Finite Group Databasebeta

← #13139419

Subgroups — #13139419

Order 1728.

626 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic
2C3×#63488C211yesmaximal, characteristic, complemented
3C2×C9×#208C211yesmaximal, characteristic, complemented
4#11720292C214yesmaximal, complemented
5#10591113—31nomaximal
6#10588819—31nomaximal
7C22×C3×C4×C9C2211yescharacteristic
8C2×#6233—21no—
9#88736—21no—
10#88736—21no—
11#88736C412yescomplemented
12#88736C2212yes—
13C2×C4⋊(C3⋊D9)C411yescharacteristic, complemented
14C2×C4×C3⋊D9—21no—
15C2×#6226C411yescharacteristic, complemented
16C3×#3676—21no—
17C3×C4⋊Dic9—21no—
18C3×C4×Dic9C2212yescomplemented
19C9×#208—21no—
20C9×#208C2212yescomplemented
21C2×C9×C3⋊C8—21no—
22C2×#3828—31no—
23C3×#926—31no—
24C2×C3×#208D311yescharacteristic
25C2×#3697—31no—
26C2×#3697—61no—
27#63570—34no—
28C2×C9×M16—31no—
29#63488D311yescharacteristic, complemented
30#63384—34no—
31C23×C3×C9D411yescharacteristic
32C22×C3⋊D9—21no—
33C2×C3×C4×C9C2×C411yescharacteristic
34C2×C3×C4×C9C2×C411yescharacteristic
35C2×C3×C4×C9D411yescharacteristic
36C2×C3×C4×C9D411yescharacteristic
37C2×C3×C4×C9D411yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39#6233—22no—
40#6233—22no—
41C2×C3⋊Dic9—21no—
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)C2×C412yes—
44C4×C3⋊D9—22no—
45C4×C3⋊D9—22no—
46#6226—21no—
47#6226C2×C412yes—
48C2×C3×Dic9—22no—
49C9×C3⋊C8—22no—
50#5139—91no—
51#5005—91no—
52C22×C32×C4C2×D311yescharacteristic
53C2×#638—61no—
54#3828—32no—
55#3828—32no—
56#3828—61no—
57#3828—61no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4×C3⋊D3—61no—
60C2×#634—31no—
61C3×#217—61no—
62C3×C4⋊Dic3—61no—
63C3×C4×Dic3—32no—
64C3×#208—21no—
65C3×#208C2×D312yes—
66C2×C3×C3⋊C8—21no—
67C22×C4×C9—21no—
68C22×C4×C9C2×D311yescharacteristic
69C2×#611—62no—
70C2×#611—61no—
71#3697—32no—
72#3697—32no—
73#3697—62no—
74#3697—62no—
75#3697—62no—
76#3697—62no—
77#3697—61no—
78#3697—61no—
79C2×D36—31no—
80C2×D36—61no—
81C2×C4×D9—62no—
82C2×C4×D9—61no—
83C2×Dic18—31no—
84C2×Dic18—61no—
85C9×M16—32no—
86C9×M16—61no—
87C2×C8×C9—61no—
88#3676—21no—
89C4⋊Dic9—21no—
90C4×Dic9C2×D312yescomplemented
91C22×C3×C9—41no—
92C22×C3×C9C22⋊C411yescharacteristic
93C22×C3×C9C22⋊C411yescharacteristic
94C22×C3×C9C2×D411yescharacteristic
95C2×C3⋊D9—22no—
96C2×C3⋊D9—42no—
97C3×C4×C9C22⋊C411yescharacteristic
98C3×C4×C9C22⋊C411yescharacteristic
99C3×C4×C9C22×C411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3⋊Dic9—22no—
102C3×Dic9—42no—
103C2×#235—91no—
104C2×#235—91no—
105C2×#235—181no—
106C2×C3×M16—31no—
107C3×#117—91no—
108#926—31no—
109C2×#208D911yescharacteristic, complemented
110#825—94no—
111#805—94no—
112C23×C32D1211yescharacteristic
113C22×C3⋊D3—61no—
114C2×C32×C4C4×D311yescharacteristic
115C2×C32×C4C4×D311yescharacteristic
116C2×C32×C4D1211yescharacteristic
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120#638—62no—
121#638—62no—
122C2×C3⋊Dic3—61no—
123C4⋊(C3⋊D3)—32no—
124C4⋊(C3⋊D3)—61no—
125C4×C3⋊D3—62no—
126C4×C3⋊D3—62no—
127#634—32no—
128#634—61no—
129C2×C3×Dic3—62no—
130C23×C9—21no—
131C23×C9#6711yescharacteristic
132C22×D9—62no—
133C22×D9—61no—
134C3×C3⋊C8—22no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9C4×D311yescharacteristic
142C2×C4×C9C4×D311yescharacteristic
143C2×C4×C9D1211yescharacteristic
144C2×C4×C9D1211yescharacteristic
145C2×C4×C9#6711yescharacteristic
146C2×C4×C9C22×D311yescharacteristic
147#611—64no—
148#611—64no—
149#611—62no—
150#611—62no—
151C2×Dic9—22no—
152C2×Dic9—62no—
153C2×Dic9—61no—
154D36—32no—
155D36—62no—
156D36—62no—
157D36—61no—
158C4×D9—64no—
159C4×D9—64no—
160C4×D9—62no—
161C4×D9—62no—
162Dic18—32no—
163Dic18—62no—
164Dic18—62no—
165Dic18—61no—
166C8×C9—62no—
167#421—271noSylow
168C2×C3×C9—21no—
169C2×C3×C9—21no—
170C2×C3×C9—21no—
171C2×C3×C9C2×C22⋊C411yescharacteristic
172C3⋊D9—42no—
173C22×C3×C4—21no—
174C22×C3×C4C2×D911yescharacteristic
175C22×C3×C4D3211yescharacteristic
176C2×#67—182no—
177C2×#67—181no—
178C2×#67—181no—
179#235—92no—
180#235—92no—
181#235—92no—
182#235—92no—
183#235—182no—
184#235—182no—
185#235—182no—
186#235—182no—
187#235—181no—
188#235—181no—
189#235—181no—
190#235—181no—
191C2×D12—91no—
192C2×D12—91no—
193C2×D12—181no—
194C2×C4×D3—182no—
195C2×C4×D3—181no—
196C2×C4×D3—181no—
197C2×Dic6—91no—
198C2×Dic6—91no—
199C2×Dic6—181no—
200C3×M16—32no—
201C3×M16—61no—
202C2×C3×C8—61no—
203C3×C4⋊C4—181no—
204C3×C22⋊C4—181no—
205C3×C42—92no—
206#217—61no—
207C4⋊Dic3—61no—
208C4×Dic3—32no—
209#208—21no—
210#208C2×D912yescomplemented
211C2×C3⋊C8—21no—
212C22×C32—41no—
213C22×C32(C2×D3)⋊C411yescharacteristic
214C22×C32(C2×D3)⋊C411yescharacteristic
215C22×C32C2×D1211yescharacteristic
216C2×C3⋊D3—62no—
217C2×C3⋊D3—122no—
218C32×C4(C2×D3)⋊C411yescharacteristic
219C32×C4(C2×D3)⋊C411yescharacteristic
220C32×C4C2×C4×D311yescharacteristic
221C32×C4C2×#6711yescharacteristic
222C3⋊Dic3—62no—
223C3×Dic3—122no—
224C22×C9—21no—
225C22×C9—21no—
226C22×C9—21no—
227C22×C9—41no—
228C22×C9—42no—
229C22×C9(C2×D3)⋊C411yescharacteristic
230C22×C9(C2×D3)⋊C411yescharacteristic
231C22×C9C2×#6711yescharacteristic
232C2×D9—64no—
233C2×D9—62no—
234C2×D9—124no—
235C2×D9—122no—
236C4×C9—21no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—21no—
240C4×C9(C2×D3)⋊C411yescharacteristic
241C4×C9(C2×D3)⋊C411yescharacteristic
242C4×C9C2×C4×D311yescharacteristic
243C4×C9C2×D1211yescharacteristic
244Dic9—42no—
245Dic9—64no—
246Dic9—62no—
247C2×#41—271no—
248C2×M16—31no—
249#117—91no—
250#104—274no—
251C3×C9#42111yescharacteristic, complemented, Sylow
252C23×C3—21no—
253C23×C3D3611yescharacteristic
254C23×C3#62611yescharacteristic
255C22×D3—182no—
256C22×D3—181no—
257C22×D3—181no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—182no—
265C2×C3×C4C4×D911yescharacteristic
266C2×C3×C4C4×D911yescharacteristic
267C2×C3×C4D3611yescharacteristic
268C2×C3×C4#61111yescharacteristic
269C2×C3×C4#61111yescharacteristic
270C2×C3×C4C22×D911yescharacteristic
271C2×C3×C4#62411yescharacteristic
272C2×C3×C4#62411yescharacteristic
273C2×C3×C4#62611yescharacteristic
274C2×C3×C4#62611yescharacteristic
275C2×C3×C4#62611yescharacteristic
276C2×C3×C4C2×D3211yescharacteristic
277#67—184no—
278#67—184no—
279#67—182no—
280#67—182no—
281#67—182no—
282#67—182no—
283C2×Dic3—62no—
284C2×Dic3—182no—
285C2×Dic3—181no—
286C2×Dic3—181no—
287D12—92no—
288D12—92no—
289D12—182no—
290D12—182no—
291D12—181no—
292D12—181no—
293C4×D3—184no—
294C4×D3—184no—
295C4×D3—182no—
296C4×D3—182no—
297C4×D3—182no—
298C4×D3—182no—
299Dic6—92no—
300Dic6—92no—
301Dic6—182no—
302Dic6—182no—
303Dic6—181no—
304Dic6—181no—
305C3×C8—62no—
306C3⋊C8—22no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32C2×(C2×D3)⋊C411yescharacteristic
311C3⋊D3—122no—
312C2×C9—22no—
313C2×C9—21no—
314C2×C9—21no—
315C2×C9—21no—
316C2×C9—21no—
317C2×C9—41no—
318C2×C9—41no—
319C2×C9C2×(C2×D3)⋊C411yescharacteristic
320D9—124no—
321D9—122no—
322#41—272no—
323#41—272no—
324#41—541no—
325#41—541no—
326C2×Q8—271no—
327C2×D4—271no—
328C2×D4—541no—
329C22×C4—541no—
330C22×C4D3×D911yescharacteristic
331M16—32no—
332M16—61no—
333C2×C8—61no—
334C4⋊C4—181no—
335C22⋊C4—181no—
336C42—92no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—42no—
343C22×C3(C2×D9)⋊C411yescharacteristic
344C22×C3(C2×D9)⋊C411yescharacteristic
345C22×C3C2×D3611yescharacteristic
346C22×C3#372211yescharacteristic
347C22×C3#372211yescharacteristic
348C22×C3C2×#62611yescharacteristic
349C2×D3—184no—
350C2×D3—182no—
351C2×D3—182no—
352C2×D3—362no—
353C2×D3—364no—
354C2×D3—362no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—362no—
360C3×C4(C2×D9)⋊C411yescharacteristic
361C3×C4(C2×D9)⋊C411yescharacteristic
362C3×C4C2×C4×D911yescharacteristic
363C3×C4C2×#61111yescharacteristic
364C3×C4#372211yescharacteristic
365C3×C4#372211yescharacteristic
366C3×C4C2×#62411yescharacteristic
367C3×C4C2×#62611yescharacteristic
368Dic3—122no—
369Dic3—184no—
370Dic3—182no—
371Dic3—182no—
372C32#513911yescharacteristic
373C9—21no—
374C9#500511yescharacteristic, complemented
375C23—541no—
376C23#619211yescharacteristic
377Q8—272no—
378Q8—541no—
379D4—272no—
380D4—542no—
381D4—542no—
382D4—541no—
383C2×C4—182no—
384C2×C4—541no—
385C2×C4—542no—
386C2×C4—542no—
387C2×C4(C3⋊D9)⋊C411yescharacteristic
388C2×C4(C3⋊D9)⋊C411yescharacteristic
389C2×C4#619211yescharacteristic
390C2×C4#619511yescharacteristic
391C2×C4#619511yescharacteristic
392C2×C4C2×D3×D911yescharacteristic
393C8—62no—
394C2×C3—22no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3C2×(C2×D9)⋊C411yescharacteristic
405C2×C3C2×#372211yescharacteristic
406D3—364no—
407D3—362no—
408D3—362no—
409C22—41no—
410C22—542no—
411C22—1082no—
412C22#8844511yescharacteristic
413C22#8844511yescharacteristic
414C22C2×#619211yescharacteristic
415C4—362no—
416C4—542no—
417C4#8844511yescharacteristic
418C4#8844511yescharacteristic
419C4C2×(C3⋊D9)⋊C411yescharacteristic
420C4C2×#619511yescharacteristic
421C3—21no—
422C3#1058881911yescharacteristic, complemented
423C3#1059111311yescharacteristic
424C2—21no—
425C2—21no—
426C2—21no—
427C2—1082no—
428C2C2×#8844511yescharacteristic
subgroupG/H
centerC4C2×#6195
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC2×C32C2×(C2×D3)⋊C4
prime radicalC2×#88736C2
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#421
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63488C2semidirect
3. C2×C9×#208C2semidirect
4. #11720292C2semidirect
11. #88736C4semidirect
13. C2×C4⋊(C3⋊D9)C4semidirect
15. C2×#6226C4semidirect
18. C3×C4×Dic9C22semidirect
20. C9×#208C22semidirect
29. #63488D3semidirect
90. C4×Dic9C2×D3semidirect
109. C2×#208D9semidirect
210. #208C2×D9semidirect
251. C3×C9#421semidirect
374. C9#5005semidirect
422. C3#10588819semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C3×#634882
C2×C9×#2082
#117202922
#105911133
#105888193