Finite Group Databasebeta

← C2×C3×#1605 #10636718

Subgroups — C2×C3×#1605 #10636718

Order 768.

1200 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic
2C2×C3×#451C212yesmaximal
3C2×C3×#395C211yesmaximal, characteristic
4C3×#1605C218yesmaximal, direct product factor
5C2×C3×#328C212yesmaximal
6C2×C3×#324C211yesmaximal, characteristic
7C2×#1605C311yesmaximal, characteristic, direct product factor, Sylow
8C23×C3×Q8C2211yescharacteristic
9C22×C3×Q16—22no—
10C3×#544—24no—
11C3×#544—22no—
12C3×#544C2214yes—
13C2×C3×C4⋊Q8—22no—
14C2×C3×#125—22no—
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×#117—21no—
18C22×C3×C4⋊C4—21no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C3×#451—24no—
21C3×#451C2218yes—
22C2×C3×#103—22no—
23C2×C3×C22⋊C8C2212yes—
24C3×#395—22no—
25C3×#395C2214yes—
26C2×C3×#95—21no—
27C2×C3×#95C2211yescharacteristic
28C3×#328C2218yes—
29C3×#324C2214yes—
30C2×#544C2×C311yescharacteristic
31C2×#451C2×C312yes—
32C2×#395C2×C311yescharacteristic
33#1605C2×C318yesdirect product factor
34C2×#328C2×C312yes—
35C2×#324C2×C311yescharacteristic
36C22×C3×Q8—21no—
37C22×C3×Q8—21no—
38C22×C3×Q8—22no—
39C22×C3×Q8—22no—
40C22×C3×Q8—44no—
41C22×C3×Q8D412yes—
42C22×C3×Q8C2312yes—
43C23×C3×C4—21no—
44C23×C3×C4—21no—
45C23×C3×C4D411yescharacteristic
46C23×C3×C4D411yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C2×C3×Q16—28no—
49C2×C3×Q16—48no—
50C22×C3×C8—22no—
51C3×C4⋊Q8—28no—
52C3×C4⋊Q8—28no—
53C3×#125—28no—
54C3×#125—28no—
55C3×C22⋊Q8—24no—
56C3×C22⋊Q8—24no—
57C3×C22⋊Q8—22no—
58C3×C22⋊Q8—22no—
59C3×C22⋊Q8D414yes—
60C3×C22⋊Q8C2314yes—
61C3×#117—24no—
62C3×#117—24no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—44no—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C42—22no—
77C3×#103—28no—
78C3×C22⋊C8—22no—
79C3×C22⋊C8C2314yes—
80C3×#95—24no—
81C3×#95—21no—
82C3×#95C2312yes—
83C23×Q8C22×C311yescharacteristic
84C22×Q16—22no—
85#544—24no—
86#544—22no—
87#544C22×C314yes—
88C2×C4⋊Q8—22no—
89C2×#125—22no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8C22×C312yes—
92C2×#117—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C22×C311yescharacteristic
95#451—24no—
96#451C22×C318yes—
97C2×#103—22no—
98C2×C22⋊C8C22×C312yes—
99#395—22no—
100#395C22×C314yes—
101C2×#95—21no—
102C2×#95C22×C311yescharacteristic
103#328C22×C318yes—
104#324C22×C314yes—
105C24×C3C2×D411yescharacteristic
106C2×C3×Q8—24no—
107C2×C3×Q8—22no—
108C2×C3×Q8—28no—
109C2×C3×Q8—24no—
110C2×C3×Q8—44no—
111C2×C3×Q8—44no—
112C2×C3×Q8—44no—
113C2×C3×Q8—44no—
114C2×C3×Q8—48no—
115C2×C3×Q8—42no—
116C2×C3×Q8—82no—
117C2×C3×Q8C2×D414yes—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—24no—
122C22×C3×C4—24no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—42no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4Q1614yescomplemented
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2411yescharacteristic
143C3×Q16—416no—
144C2×C3×C8—24no—
145C2×C3×C8—44no—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—48no—
156C3×C4⋊C4C2×D414yes—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—24no—
159C3×C42—28no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—44no—
165C22×Q8C3×D412yes—
166C22×Q8C23×C312yes—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C3×D411yescharacteristic
170C23×C4C3×D411yescharacteristic
171C23×C4C23×C311yescharacteristic
172C2×Q16—28no—
173C2×Q16—48no—
174C22×C8—22no—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177#125—28no—
178#125—28no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8C3×D414yes—
184C22⋊Q8C23×C314yes—
185#117—24no—
186#117—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4C3×D412yes—
197C2×C4⋊C4C3×D412yes—
198C2×C4⋊C4C23×C312yes—
199C2×C22⋊C4—22no—
200C2×C42—22no—
201#103—28no—
202C22⋊C8—22no—
203C22⋊C8C23×C314yes—
204#95—24no—
205#95—21no—
206#95C23×C312yes—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3#12011yescharacteristic
210C23×C3#12011yescharacteristic
211C23×C3C2×Q1612yes—
212C23×C3#13712yes—
213C23×C3C22×D411yescharacteristic
214C3×Q8—28no—
215C3×Q8—44no—
216C3×Q8—44no—
217C3×Q8—48no—
218C3×Q8—82no—
219C3×Q8—84no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—48no—
235C2×C3×C4—42no—
236C2×C3×C4—44no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—44no—
240C2×C3×C4—44no—
241C2×C3×C4—42no—
242C2×C3×C4—44no—
243C2×C3×C4—42no—
244C2×C3×C4—81no—
245C2×C3×C4#12012yes—
246C2×C3×C4C2×Q1614yescomplemented
247C2×C3×C4C22×D412yes—
248C3×C8—44no—
249C24C2×C3×D411yescharacteristic
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—28no—
253C2×Q8—24no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—48no—
259C2×Q8—42no—
260C2×Q8—82no—
261C2×Q8C2×C3×D414yes—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4C3×Q1614yescomplemented
281C22×C4C2×C3×D412yes—
282C22×C4C2×C3×D412yes—
283C22×C4C2×C3×D412yes—
284C22×C4C2×C3×D411yescharacteristic
285C22×C4C2×C3×D412yes—
286C22×C4C24×C311yescharacteristic
287Q16—416no—
288C2×C8—24no—
289C2×C8—44no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—28no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—48no—
300C4⋊C4C2×C3×D414yes—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C42—28no—
304C22×C3—24no—
305C22×C3—24no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3#45112yes—
311C22×C3#45112yes—
312C22×C3#45612yes—
313C22×C3B(4,𝔽2)12yes—
314C22×C3C2×#12011yescharacteristic
315C22×C3C22×Q1611yescharacteristic
316C22×C3C2×#13711yescharacteristic
317C3×C4—24no—
318C3×C4—22no—
319C3×C4—22no—
320C3×C4—44no—
321C3×C4—44no—
322C3×C4—44no—
323C3×C4—42no—
324C3×C4—44no—
325C23—22no—
326C23—22no—
327C23C3×#12011yescharacteristic
328C23C3×#12011yescharacteristic
329C23C2×C3×Q1612yes—
330C23C3×#13712yes—
331C23C22×C3×D411yescharacteristic
332Q8—28no—
333Q8—44no—
334Q8—44no—
335Q8—48no—
336Q8—82no—
337Q8—84no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—81no—
363C2×C4C3×#12012yes—
364C2×C4C2×C3×Q1614yescomplemented
365C2×C4C22×C3×D412yes—
366C8—44no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—22no—
370C2×C3#160514yesdirect product factor
371C2×C3C2×#45111yescharacteristic
372C2×C3C2×#45611yescharacteristic
373C2×C3C2×B(4,𝔽2)11yescharacteristic
374C22—24no—
375C22—24no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22C3×#45112yes—
381C22C3×#45112yes—
382C22C3×#45612yes—
383C22C3×B(4,𝔽2)12yes—
384C22C2×C3×#12011yescharacteristic
385C22C22×C3×Q1611yescharacteristic
386C22C2×C3×#13711yescharacteristic
387C4—24no—
388C4—22no—
389C4—22no—
390C4—44no—
391C4—44no—
392C4—44no—
393C4—42no—
394C4—44no—
395C3C2×#160511yescharacteristic, direct product factor, Sylow
396C2—21no—
397C2—21no—
398C2—22no—
399C2C3×#160514yesdirect product factor
400C2C2×C3×#45111yescharacteristic
401C2C2×C3×#45611yescharacteristic
402C2C2×C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#1605C2direct (N × H)
7. C2×#1605C3direct (N × H)
33. #1605C2×C3direct (N × H)
136. C22×C3×C4Q16semidirect
246. C2×C3×C4C2×Q16semidirect
280. C22×C4C3×Q16semidirect
364. C2×C4C2×C3×Q16semidirect
370. C2×C3#1605direct (N × H)
395. C3C2×#1605direct (N × H)
399. C2C3×#1605direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×C3×#4512
C2×C3×#3952
C3×#16052
C2×C3×#3282
C2×C3×#3242
C2×#16053