Finite Group Databasebeta

← C3×#13329 #10631936

Subgroups — C3×#13329 #10631936

Order 768.

578 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Q32C211yesmaximal, characteristic, complemented
2C22×C3×SD32C211yesmaximal, characteristic
3C2×C3×#495C211yesmaximal, characteristic
4C2×C3×#358C211yesmaximal, characteristic, complemented
5C2×C3×#357C211yesmaximal, characteristic
6C3×#1969C211yesmaximal, characteristic, complemented
7C3×#1374C211yesmaximal, characteristic, complemented
8#13329C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×Q16C2211yescharacteristic
10C22×C3×Q16C2211yescharacteristic
11C22×C3×SD16—21no—
12C22×C3×D8C2211yescharacteristic
13C2×C3×#124—21no—
14C2×C3×Q32—22no—
15C2×C3×Q32—22no—
16C2×C3×Q32—22no—
17C2×C3×SD32—22no—
18C2×C3×SD32—22no—
19C2×C3×SD32—22no—
20C22×C3×C16C2211yescharacteristic
21C22×C3×C16C2211yescharacteristic
22C3×#495—22no—
23C3×#495—22no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103—21no—
26C2×C3×C4×C8C2211yescharacteristic
27C3×#391—21no—
28C3×#358—22no—
29C3×#357—22no—
30C22×Q32C2×C311yescharacteristic, complemented
31C22×SD32C2×C311yescharacteristic
32C2×#495C2×C311yescharacteristic
33C2×#358C2×C311yescharacteristic, complemented
34C2×#357C2×C311yescharacteristic
35#1969C2×C311yescharacteristic, complemented
36#1374C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×D4—21no—
40C2×C3×Q16—21no—
41C2×C3×Q16—21no—
42C2×C3×Q16—21no—
43C2×C3×Q16—21no—
44C2×C3×Q16—21no—
45C2×C3×Q16—21no—
46C2×C3×Q16—21no—
47C2×C3×Q16—21no—
48C2×C3×Q16—21no—
49C2×C3×Q16—21no—
50C2×C3×Q16D412yes—
51C2×C3×Q16D412yes—
52C2×C3×SD16—24no—
53C2×C3×SD16—42no—
54C2×C3×SD16—42no—
55C2×C3×D8—21no—
56C2×C3×D8—21no—
57C2×C3×D8—21no—
58C2×C3×D8—21no—
59C2×C3×D8—21no—
60C2×C3×D8D412yes—
61C22×C3×C8D411yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#124—24no—
64C3×#124—42no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C22⋊C4—42no—
67C2×C3×C42D411yescharacteristic
68C3×Q32—28no—
69C3×SD32—28no—
70C2×C3×C16—21no—
71C2×C3×C16—21no—
72C2×C3×C16—21no—
73C2×C3×C16—21no—
74C2×C3×C16D412yes—
75C2×C3×C16D412yes—
76C3×#107—21no—
77C3×#107—21no—
78C3×#103—42no—
79C3×C4×C8—21no—
80C3×C4×C8—21no—
81C3×#95—42no—
82C22×Q16C22×C311yescharacteristic
83C22×Q16C22×C311yescharacteristic
84C22×SD16—21no—
85C22×D8C22×C311yescharacteristic
86C2×#124—21no—
87C2×Q32—22no—
88C2×Q32—22no—
89C2×Q32—22no—
90C2×SD32—22no—
91C2×SD32—22no—
92C2×SD32—22no—
93C22×C16C22×C311yescharacteristic
94C22×C16C22×C311yescharacteristic
95#495—22no—
96#495—22no—
97C2×#107C22×C311yescharacteristic
98C2×#103—21no—
99C2×C4×C8C22×C311yescharacteristic
100#391—21no—
101#358—22no—
102#357—22no—
103C24×C3—41no—
104C2×C3×Q8—22no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×D4—22no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4D812yes—
126C22×C3×C4C2×D411yescharacteristic
127C3×Q16—22no—
128C3×Q16—22no—
129C3×Q16—22no—
130C3×Q16—22no—
131C3×Q16—41no—
132C3×Q16—41no—
133C3×Q16—41no—
134C3×Q16—41no—
135C3×SD16—48no—
136C3×D8—22no—
137C3×D8—22no—
138C3×D8—41no—
139C3×D8—41no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C2×C3×C8D812yes—
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8C2×D411yescharacteristic
146C2×C3×C8C2×D411yescharacteristic
147C2×C3×C8C2×D411yescharacteristic
148C2×C3×C8#4111yescharacteristic
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C22⋊C4—48no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C16—22no—
155C3×C16—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×D4—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16C3×D412yes—
170C2×Q16C3×D412yes—
171C2×SD16—24no—
172C2×SD16—42no—
173C2×SD16—42no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8C3×D412yes—
180C22×C8C3×D411yescharacteristic
181C22×C8C23×C311yescharacteristic
182#124—24no—
183#124—42no—
184C2×C4⋊C4—21no—
185C2×C22⋊C4—42no—
186C2×C42C3×D411yescharacteristic
187Q32—28no—
188SD32—28no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16C3×D412yes—
194C2×C16C3×D412yes—
195#107—21no—
196#107—21no—
197#103—42no—
198C4×C8—21no—
199C4×C8—21no—
200#95—42no—
201C23×C3—42no—
202C23×C3—81no—
203C23×C3—81no—
204C23×C3—82no—
205C23×C3—82no—
206C23×C3C2×D811yescharacteristic
207C3×Q8—42no—
208C3×Q8—42no—
209C3×Q8—42no—
210C3×Q8—42no—
211C3×Q8—81no—
212C3×Q8—81no—
213C3×Q8—81no—
214C3×Q8—81no—
215C3×D4—42no—
216C3×D4—42no—
217C3×D4—81no—
218C3×D4—81no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×C3×C4—82no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4C2×D811yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C3×C8—22no—
237C3×C8#12011yescharacteristic
238C3×C8#12011yescharacteristic
239C3×C8#12111yescharacteristic
240C3×C8#12111yescharacteristic
241C24—41no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4C3×D812yes—
264C22×C4C2×C3×D411yescharacteristic
265Q16—22no—
266Q16—22no—
267Q16—22no—
268Q16—22no—
269Q16—41no—
270Q16—41no—
271Q16—41no—
272Q16—41no—
273SD16—48no—
274D8—22no—
275D8—22no—
276D8—41no—
277D8—41no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8C3×D812yes—
281C2×C8C2×C3×D411yescharacteristic
282C2×C8C2×C3×D411yescharacteristic
283C2×C8C2×C3×D411yescharacteristic
284C2×C8C2×C3×D411yescharacteristic
285C2×C8C2×C3×D411yescharacteristic
286C2×C8C3×#4111yescharacteristic
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—48no—
290C42—21no—
291C42—21no—
292C16—22no—
293C16—22no—
294C22×C3—84no—
295C22×C3—82no—
296C22×C3—162no—
297C22×C3—161no—
298C22×C3—161no—
299C22×C3#44711yescharacteristic
300C22×C3#45911yescharacteristic
301C22×C3#49311yescharacteristic
302C22×C3#50812yes—
303C22×C3#51012yes—
304C3×C4—42no—
305C3×C4—82no—
306C3×C4—82no—
307C3×C4—161no—
308C3×C4#39211yescharacteristic
309C3×C4#44711yescharacteristic
310C3×C4#45911yescharacteristic
311C3×C4#49611yescharacteristic
312C23—42no—
313C23—82no—
314C23—82no—
315C23—81no—
316C23—81no—
317C23C2×C3×D811yescharacteristic
318Q8—42no—
319Q8—42no—
320Q8—42no—
321Q8—42no—
322Q8—81no—
323Q8—81no—
324Q8—81no—
325Q8—81no—
326D4—42no—
327D4—42no—
328D4—81no—
329D4—81no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4C3×#12011yescharacteristic
342C2×C4C3×#12111yescharacteristic
343C2×C4C3×#12711yescharacteristic
344C2×C4C2×C3×D811yescharacteristic
345C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
346C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
347C8—22no—
348C8C3×#12011yescharacteristic
349C8C3×#12011yescharacteristic
350C8C3×#12111yescharacteristic
351C8C3×#12111yescharacteristic
352C2×C3—162no—
353C2×C3#201111yescharacteristic
354C2×C3#218912yes—
355C2×C3#221112yes—
356C2×C3#224911yescharacteristic
357C2×C3#225111yescharacteristic
358C22—82no—
359C22—84no—
360C22—161no—
361C22—161no—
362C22—162no—
363C22C3×#44711yescharacteristic
364C22C3×#45911yescharacteristic
365C22C3×#49311yescharacteristic
366C22C3×#50812yes—
367C22C3×#51012yes—
368C4—42no—
369C4—82no—
370C4—82no—
371C4—161no—
372C4C3×#39211yescharacteristic
373C4C3×#44711yescharacteristic
374C4C3×#45911yescharacteristic
375C4C3×#49611yescharacteristic
376C3#1332911yescharacteristic, direct product factor, Sylow
377C2—162no—
378C2C3×#201111yescharacteristic
379C2C3×#218912yes—
380C2C3×#221112yes—
381C2C3×#224911yescharacteristic
382C2C3×#225111yescharacteristic
subgroupG/H
center, socleC23×C3C2×D8
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q32C2semidirect
4. C2×C3×#358C2semidirect
6. C3×#1969C2semidirect
7. C3×#1374C2semidirect
8. #13329C3direct (N × H)
30. C22×Q32C2×C3semidirect
33. C2×#358C2×C3semidirect
35. #1969C2×C3semidirect
36. #1374C2×C3semidirect
376. C3#13329direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q322
C22×C3×SD322
C2×C3×#4952
C2×C3×#3582
C2×C3×#3572
C3×#19692
C3×#13742
#133293