Finite Group Databasebeta

← C32×#1854 #11768903

Subgroups — C32×#1854 #11768903

Order 1152.

948 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#117C211yesmaximal, characteristic
2C2×C32×#107C211yesmaximal, characteristic, complemented
3C2×C32×#106C211yesmaximal, characteristic, complemented
4C2×C32×C22⋊C8C211yesmaximal, characteristic
5C32×#380C211yesmaximal, characteristic
6C32×#340C211yesmaximal, characteristic, complemented
7C32×#340C211yesmaximal, characteristic, complemented
8C3×#1854C314yesmaximal, direct product factor
9C23×C32×C4C2211yescharacteristic
10C22×C32×C8C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C32×#117—22no—
13C32×#117—22no—
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4—21no—
19C2×C32×C22⋊C4—21no—
20C2×C32×C42—21no—
21C32×#107—22no—
22C32×#106—22no—
23C32×C22⋊C8C414yescomplemented
24C32×#95—21no—
25C2×C3×#117C2×C314yes—
26C2×C3×#107C2×C314yescomplemented
27C2×C3×#106C2×C314yescomplemented
28C2×C3×C22⋊C8C2×C314yes—
29C3×#380C2×C314yes—
30C3×#340C2×C314yescomplemented
31C3×#340C2×C314yescomplemented
32C24×C32D411yescharacteristic
33C22×C32×C4—21no—
34C22×C32×C4—21no—
35C22×C32×C4—21no—
36C22×C32×C4—21no—
37C22×C32×C4—21no—
38C22×C32×C4—21no—
39C22×C32×C4—21no—
40C22×C32×C4—21no—
41C22×C32×C4C2×C412yes—
42C22×C32×C4D411yescharacteristic
43C22×C32×C4Q812yes—
44C22×C32×C4C2311yescharacteristic
45C2×C32×C8—21no—
46C2×C32×C8—21no—
47C2×C32×C8—21no—
48C2×C32×C8—21no—
49C2×C32×C8C2×C412yes—
50C2×C32×C8C2×C412yes—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4D412yes—
58C32×C4⋊C4D412yes—
59C32×C22⋊C4—24no—
60C32×C22⋊C4—42no—
61C32×C42—24no—
62#1854C3211yescharacteristic, direct product factor, Sylow
63C23×C3×C4C22×C314yes—
64C22×C3×C8C22×C314yes—
65C22×C3×C8C22×C314yes—
66C3×#117—28no—
67C3×#117—28no—
68C2×C3×C4⋊C4C22×C314yes—
69C2×C3×C4⋊C4C22×C314yes—
70C2×C3×C4⋊C4C22×C314yes—
71C2×C3×C4⋊C4C22×C314yes—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C42—24no—
75C3×#107—28no—
76C3×#106—28no—
77C3×C22⋊C8C3×C4116yescomplemented
78C3×#95—24no—
79C23×C32—21no—
80C23×C32—21no—
81C23×C32—21no—
82C23×C32—21no—
83C23×C32—21no—
84C23×C32—21no—
85C23×C32C4⋊C412yes—
86C23×C32C2×D411yescharacteristic
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—41no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4—41no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4—41no—
105C2×C32×C4C4⋊C412yes—
106C2×C32×C4C22×C411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×Q811yescharacteristic
110C2×C32×C4#4111yescharacteristic
111C2×C32×C4#4111yescharacteristic
112C32×C8—22no—
113C32×C8—22no—
114C2×#117C2×C3211yescharacteristic
115C2×#107C2×C3211yescharacteristic, complemented
116C2×#106C2×C3211yescharacteristic, complemented
117C2×C22⋊C8C2×C3211yescharacteristic
118#380C2×C3211yescharacteristic
119#340C2×C3211yescharacteristic, complemented
120#340C2×C3211yescharacteristic, complemented
121C24×C3C3×D414yes—
122C22×C3×C4—24no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—24no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4C2×C3×C418yes—
131C22×C3×C4C3×D414yes—
132C22×C3×C4C3×Q818yes—
133C22×C3×C4C23×C314yes—
134C2×C3×C8—24no—
135C2×C3×C8—24no—
136C2×C3×C8—24no—
137C2×C3×C8—24no—
138C2×C3×C8C2×C3×C418yes—
139C2×C3×C8C2×C3×C418yes—
140C3×C4⋊C4—24no—
141C3×C4⋊C4—24no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4C3×D418yes—
147C3×C4⋊C4C3×D418yes—
148C3×C22⋊C4—216no—
149C3×C22⋊C4—48no—
150C3×C42—216no—
151C22×C32—22no—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32—41no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—41no—
158C22×C32C2×C4⋊C411yescharacteristic
159C22×C32#12011yescharacteristic
160C22×C32#12311yescharacteristic
161C22×C32#13511yescharacteristic
162C22×C32#13511yescharacteristic
163C22×C32AGL(1,ℤ/8)11yescharacteristic
164C22×C32#13711yescharacteristic
165C32×C4—22no—
166C32×C4—42no—
167C32×C4—42no—
168C32×C4—81no—
169C32×C4—81no—
170C32×C4C4×D411yescharacteristic
171C32×C4C4×D411yescharacteristic
172C32×C4C22⋊Q811yescharacteristic
173C32×C4C22⋊Q811yescharacteristic
174C23×C4C22×C3211yescharacteristic
175C22×C8C22×C3211yescharacteristic
176C22×C8C22×C3211yescharacteristic
177#117—22no—
178#117—22no—
179C2×C4⋊C4C22×C3211yescharacteristic
180C2×C4⋊C4C22×C3211yescharacteristic
181C2×C4⋊C4C22×C3211yescharacteristic
182C2×C4⋊C4C22×C3211yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42—21no—
186#107—22no—
187#106—22no—
188C22⋊C8C32×C414yescomplemented
189#95—21no—
190C23×C3—24no—
191C23×C3—24no—
192C23×C3—24no—
193C23×C3—24no—
194C23×C3—24no—
195C23×C3—24no—
196C23×C3C3×C4⋊C418yes—
197C23×C3C2×C3×D414yes—
198C2×C3×C4—24no—
199C2×C3×C4—24no—
200C2×C3×C4—28no—
201C2×C3×C4—28no—
202C2×C3×C4—28no—
203C2×C3×C4—28no—
204C2×C3×C4—28no—
205C2×C3×C4—28no—
206C2×C3×C4—44no—
207C2×C3×C4—44no—
208C2×C3×C4—44no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4—44no—
212C2×C3×C4—44no—
213C2×C3×C4—44no—
214C2×C3×C4—44no—
215C2×C3×C4—44no—
216C2×C3×C4C3×C4⋊C418yes—
217C2×C3×C4C22×C3×C414yes—
218C2×C3×C4C2×C3×D414yes—
219C2×C3×C4C2×C3×D414yes—
220C2×C3×C4C2×C3×Q814yes—
221C2×C3×C4C3×#4114yes—
222C2×C3×C4C3×#4114yes—
223C3×C8—28no—
224C3×C8—28no—
225C2×C32—42no—
226C2×C32#38511yescharacteristic
227C2×C32#42711yescharacteristic
228C2×C32#42811yescharacteristic
229C2×C32#44911yescharacteristic
230C2×C32#45211yescharacteristic
231C2×C32#48211yescharacteristic
232C2×C32#48511yescharacteristic
233C24C32×D411yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C2×C32×C412yes—
243C22×C4C32×D411yescharacteristic
244C22×C4C32×Q812yes—
245C22×C4C23×C3211yescharacteristic
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8C2×C32×C412yes—
251C2×C8C2×C32×C412yes—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4C32×D412yes—
259C4⋊C4C32×D412yes—
260C22⋊C4—24no—
261C22⋊C4—42no—
262C42—24no—
263C22×C3—28no—
264C22×C3—28no—
265C22×C3—28no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3—44no—
269C22×C3—44no—
270C22×C3C2×C3×C4⋊C414yes—
271C22×C3C3×#12014yes—
272C22×C3C3×#12314yes—
273C22×C3C3×#13514yes—
274C22×C3C3×#13514yes—
275C22×C3C3×AGL(1,ℤ/8)14yes—
276C22×C3C3×#13714yes—
277C3×C4—28no—
278C3×C4—48no—
279C3×C4—48no—
280C3×C4—84no—
281C3×C4—84no—
282C3×C4C3×C4×D414yes—
283C3×C4C3×C4×D414yes—
284C3×C4C3×C22⋊Q814yes—
285C3×C4C3×C22⋊Q814yes—
286C32#185411yescharacteristic, direct product factor, Sylow
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23C32×C4⋊C412yes—
294C23C2×C32×D411yescharacteristic
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4C32×C4⋊C412yes—
314C2×C4C22×C32×C411yescharacteristic
315C2×C4C2×C32×D411yescharacteristic
316C2×C4C2×C32×D411yescharacteristic
317C2×C4C2×C32×Q811yescharacteristic
318C2×C4C32×#4111yescharacteristic
319C2×C4C32×#4111yescharacteristic
320C8—22no—
321C8—22no—
322C2×C3—48no—
323C2×C3C3×#38514yes—
324C2×C3C3×#42714yes—
325C2×C3C3×#42814yes—
326C2×C3C3×#44914yes—
327C2×C3C3×#45214yes—
328C2×C3C3×#48214yes—
329C2×C3C3×#48514yes—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C2×C32×C4⋊C411yescharacteristic
338C22C32×#12011yescharacteristic
339C22C32×#12311yescharacteristic
340C22C32×#13511yescharacteristic
341C22C32×#13511yescharacteristic
342C22C32×AGL(1,ℤ/8)11yescharacteristic
343C22C32×#13711yescharacteristic
344C4—22no—
345C4—42no—
346C4—42no—
347C4—81no—
348C4—81no—
349C4C32×C4×D411yescharacteristic
350C4C32×C4×D411yescharacteristic
351C4C32×C22⋊Q811yescharacteristic
352C4C32×C22⋊Q811yescharacteristic
353C3C3×#185414yesdirect product factor
354C2—42no—
355C2C32×#38511yescharacteristic
356C2C32×#42711yescharacteristic
357C2C32×#42811yescharacteristic
358C2C32×#44911yescharacteristic
359C2C32×#45211yescharacteristic
360C2C32×#48211yescharacteristic
361C2C32×#48511yescharacteristic
subgroupG/H
center, socleC23×C32C2×D4
commutatorC2×C4C22×C32×C4
Frattini, cosocleC22×C4C23×C32
prime radicalC24×C32D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C32 ◁ C23×C32×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C32×#107C2semidirect
3. C2×C32×#106C2semidirect
6. C32×#340C2semidirect
7. C32×#340C2semidirect
8. C3×#1854C3direct (N × H)
23. C32×C22⋊C8C4semidirect
26. C2×C3×#107C2×C3semidirect
27. C2×C3×#106C2×C3semidirect
30. C3×#340C2×C3semidirect
31. C3×#340C2×C3semidirect
62. #1854C32direct (N × H)
77. C3×C22⋊C8C3×C4semidirect
115. C2×#107C2×C32semidirect
116. C2×#106C2×C32semidirect
119. #340C2×C32semidirect
120. #340C2×C32semidirect
188. C22⋊C8C32×C4semidirect
286. C32#1854direct (N × H)
353. C3C3×#1854direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C2×C32×#1072
C2×C32×#1062
C2×C32×C22⋊C82
C32×#3802
C32×#3402
C3×#18543