Finite Group Databasebeta

← #13139125

Subgroups — #13139125

Order 1728.

488 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88641C211yesmaximal, characteristic
2D3×C9×M16C211yesmaximal, characteristic, complemented
3D3×#3667C211yesmaximal, characteristic, complemented
4#11720556C211yesmaximal, characteristic, complemented
5C3×#63383C211yesmaximal, characteristic, complemented
6#11720290C211yesmaximal, characteristic, complemented
7#11720287C211yesmaximal, characteristic, complemented
8#10591006—31nomaximal
9#10588814—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C4×D3×C9C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×#6194C412yes—
14C2×Dic3×D9C412yes—
15#88641—21no—
16#88641—21no—
17#88641—21no—
18#88641—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×#3795—31no—
28C3×D3×M16D311yescharacteristic
29D3×#208—31no—
30C2×#3697—31no—
31#63651—31no—
32C3×#823—31no—
33#63561—31no—
34#63558—31no—
35C2×C9×M16—31no—
36C2×#3667—31no—
37#63383—31no—
38#63383D311yescharacteristic, complemented
39#63382—31no—
40#63382—31no—
41#63382—61no—
42C22×D3×C9C2×C411yescharacteristic
43C22×C3×D9C2×C412yes—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9C2×C412yes—
46#6226—21no—
47#6226—21no—
48C2×C9×Dic3C2×C411yescharacteristic
49C4×D3×C9D411yescharacteristic
50C4×D3×C9D411yescharacteristic
51C4×D3×C9D411yescharacteristic
52C4×D3×C9D411yescharacteristic
53C3×D36—21no—
54C3×D36—21no—
55#6194—22no—
56#6194—22no—
57Dic3×D9—22no—
58Dic3×D9—22no—
59C3×C8×C9—21no—
60C3⋊(C9⋊C8)—21no—
61C9×C3⋊C8—21no—
62C3×C9⋊C8—21no—
63#5132—91no—
64#4745—91no—
65C2×#634—31no—
66C2×C3×D12—31no—
67C2×C3×C4×D3C2×D311yescharacteristic
68C2×#625—32no—
69C2×D3×Dic3—32no—
70#3795—61no—
71#3795—61no—
72#3795—61no—
73#3795—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×D36C2×D311yescharacteristic
89C2×C4×D9—32no—
90C2×Dic18—31no—
91C2×Dic18—61no—
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×D3×C9—41no—
104C2×D3×C9C22⋊C411yescharacteristic
105C2×D3×C9C22⋊C411yescharacteristic
106C2×C3×D9—22no—
107C2×C3×D9—42no—
108C3×C4×C9C2×D411yescharacteristic
109C3×C4×C9C2×D411yescharacteristic
110C3⋊Dic9—22no—
111C9×Dic3C22⋊C411yescharacteristic
112C9×Dic3C22⋊C411yescharacteristic
113C2×#237—91no—
114C2×#235—91no—
115C2×C3×M16—31no—
116C2×#208—91no—
117D3×M16—91no—
118D3×M16D911yescharacteristic, complemented
119C3×#100—91no—
120#833—91no—
121#824—91no—
122#824—91no—
123#824—91no—
124#824—91no—
125#824—181no—
126#823—31no—
127#823—91no—
128C22×C3×D3—32no—
129C22×C3×D3C4×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—32no—
132#634—61no—
133#634—61no—
134C2×C3×Dic3C4×D311yescharacteristic
135C3×D12—61no—
136C3×D12—61no—
137C3×C4×D3D1211yescharacteristic
138C3×C4×D3D1211yescharacteristic
139C3×C4×D3#6711yescharacteristic
140C3×C4×D3#6711yescharacteristic
141#625—62no—
142#625—62no—
143D3×Dic3—62no—
144D3×Dic3—62no—
145C23×C9—31no—
146C22×D9C4×D312yes—
147C32×C8—21no—
148C3⋊(C3⋊C8)—61no—
149C3×C3⋊C8—21no—
150C3×C3⋊C8—61no—
151C2×C4×C9—21no—
152C2×C4×C9—31no—
153C2×C4×C9—31no—
154C2×C4×C9—31no—
155C2×C4×C9—31no—
156C2×C4×C9—31no—
157C2×C4×C9C22×D311yescharacteristic
158#611—62no—
159#611—62no—
160C2×Dic9—32no—
161C2×Dic9—62no—
162D36—21no—
163D36—21no—
164C4×D9—62no—
165C4×D9—62no—
166Dic18—61no—
167Dic18—61no—
168Dic18—121no—
169Dic18—121no—
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
179D3×C9—21no—
180D3×C9—21no—
181C3×D9—42no—
182C2×C3×D4—91no—
183C22×C3×C4—31no—
184C2×#67—92no—
185C2×#67—92no—
186C22×Dic3—92no—
187#237—181no—
188#237—181no—
189#237—181no—
190#237—181no—
191#235—181no—
192#235—181no—
193#235—181no—
194#235—181no—
195C2×D12—31no—
196C2×C4×D3—92no—
197C2×C4×D3C2×D911yescharacteristic
198C2×Dic6—91no—
199C2×Dic6—91no—
200C2×Dic6—181no—
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—41no—
221C2×C3×D3—62no—
222C2×C3×D3—122no—
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—22no—
236C2×D9—42no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—31no—
240C4×C9—31no—
241C4×C9D3×D411yescharacteristic
242C4×C9D3×D411yescharacteristic
243Dic9—62no—
244Dic9—122no—
245C2×#41—271no—
246C2×M16—31no—
247C2×M16—271no—
248#101—271no—
249#101—271no—
250#100—91no—
251#100—271no—
252C3×C9#41311yescharacteristic, complemented, Sylow
253C23×C3—31no—
254C23×C3—92no—
255C22×D3—32no—
256C22×D3C4×D911yescharacteristic
257C3×D4—181no—
258C3×D4—181no—
259C2×C3×C4—21no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4C22×D911yescharacteristic
266C2×C3×C4C2×D3211yescharacteristic
267#67—182no—
268#67—182no—
269#67—182no—
270#67—182no—
271C2×Dic3—92no—
272C2×Dic3—92no—
273C2×Dic3—182no—
274C2×Dic3—182no—
275C2×Dic3—182no—
276C2×Dic3C4×D911yescharacteristic
277D12—61no—
278D12—61no—
279C4×D3—182no—
280C4×D3—182no—
281C4×D3D3611yescharacteristic
282C4×D3D3611yescharacteristic
283C4×D3#61111yescharacteristic
284C4×D3#61111yescharacteristic
285Dic6—181no—
286Dic6—181no—
287Dic6—181no—
288Dic6—181no—
289Dic6—361no—
290Dic6—361no—
291C3×C8—21no—
292C3×C8—21no—
293C3×C8—41no—
294C3×C8—61no—
295C3×C8—181no—
296C3⋊C8—21no—
297C3⋊C8—61no—
298C3⋊C8—181no—
299C3⋊C8—181no—
300C3⋊C8—361no—
301C2×C32—21no—
302C2×C32C2×(C2×D3)⋊C411yescharacteristic
303C3×D3—21no—
304C3×D3—21no—
305C3×D3—122no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—41no—
309C2×C9—61no—
310C2×C9—61no—
311C2×C9D3×C22⋊C411yescharacteristic
312D9—42no—
313#41—541no—
314#41—541no—
315#41—541no—
316#41—541no—
317C2×Q8—271no—
318C2×D4—91no—
319C2×D4—272no—
320C22×C4—31no—
321C22×C4—272no—
322M16—31no—
323M16—61no—
324M16—91no—
325M16—271no—
326M16—541no—
327M16D3×D911yescharacteristic
328C2×C8—61no—
329C2×C8—541no—
330C22×C3—21no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—121no—
334C22×C3—182no—
335C22×C3—362no—
336C22×C3C2×C4×D911yescharacteristic
337C22×C3C4×D3211yescharacteristic
338C2×D3—41no—
339C2×D3—62no—
340C2×D3—122no—
341C2×D3(C2×D9)⋊C411yescharacteristic
342C2×D3(C2×D9)⋊C411yescharacteristic
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4C2×D3611yescharacteristic
348C3×C4C2×#61111yescharacteristic
349C3×C4D3×D1211yescharacteristic
350C3×C4D3×#6711yescharacteristic
351Dic3—182no—
352Dic3—182no—
353Dic3—362no—
354Dic3(C2×D9)⋊C411yescharacteristic
355Dic3(C2×D9)⋊C411yescharacteristic
356C32#513211yescharacteristic
357C9—21no—
358C9#474511yescharacteristic, complemented
359C23—31no—
360C23—92no—
361Q8—541no—
362Q8—541no—
363D4—181no—
364D4—181no—
365D4—542no—
366D4—542no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—272no—
373C2×C4—542no—
374C2×C4—542no—
375C2×C4C2×D3×D911yescharacteristic
376C8—21no—
377C8—61no—
378C8—181no—
379C8—541no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—41no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—362no—
387C2×C3C2×(C2×D9)⋊C411yescharacteristic
388C2×C3D3×(C2×D3)⋊C411yescharacteristic
389D3—21no—
390D3—21no—
391D3—122no—
392C22—31no—
393C22—31no—
394C22—121no—
395C22—182no—
396C22—362no—
397C22C4×D3×D911yescharacteristic
398C4—31no—
399C4—31no—
400C4—542no—
401C4D3×D3611yescharacteristic
402C4D3×#61111yescharacteristic
403C3—21no—
404C3#1058881411yescharacteristic, complemented
405C3#1059100611yescharacteristic
406C2—21no—
407C2—61no—
408C2—61no—
409C2—362no—
410C2D3×(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×#88641C2
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. #11720556C2semidirect
5. C3×#63383C2semidirect
6. #11720290C2semidirect
7. #11720287C2semidirect
38. #63383D3semidirect
118. D3×M16D9semidirect
252. C3×C9#413semidirect
358. C9#4745semidirect
404. C3#10588814semidirect

Maximal subgroups

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

subgroupindex
C2×#886412
D3×C9×M162
D3×#36672
#117205562
C3×#633832
#117202902
#117202872
#105910063
#105888143