⋊C4C412yes—
15#88642—21no—
16#88642—21no—
17#88642—21no—
18#88642—21no—
19C3×C9×M16C2211yescharacteristic
20#88532C2211yescharacteristic
21C9×#208C2211yescharacteristic
22C3×#3667C2211yescharacteristic
23C9×ΓL(1,𝔽25)—21no—
24D3×C8×C9—21no—
25#88420—21no—
26D3×C9⋊C8—21no—
27C2×#3799—31no—
28C3×D3×M16D311yescharacteristic
29D3×#208—31no—
30C2×#3697—31no—
31#63650—31no—
32C3×#824—31no—
33#63560—31no—
34#63559—31no—
35C2×C9×M16—31no—
36C2×#3667—31no—
37#63383—31no—
38#63383—31no—
39#63383—61no—
40#63382—31no—
41#63382D311yescharacteristic, complemented
42C22×C3⋊D9C2×C412yes—
43C22×D3×C9C2×C411yescharacteristic
44C2×C3×C4×C9C2311yescharacteristic
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—21no—
47C2×C9×Dic3C2×C411yescharacteristic
48C2×C3×Dic9C2×C412yes—
49C4×D3×C9D411yescharacteristic
50C4×D3×C9D411yescharacteristic
51C4×D3×C9D411yescharacteristic
52C4×D3×C9D411yescharacteristic
53C3×Dic18—21no—
54C3×Dic18—21no—
55#6195—22no—
56#6195—22no—
57(C3⋊D9)⋊C4—22no—
58(C3⋊D9)⋊C4—22no—
59C3×C8×C9—21no—
60C3⋊(C9⋊C8)—21no—
61C9×C3⋊C8—21no—
62C3×C9⋊C8—21no—
63#5132—91no—
64#4751—91no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C3×C4×D3C2×D311yescharacteristic
67C2×C3×Dic6—31no—
68C2×#626—32no—
69C2×#624—32no—
70#3799—61no—
71#3799—61no—
72#3799—61no—
73#3799—61no—
74C32×M16C2×D311yescharacteristic
75#3748—31no—
76C3×#208—31no—
77C3×#208C2×D311yescharacteristic
78C3×ΓL(1,𝔽25)—21no—
79C3×D3×C8—21no—
80#3711—61no—
81D3×C3⋊C8—61no—
82C22×C4×C9—31no—
83C2×#611—32no—
84#3697—61no—
85#3697—61no—
86#3697—61no—
87#3697—61no—
88C2×D36—31no—
89C2×D36—61no—
90C2×C4×D9—32no—
91C2×Dic18C2×D311yescharacteristic
92C9×M16—21no—
93C9×M16—31no—
94C9×M16—61no—
95C9×M16C2×D311yescharacteristic
96C2×C8×C9—61no—
97#3667—31no—
98#3667—61no—
99#3667—61no—
100#3667C2×D311yescharacteristic
101C2×C9⋊C8—61no—
102C22×C3×C9C22×C411yescharacteristic
103C2×C3⋊D9—22no—
104C2×C3⋊D9—42no—
105C2×D3×C9—41no—
106C2×D3×C9C22⋊C411yescharacteristic
107C2×D3×C9C22⋊C411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3×C4×C9C2×D411yescharacteristic
110C9×Dic3C22⋊C411yescharacteristic
111C9×Dic3C22⋊C411yescharacteristic
112C3×Dic9—22no—
113C2×#239—91no—
114C2×#235—91no—
115C2×C3×M16—31no—
116C2×#208—91no—
117D3×M16—91no—
118D3×M16D911yescharacteristic, complemented
119C3×#101—91no—
120#836—91no—
121#824—31no—
122#824—91no—
123#823—91no—
124#823—91no—
125#823—91no—
126#823—91no—
127#823—181no—
128C22×C3⋊D3—32no—
129C22×C3×D3C4×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—61no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3C4×D311yescharacteristic
135C3×C4×D3D1211yescharacteristic
136C3×C4×D3D1211yescharacteristic
137C3×C4×D3#6711yescharacteristic
138C3×C4×D3#6711yescharacteristic
139C3×Dic6—61no—
140C3×Dic6—61no—
141#626—62no—
142#626—62no—
143#624—62no—
144#624—62no—
145C23×C9—31no—
146C22×D9—32no—
147C22×D9—62no—
148C32×C8—21no—
149C3⋊(C3⋊C8)—61no—
150C3×C3⋊C8—21no—
151C3×C3⋊C8—61no—
152C2×C4×C9—21no—
153C2×C4×C9—31no—
154C2×C4×C9—31no—
155C2×C4×C9—31no—
156C2×C4×C9—31no—
157C2×C4×C9—31no—
158C2×C4×C9C22×D311yescharacteristic
159#611—62no—
160#611—62no—
161C2×Dic9C4×D312yes—
162D36—61no—
163D36—61no—
164D36—121no—
165D36—121no—
166C4×D9—62no—
167C4×D9—62no—
168Dic18—21no—
169Dic18—21no—
170C8×C9—21no—
171C8×C9—41no—
172C8×C9—61no—
173C9⋊C8—21no—
174C9⋊C8—61no—
175C9⋊C8—121no—
176#413—271noSylow
177C2×C3×C9—21no—
178C2×C3×C9C2×C22⋊C411yescharacteristic
179C3⋊D9—42no—
180D3×C9—21no—
181D3×C9—21no—
182C2×C3×Q8—91no—
183C22×C3×C4—31no—
184C2×#67—92no—
185#239—181no—
186#239—181no—
187#239—181no—
188#239—181no—
189#235—181no—
190#235—181no—
191#235—181no—
192#235—181no—
193C2×D12—92no—
194C2×D12—91no—
195C2×D12—91no—
196C2×D12—181no—
197C2×C4×D3—92no—
198C2×C4×D3—92no—
199C2×C4×D3C2×D911yescharacteristic
200C2×Dic6—31no—
201C3×M16—21no—
202C3×M16—31no—
203C3×M16—61no—
204C3×M16—91no—
205C3×M16C2×D911yescharacteristic
206C3×M16D3211yescharacteristic
207C2×C3×C8—61no—
208#208—31no—
209#208—91no—
210#208—91no—
211#208—181no—
212#208—181no—
213#208C2×D911yescharacteristic
214C2×C3⋊C8—181no—
215ΓL(1,𝔽25)—21no—
216ΓL(1,𝔽25)—181no—
217D3×C8—21no—
218D3×C8—181no—
219C22×C32C2×C4×D311yescharacteristic
220C2×C3⋊D3—62no—
221C2×C3⋊D3—122no—
222C2×C3×D3—41no—
223C2×C3×D3(C2×D3)⋊C411yescharacteristic
224C2×C3×D3(C2×D3)⋊C411yescharacteristic
225C32×C4C2×D1211yescharacteristic
226C32×C4C2×#6711yescharacteristic
227C3×Dic3—62no—
228C3×Dic3(C2×D3)⋊C411yescharacteristic
229C3×Dic3(C2×D3)⋊C411yescharacteristic
230C22×C9—21no—
231C22×C9—31no—
232C22×C9—31no—
233C22×C9—121no—
234C22×C9C2×C4×D311yescharacteristic
235C2×D9—62no—
236C2×D9—122no—
237C2×D9—122no—
238C2×D9—242no—
239C4×C9—21no—
240C4×C9—21no—
241C4×C9—31no—
242C4×C9—31no—
243C4×C9D3×D411yescharacteristic
244C4×C9D3×D411yescharacteristic
245Dic9—22no—
246C2×#41—271no—
247C2×M16—31no—
248C2×M16—271no—
249#101—91no—
250#101—271no—
251#100—271no—
252#100—271no—
253C3×C9#41311yescharacteristic, complemented, Sylow
254C23×C3—31no—
255C22×D3—92no—
256C22×D3—92no—
257C22×D3—182no—
258C22×D3C4×D911yescharacteristic
259C3×Q8—181no—
260C3×Q8—181no—
261C2×C3×C4—21no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—92no—
268C2×C3×C4C22×D911yescharacteristic
269C2×C3×C4C2×D3211yescharacteristic
270#67—182no—
271#67—182no—
272C2×Dic3—32no—
273C2×Dic3C4×D911yescharacteristic
274D12—182no—
275D12—182no—
276D12—181no—
277D12—181no—
278D12—181no—
279D12—181no—
280D12—361no—
281D12—361no—
282C4×D3—182no—
283C4×D3—182no—
284C4×D3—182no—
285C4×D3—182no—
286C4×D3D3611yescharacteristic
287C4×D3D3611yescharacteristic
288C4×D3#61111yescharacteristic
289C4×D3#61111yescharacteristic
290Dic6—61no—
291Dic6—61no—
292C3×C8—21no—
293C3×C8—21no—
294C3×C8—41no—
295C3×C8—61no—
296C3×C8—181no—
297C3⋊C8—21no—
298C3⋊C8—61no—
299C3⋊C8—181no—
300C3⋊C8—181no—
301C3⋊C8—361no—
302C2×C32—21no—
303C2×C32C2×(C2×D3)⋊C411yescharacteristic
304C3⋊D3—122no—
305C3×D3—21no—
306C3×D3—21no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—41no—
310C2×C9—61no—
311C2×C9—61no—
312C2×C9D3×C22⋊C411yescharacteristic
313D9—122no—
314D9—242no—
315#41—541no—
316#41—541no—
317#41—541no—
318#41—541no—
319C2×Q8—91no—
320C2×D4—272no—
321C2×D4—271no—
322C22×C4—31no—
323C22×C4—272no—
324M16—31no—
325M16—61no—
326M16—91no—
327M16—271no—
328M16—541no—
329M16D3×D911yescharacteristic
330C2×C8—61no—
331C2×C8—541no—
332C22×C3—21no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—121no—
336C22×C3C2×C4×D911yescharacteristic
337C22×C3C4×D3211yescharacteristic
338C2×D3—41no—
339C2×D3—182no—
340C2×D3—182no—
341C2×D3—362no—
342C2×D3—362no—
343C2×D3—362no—
344C2×D3—722no—
345C2×D3(C2×D9)⋊C411yescharacteristic
346C2×D3(C2×D9)⋊C411yescharacteristic
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—182no—
352C3×C4C2×D3611yescharacteristic
353C3×C4C2×#61111yescharacteristic
354C3×C4D3×D1211yescharacteristic
355C3×C4D3×#6711yescharacteristic
356Dic3—62no—
357Dic3(C2×D9)⋊C411yescharacteristic
358Dic3(C2×D9)⋊C411yescharacteristic
359C32#513211yescharacteristic
360C9—21no—
361C9#475111yescharacteristic, complemented
362C23—31no—
363C23—272no—
364Q8—181no—
365Q8—181no—
366D4—542no—
367D4—542no—
368D4—541no—
369D4—541no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—92no—
376C2×C4—542no—
377C2×C4—542no—
378C2×C4C2×D3×D911yescharacteristic
379C8—21no—
380C8—61no—
381C8—181no—
382C8—541no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—41no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3C2×(C2×D9)⋊C411yescharacteristic
390C2×C3D3×(C2×D3)⋊C411yescharacteristic
391D3—21no—
392D3—21no—
393D3—362no—
394D3—362no—
395D3—722no—
396C22—31no—
397C22—31no—
398C22—121no—
399C22—542no—
400C22—1082no—
401C22C4×D3×D911yescharacteristic
402C4—31no—
403C4—31no—
404C4—182no—
405C4D3×D3611yescharacteristic
406C4D3×#61111yescharacteristic
407C3—21no—
408C3#1058881411yescharacteristic, complemented
409C3#1059100711yescharacteristic
410C2—21no—
411C2—61no—
412C2—61no—
413C2—1082no—
414C2D3×(C2×D9)⋊C411yescharacteristic
subgroupG/H
centerC2D3×(C2×D9)⋊C4
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32C2×(C2×D3)⋊C4
prime radicalC2×#88642C2
cosocleC2×C3×C4×C9C23
hypercenterM16D3×D9
hypocenterC3×C9#413
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. D3×C9×M16C2semidirect
3. D3×#3667C2semidirect
4. #11720557C2semidirect
5. C3×#63382C2semidirect
6. #11720289C2semidirect
7. #11720288C2semidirect
41. #63382D3semidirect
118. D3×M16D9semidirect
253. C3×C9#413semidirect
361. C9#4751semidirect
408. C3#10588814semidirect

Maximal subgroups

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

subgroupindex
C2×#886422
D3×C9×M162
D3×#36672
#117205572
C3×#633822
#117202892
#117202882
#105910073
#105888143