Finite Group Databasebeta

← C3×#33753 #10652360

Subgroups — C3×#33753 #10652360

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#467C211yesmaximal, characteristic, complemented
2C2×C3×#466C211yesmaximal, characteristic
3C3×#2235C212yesmaximal, complemented
4C3×#2233C212yesmaximal, complemented
5C3×#2217C214yesmaximal, complemented
6C3×#2187C212yesmaximal, complemented
7C3×#2185C212yesmaximal, complemented
8C2×C3×C22⋊C16C211yesmaximal, characteristic, complemented
9#33753C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×Q16C2211yescharacteristic, complemented
11C22×C3×D8C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C2×C3×C22⋊Q8—21no—
14C2×C3×#121—21no—
15C2×C3×SD32C2214yescomplemented
16C22×C3×C16C2212yescomplemented
17C3×#467—21no—
18C3×#467—21no—
19C3×#467C2212yes—
20C3×#467C2212yescomplemented
21C3×#466—21no—
22C3×#466—21no—
23C3×#466C2212yes—
24C3×#466C2212yes—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic, complemented
27C2×C3×#103—21no—
28C2×C3×#102—21no—
29C3×#367C2214yescomplemented
30C3×#358C2214yescomplemented
31C3×#357C2214yescomplemented
32C3×C22⋊C16C2212yes—
33C3×C22⋊C16C2212yescomplemented
34C2×#467C2×C311yescharacteristic, complemented
35C2×#466C2×C311yescharacteristic
36#2235C2×C312yescomplemented
37#2233C2×C312yescomplemented
38#2217C2×C314yescomplemented
39#2187C2×C312yescomplemented
40#2185C2×C312yescomplemented
41C2×C22⋊C16C2×C311yescharacteristic, complemented
42C22×C3×Q8—21no—
43C22×C3×D4—21no—
44C22×C3×D4—41no—
45C23×C3×C4D411yescharacteristic
46C2×C3×Q16—21no—
47C2×C3×Q16—42no—
48C2×C3×Q16C2312yes—
49C2×C3×D8—21no—
50C2×C3×D8—42no—
51C2×C3×D8C2312yes—
52C22×C3×C8—42no—
53C22×C3×C8D411yescharacteristic
54C22×C3×C8D411yescharacteristic
55C22×C3×C8D411yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8—22no—
61C3×C22⋊Q8—41no—
62C3×C22⋊Q8—41no—
63C3×#121—22no—
64C3×#121—22no—
65C3×#121—41no—
66C3×#121—41no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—41no—
70C2×C3×C22⋊C4—41no—
71C2×C3×C22⋊C4—41no—
72C3×SD32—44no—
73C2×C3×C16—42no—
74C2×C3×C16C2314yes—
75C3×#107—21no—
76C3×#107—21no—
77C3×#107C2312yes—
78C3×#107C2312yes—
79C3×#103—22no—
80C3×#103—22no—
81C3×#102—22no—
82C3×#102—22no—
83C22×Q16C22×C311yescharacteristic, complemented
84C22×D8C22×C311yescharacteristic
85C23×C8C22×C311yescharacteristic
86C2×C22⋊Q8—21no—
87C2×#121—21no—
88C2×SD32C22×C314yescomplemented
89C22×C16C22×C312yescomplemented
90#467—21no—
91#467—21no—
92#467C22×C312yes—
93#467C22×C312yescomplemented
94#466—21no—
95#466—21no—
96#466C22×C312yes—
97#466C22×C312yes—
98C2×#107C22×C311yescharacteristic
99C2×#107C22×C311yescharacteristic, complemented
100C2×#103—21no—
101C2×#102—21no—
102#367C22×C314yescomplemented
103#358C22×C314yescomplemented
104#357C22×C314yescomplemented
105C22⋊C16C22×C312yes—
106C22⋊C16C22×C312yescomplemented
107C24×C3—41no—
108C24×C3D811yescharacteristic
109C2×C3×Q8—22no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C2×C3×Q8—41no—
113C2×C3×D4—22no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C2×C3×D4—42no—
117C2×C3×D4—42no—
118C2×C3×D4—41no—
119C2×C3×D4—84no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4D811yescharacteristic
126C22×C3×C4D811yescharacteristic
127C22×C3×C4D811yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C3×Q16—24no—
132C3×Q16—42no—
133C3×D8—24no—
134C3×D8—42no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C2×C3×C8C2×D411yescharacteristic
142C2×C3×C8C2×D411yescharacteristic
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8C2411yescharacteristic
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—42no—
150C3×C4⋊C4—42no—
151C3×C22⋊C4—42no—
152C3×C22⋊C4—42no—
153C3×C22⋊C4—42no—
154C3×C22⋊C4—42no—
155C3×C16—24no—
156C22×Q8—21no—
157C22×D4—21no—
158C22×D4—41no—
159C23×C4C3×D411yescharacteristic
160C2×Q16—21no—
161C2×Q16—42no—
162C2×Q16C23×C312yes—
163C2×D8—21no—
164C2×D8—42no—
165C2×D8C23×C312yes—
166C22×C8—42no—
167C22×C8C3×D411yescharacteristic
168C22×C8C3×D411yescharacteristic
169C22×C8C3×D411yescharacteristic
170C22×C8C23×C311yescharacteristic
171C22×C8C23×C311yescharacteristic
172C22×C8C23×C311yescharacteristic
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177#121—22no—
178#121—22no—
179#121—41no—
180#121—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186SD32—44no—
187C2×C16—42no—
188C2×C16C23×C314yes—
189#107—21no—
190#107—21no—
191#107C23×C312yes—
192#107C23×C312yes—
193#103—22no—
194#103—22no—
195#102—22no—
196#102—22no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3—82no—
201C23×C3—82no—
202C23×C3—82no—
203C23×C3SD3214yescomplemented
204C23×C3C2×D811yescharacteristic
205C23×C3C2×D811yescharacteristic
206C23×C3C2×D811yescharacteristic
207C3×Q8—42no—
208C3×Q8—42no—
209C3×Q8—82no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—82no—
213C3×D4—84no—
214C3×D4—84no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—82no—
228C2×C3×C4—82no—
229C2×C3×C4—82no—
230C2×C3×C4C2×D811yescharacteristic
231C2×C3×C4C2×D811yescharacteristic
232C2×C3×C4C2×D811yescharacteristic
233C2×C3×C4C22×D411yescharacteristic
234C3×C8—21no—
235C3×C8—21no—
236C3×C8—21no—
237C3×C8H(5,𝔽2)12yes—
238C24—41no—
239C24C3×D811yescharacteristic
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×D4—22no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—84no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C3×D811yescharacteristic
257C22×C4C3×D811yescharacteristic
258C22×C4C3×D811yescharacteristic
259C22×C4C2×C3×D411yescharacteristic
260C22×C4C2×C3×D411yescharacteristic
261C22×C4C2×C3×D411yescharacteristic
262Q16—24no—
263Q16—42no—
264D8—24no—
265D8—42no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8C2×C3×D411yescharacteristic
273C2×C8C2×C3×D411yescharacteristic
274C2×C8C2×C3×D411yescharacteristic
275C2×C8C24×C311yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C16—24no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—44no—
294C22×C3—82no—
295C22×C3—82no—
296C22×C3—82no—
297C22×C3—164no—
298C22×C3C2×SD3212yes—
299C22×C3C2×SD3212yes—
300C22×C3C2×SD3212yes—
301C22×C3C22×D811yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—82no—
306C3×C4—82no—
307C3×C4—82no—
308C3×C4#53411yescharacteristic
309C3×C4#57511yescharacteristic
310C23—41no—
311C23—41no—
312C23—42no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23C3×SD3214yescomplemented
317C23C2×C3×D811yescharacteristic
318C23C2×C3×D811yescharacteristic
319C23C2×C3×D811yescharacteristic
320Q8—42no—
321Q8—42no—
322Q8—82no—
323D4—42no—
324D4—42no—
325D4—82no—
326D4—84no—
327D4—84no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4C2×C3×D811yescharacteristic
344C2×C4C2×C3×D811yescharacteristic
345C2×C4C2×C3×D811yescharacteristic
346C2×C4C22×C3×D411yescharacteristic
347C8—21no—
348C8—21no—
349C8—21no—
350C8C3×H(5,𝔽2)12yes—
351C2×C3—22no—
352C2×C3—22no—
353C2×C3—22no—
354C2×C3—162no—
355C2×C3#318611yescharacteristic
356C2×C3C22×SD3211yescharacteristic
357C2×C3#341411yescharacteristic
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—164no—
369C22C2×C3×SD3212yes—
370C22C2×C3×SD3212yes—
371C22C2×C3×SD3212yes—
372C22C22×C3×D811yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—82no—
377C4—82no—
378C4—82no—
379C4C3×#53411yescharacteristic
380C4C3×#57511yescharacteristic
381C3#3375311yescharacteristic, direct product factor, Sylow
382C2—22no—
383C2—22no—
384C2—22no—
385C2—162no—
386C2C3×#318611yescharacteristic
387C2C22×C3×SD3211yescharacteristic
388C2C3×#341411yescharacteristic
subgroupG/H
center, socleC22×C3C22×D8
commutator, Frattini, cosocleC2×C8C24×C3
prime radicalC2×C3×#466C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#467C2semidirect
3. C3×#2235C2semidirect
4. C3×#2233C2semidirect
5. C3×#2217C2semidirect
6. C3×#2187C2semidirect
7. C3×#2185C2semidirect
8. C2×C3×C22⋊C16C2semidirect
9. #33753C3direct (N × H)
10. C22×C3×Q16C22semidirect
15. C2×C3×SD32C22semidirect
16. C22×C3×C16C22semidirect
20. C3×#467C22semidirect
26. C2×C3×#107C22semidirect
29. C3×#367C22semidirect
30. C3×#358C22semidirect
31. C3×#357C22semidirect
33. C3×C22⋊C16C22semidirect
34. C2×#467C2×C3semidirect
36. #2235C2×C3semidirect
37. #2233C2×C3semidirect
38. #2217C2×C3semidirect
39. #2187C2×C3semidirect
40. #2185C2×C3semidirect
41. C2×C22⋊C16C2×C3semidirect
83. C22×Q16C22×C3semidirect
88. C2×SD32C22×C3semidirect
89. C22×C16C22×C3semidirect
93. #467C22×C3semidirect
99. C2×#107C22×C3semidirect
102. #367C22×C3semidirect
103. #358C22×C3semidirect
104. #357C22×C3semidirect
106. C22⋊C16C22×C3semidirect
203. C23×C3SD32semidirect
316. C23C3×SD32semidirect
381. C3#33753direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4672
C2×C3×#4662
C3×#22352
C3×#22332
C3×#22172
C3×#21872
C3×#21852
C2×C3×C22⋊C162
#337533