Finite Group Databasebeta

← C3×#23556 #10642163

Subgroups — C3×#23556 #10642163

Order 768.

1110 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic
2C2×C3×#428C211yesmaximal, characteristic
3C22×C3×#106C211yesmaximal, characteristic
4C3×#2317C211yesmaximal, characteristic
5C3×#2059C214yesmaximal, complemented
6C3×#2058C214yesmaximal, complemented
7C3×#1736C211yesmaximal, characteristic
8C3×#1735C211yesmaximal, characteristic
9C3×#1729C211yesmaximal, characteristic
10#23556C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×M16C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C3×#544—24no—
14C3×#544—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×#125—21no—
18C2×C3×#125C2212yes—
19C2×C3×C22⋊Q8—22no—
20C2×C3×#117C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C2×C3×#117C2211yescharacteristic
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C4⋊C4C2211yescharacteristic
25C3×#428—24no—
26C2×C3×#107C2212yes—
27C2×C3×#106—24no—
28C2×C3×#106C2214yes—
29C2×C3×#106C2212yes—
30C3×#389C2214yes—
31C3×#385—22no—
32C3×#384—21no—
33C3×#382—21no—
34C3×#340C2212yes—
35C3×#340C2212yes—
36C3×#340C2214yes—
37C3×#336C2214yes—
38C2×#544C2×C311yescharacteristic
39C2×#428C2×C311yescharacteristic
40C22×#106C2×C311yescharacteristic
41#2317C2×C311yescharacteristic
42#2059C2×C314yescomplemented
43#2058C2×C314yescomplemented
44#1736C2×C311yescharacteristic
45#1735C2×C311yescharacteristic
46#1729C2×C311yescharacteristic
47C22×C3×Q8—22no—
48C23×C3×C4—21no—
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C2×C3×M16—22no—
52C2×C3×M16—22no—
53C2×C3×M16Q814yescomplemented
54C22×C3×C8—22no—
55C22×C3×C8—22no—
56C22×C3×C8Q814yescomplemented
57C22×C3×C8C2312yes—
58C22×C3×C8C2312yes—
59C3×C4⋊Q8—28no—
60C3×C4⋊Q8—24no—
61C3×C4⋊Q8—24no—
62C3×#125—28no—
63C3×#125—24no—
64C3×#125—24no—
65C3×C22⋊Q8—216no—
66C3×#117—22no—
67C3×#117—22no—
68C3×#117—22no—
69C3×#117—22no—
70C3×#117D414yes—
71C3×#117Q814yescomplemented
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4D414yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4D412yes—
92C2×C3×C42—21no—
93C2×C3×C42—21no—
94C2×C3×C42C2312yes—
95C3×#107—24no—
96C3×#106—28no—
97C3×#106—24no—
98C3×#95—24no—
99C22×M16C22×C311yescharacteristic
100C23×C8C22×C311yescharacteristic
101#544—24no—
102#544—24no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C22×C312yes—
105C2×#125—21no—
106C2×#125C22×C312yes—
107C2×C22⋊Q8—22no—
108C2×#117C22×C311yescharacteristic
109C2×#117C22×C311yescharacteristic
110C2×#117C22×C311yescharacteristic
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C4⋊C4C22×C311yescharacteristic
113#428—24no—
114C2×#107C22×C312yes—
115C2×#106—24no—
116C2×#106C22×C312yes—
117C2×#106C22×C314yes—
118#389C22×C314yes—
119#385—22no—
120#384—21no—
121#382—21no—
122#340C22×C312yes—
123#340C22×C314yes—
124#340C22×C312yes—
125#336C22×C314yes—
126C24×C3C2×D411yescharacteristic
127C2×C3×Q8—28no—
128C2×C3×Q8—84no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—42no—
136C22×C3×C4—82no—
137C22×C3×C4—82no—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×Q812yes—
144C22×C3×C4C2×Q812yes—
145C22×C3×C4C2×Q812yes—
146C22×C3×C4#4112yes—
147C22×C3×C4C2411yescharacteristic
148C3×M16—28no—
149C2×C3×C8—28no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8C2×Q814yes—
155C2×C3×C8C2×Q814yes—
156C3×C4⋊C4—28no—
157C3×C4⋊C4—28no—
158C3×C4⋊C4—28no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—84no—
170C3×C4⋊C4C2×D414yes—
171C3×C4⋊C4C2×D412yes—
172C3×C4⋊C4C2×D412yes—
173C3×C4⋊C4C2×Q814yes—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C42—24no—
178C3×C42—24no—
179C3×C42—22no—
180C3×C42—22no—
181C22×Q8—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C23×C311yescharacteristic
185C2×M16—22no—
186C2×M16—22no—
187C2×M16C3×Q814yescomplemented
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C3×Q814yescomplemented
191C22×C8C23×C312yes—
192C22×C8C23×C312yes—
193C4⋊Q8—28no—
194C4⋊Q8—24no—
195C4⋊Q8—24no—
196#125—28no—
197#125—24no—
198#125—24no—
199C22⋊Q8—216no—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—22no—
204#117C3×D414yes—
205#117C3×Q814yescomplemented
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4C3×D411yescharacteristic
216C2×C4⋊C4C3×D411yescharacteristic
217C2×C4⋊C4C3×D414yes—
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C312yes—
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C4⋊C4C23×C311yescharacteristic
222C2×C4⋊C4C23×C312yes—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C3×D412yes—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42C23×C312yes—
229#107—24no—
230#106—28no—
231#106—24no—
232#95—24no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3#12012yes—
236C23×C3C22⋊Q812yes—
237C23×C3C4⋊Q812yes—
238C23×C3C22×D411yescharacteristic
239C3×Q8—88no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—44no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—44no—
255C2×C3×C4—84no—
256C2×C3×C4—84no—
257C2×C3×C4#12012yes—
258C2×C3×C4C22⋊Q814yes—
259C2×C3×C4C22⋊Q814yes—
260C2×C3×C4C22⋊Q812yes—
261C2×C3×C4C4⋊Q814yes—
262C2×C3×C4C4⋊Q814yes—
263C2×C3×C4C4⋊Q812yes—
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C22×D411yescharacteristic
266C2×C3×C4C22×Q811yescharacteristic
267C2×C3×C4C22×Q811yescharacteristic
268C2×C3×C4C22×Q811yescharacteristic
269C2×C3×C4C2×#4111yescharacteristic
270C3×C8—24no—
271C3×C8—24no—
272C24C2×C3×D411yescharacteristic
273C2×Q8—28no—
274C2×Q8—84no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4C2×C3×D412yes—
285C22×C4C2×C3×D412yes—
286C22×C4C2×C3×D411yescharacteristic
287C22×C4C2×C3×D412yes—
288C22×C4C2×C3×D412yes—
289C22×C4C2×C3×Q812yes—
290C22×C4C2×C3×Q812yes—
291C22×C4C2×C3×Q812yes—
292C22×C4C3×#4112yes—
293C22×C4C24×C311yescharacteristic
294M16—28no—
295C2×C8—28no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8C2×C3×Q814yes—
301C2×C8C2×C3×Q814yes—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—84no—
316C4⋊C4C2×C3×D414yes—
317C4⋊C4C2×C3×D412yes—
318C4⋊C4C2×C3×D412yes—
319C4⋊C4C2×C3×Q814yes—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C42—24no—
324C42—24no—
325C42—22no—
326C42—22no—
327C22×C3—24no—
328C22×C3—24no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3#39514yes—
332C22×C3C2×#12011yescharacteristic
333C22×C3C2×C22⋊Q811yescharacteristic
334C22×C3C2×C4⋊Q811yescharacteristic
335C22×C3#57712yes—
336C22×C3#57712yes—
337C3×C4—44no—
338C3×C4—42no—
339C3×C4—44no—
340C3×C4—42no—
341C3×C4—42no—
342C3×C4—42no—
343C3×C4—82no—
344C3×C4#39514yes—
345C3×C4C2×C22⋊Q811yescharacteristic
346C3×C4C2×C22⋊Q811yescharacteristic
347C3×C4C2×C4⋊Q811yescharacteristic
348C3×C4C2×C4⋊Q811yescharacteristic
349C23—22no—
350C23—22no—
351C23C3×#12012yes—
352C23C3×C22⋊Q812yes—
353C23C3×C4⋊Q812yes—
354C23C22×C3×D411yescharacteristic
355Q8—88no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4C3×#12012yes—
374C2×C4C3×C22⋊Q812yes—
375C2×C4C3×C22⋊Q814yes—
376C2×C4C3×C22⋊Q814yes—
377C2×C4C3×C4⋊Q812yes—
378C2×C4C3×C4⋊Q814yes—
379C2×C4C3×C4⋊Q814yes—
380C2×C4C22×C3×D411yescharacteristic
381C2×C4C22×C3×D411yescharacteristic
382C2×C4C22×C3×Q811yescharacteristic
383C2×C4C22×C3×Q811yescharacteristic
384C2×C4C22×C3×Q811yescharacteristic
385C2×C4C2×C3×#4111yescharacteristic
386C8—24no—
387C8—24no—
388C2×C3—24no—
389C2×C3C2×#39511yescharacteristic
390C2×C3#300911yescharacteristic
391C2×C3#301011yescharacteristic
392C2×C3#308412yes—
393C2×C3#316312yes—
394C22—24no—
395C22—22no—
396C22—22no—
397C22—24no—
398C22C3×#39514yes—
399C22C2×C3×#12011yescharacteristic
400C22C2×C3×C22⋊Q811yescharacteristic
401C22C2×C3×C4⋊Q811yescharacteristic
402C22C3×#57712yes—
403C22C3×#57712yes—
404C4—42no—
405C4—44no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—44no—
410C4—82no—
411C4C3×#39514yes—
412C4C2×C3×C22⋊Q811yescharacteristic
413C4C2×C3×C22⋊Q811yescharacteristic
414C4C2×C3×C4⋊Q811yescharacteristic
415C4C2×C3×C4⋊Q811yescharacteristic
416C3#2355611yescharacteristic, direct product factor, Sylow
417C2—24no—
418C2C2×C3×#39511yescharacteristic
419C2C3×#300911yescharacteristic
420C2C3×#301011yescharacteristic
421C2C3×#308412yes—
422C2C3×#316312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
5. C3×#2059C2semidirect
6. C3×#2058C2semidirect
10. #23556C3direct (N × H)
42. #2059C2×C3semidirect
43. #2058C2×C3semidirect
53. C2×C3×M16Q8semidirect
56. C22×C3×C8Q8semidirect
71. C3×#117Q8semidirect
187. C2×M16C3×Q8semidirect
190. C22×C8C3×Q8semidirect
205. #117C3×Q8semidirect
416. C3#23556direct (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×#4282
C22×C3×#1062
C3×#23172
C3×#20592
C3×#20582
C3×#17362
C3×#17352
C3×#17292
#235563