Finite Group Databasebeta

← C2×C3×#2133 #10651973

Subgroups — C2×C3×#2133 #10651973

Order 768.

1198 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2C22×C3×#101C212yesmaximal, complemented
3C3×#2133C218yesmaximal, direct product factor
4C2×C3×#356C212yesmaximal, complemented
5C2×C3×#355C212yesmaximal, complemented
6C2×#2133C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×#41C411yescharacteristic
8C23×C3×Q8C2211yescharacteristic
9C22×C3×M16—22no—
10C3×#536—24no—
11C3×#536C414yes—
12C3×#536C2214yes—
13C2×C3×C4⋊Q8C411yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#124C411yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×C3×#123—22no—
18C2×C3×C22⋊Q8—22no—
19C2×C3×#117C411yescharacteristic
20C2×C3×#101—24no—
21C2×C3×#101—22no—
22C2×C3×#101C2214yes—
23C2×C3×#101C2218yescomplemented
24C3×#356C2218yescomplemented
25C3×#355C2218yescomplemented
26C2×#536C2×C311yescharacteristic
27C22×#101C2×C312yescomplemented
28#2133C2×C318yesdirect product factor
29C2×#356C2×C312yescomplemented
30C2×#355C2×C312yescomplemented
31C2×C3×#41—22no—
32C2×C3×#41—42no—
33C2×C3×#41—42no—
34C2×C3×#41C2×C414yes—
35C22×C3×Q8—42no—
36C22×C3×Q8—42no—
37C22×C3×Q8C2×C411yescharacteristic
38C22×C3×Q8C2×C411yescharacteristic
39C22×C3×Q8C2×C412yes—
40C22×C3×Q8D412yes—
41C22×C3×Q8D414yes—
42C22×C3×Q8C2311yescharacteristic
43C22×C3×Q8C2312yes—
44C22×C3×D4—21no—
45C22×C3×D4C2×C411yescharacteristic
46C23×C3×C4—21no—
47C23×C3×C4C2×C411yescharacteristic
48C23×C3×C4D412yes—
49C2×C3×M16—24no—
50C2×C3×M16—28no—
51C2×C3×M16—44no—
52C2×C3×M16—42no—
53C22×C3×C8—42no—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8C2×C414yes—
57C3×C4⋊Q8C2×C412yes—
58C3×C4⋊Q8C2312yes—
59C3×#124—22no—
60C3×#124—22no—
61C3×#124C2×C414yes—
62C3×#124C2×C412yes—
63C3×#124C2312yes—
64C3×#123—28no—
65C3×#123—28no—
66C3×C22⋊Q8—28no—
67C3×C22⋊Q8—28no—
68C3×#117—22no—
69C3×#117C2×C414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C42C2×C411yescharacteristic
77C2×C3×C42C2×C411yescharacteristic
78C3×#101—24no—
79C3×#101—28no—
80C3×#101C2318yes—
81C22×#41C3×C411yescharacteristic
82C23×Q8C22×C311yescharacteristic
83C22×M16—22no—
84#536—24no—
85#536C3×C414yes—
86#536C22×C314yes—
87C2×C4⋊Q8C3×C411yescharacteristic
88C2×C4⋊Q8C22×C311yescharacteristic
89C2×#124C3×C411yescharacteristic
90C2×#124C22×C311yescharacteristic
91C2×#123—22no—
92C2×C22⋊Q8—22no—
93C2×#117C3×C411yescharacteristic
94C2×#101—24no—
95C2×#101—22no—
96C2×#101C22×C314yes—
97C2×#101C22×C318yescomplemented
98#356C22×C318yescomplemented
99#355C22×C318yescomplemented
100C24×C3—21no—
101C24×C3C22⋊C411yescharacteristic
102C3×#41—48no—
103C3×#41—48no—
104C2×C3×Q8—21no—
105C2×C3×Q8—41no—
106C2×C3×Q8—44no—
107C2×C3×Q8—41no—
108C2×C3×Q8—44no—
109C2×C3×Q8—42no—
110C2×C3×Q8—44no—
111C2×C3×Q8—42no—
112C2×C3×Q8—44no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×Q8C22⋊C414yes—
118C2×C3×Q8C22⋊C412yes—
119C2×C3×Q8C22×C412yes—
120C2×C3×Q8C22×C412yes—
121C2×C3×Q8C22×C411yescharacteristic
122C2×C3×Q8C2×D414yes—
123C2×C3×Q8C2×D412yes—
124C2×C3×Q8C2411yescharacteristic
125C2×C3×D4—24no—
126C2×C3×D4—21no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—42no—
130C2×C3×D4—42no—
131C2×C3×D4C22×C412yes—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—24no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4C22⋊C412yes—
144C22×C3×C4C22⋊C411yescharacteristic
145C22×C3×C4C22⋊C412yes—
146C22×C3×C4C22⋊C414yes—
147C22×C3×C4C22×C411yescharacteristic
148C22×C3×C4C22×C412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D414yes—
151C3×M16—28no—
152C3×M16—44no—
153C3×M16—48no—
154C2×C3×C8—48no—
155C2×C3×C8—44no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—28no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—24no—
160C3×C22⋊C4—24no—
161C3×C22⋊C4—28no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—24no—
164C3×C42—21no—
165C3×C42—21no—
166C3×C42C22×C412yes—
167C3×C42C22×C412yes—
168C2×#41—22no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41C2×C3×C414yes—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C2×C3×C411yescharacteristic
175C22×Q8C2×C3×C411yescharacteristic
176C22×Q8C2×C3×C412yes—
177C22×Q8C3×D412yes—
178C22×Q8C3×D414yes—
179C22×Q8C23×C311yescharacteristic
180C22×Q8C23×C312yes—
181C22×D4—21no—
182C22×D4C2×C3×C411yescharacteristic
183C23×C4—21no—
184C23×C4C2×C3×C411yescharacteristic
185C23×C4C3×D412yes—
186C2×M16—24no—
187C2×M16—28no—
188C2×M16—42no—
189C2×M16—44no—
190C22×C8—42no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8C2×C3×C414yes—
194C4⋊Q8C2×C3×C412yes—
195C4⋊Q8C23×C312yes—
196#124—22no—
197#124—22no—
198#124C2×C3×C414yes—
199#124C2×C3×C412yes—
200#124C23×C312yes—
201#123—28no—
202#123—28no—
203C22⋊Q8—28no—
204C22⋊Q8—28no—
205#117—22no—
206#117C2×C3×C414yes—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42C2×C3×C411yescharacteristic
214C2×C42C2×C3×C411yescharacteristic
215#101—24no—
216#101—28no—
217#101C23×C318yes—
218C23×C3—22no—
219C23×C3—42no—
220C23×C3—44no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3#9912yes—
224C23×C3#9912yes—
225C23×C3C2×C22⋊C411yescharacteristic
226C23×C3C2×C22⋊C412yes—
227C3×Q8—44no—
228C3×Q8—42no—
229C3×Q8—42no—
230C3×Q8—41no—
231C3×Q8—44no—
232C3×Q8—42no—
233C3×Q8—41no—
234C3×Q8—42no—
235C3×Q8—42no—
236C3×D4—42no—
237C3×D4—42no—
238C3×D4—48no—
239C2×C3×C4—24no—
240C2×C3×C4—22no—
241C2×C3×C4—24no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—48no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4C2×C22⋊C411yescharacteristic
262C2×C3×C4C2×C22⋊C414yes—
263C2×C3×C4C2×C22⋊C412yes—
264C2×C3×C4C2×C22⋊C412yes—
265C2×C3×C4C23×C411yescharacteristic
266C2×C3×C4C22×D412yes—
267C3×C8—48no—
268C24—21no—
269C24C3×C22⋊C411yescharacteristic
270#41—48no—
271#41—48no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—44no—
276C2×Q8—42no—
277C2×Q8—44no—
278C2×Q8—42no—
279C2×Q8—44no—
280C2×Q8—41no—
281C2×Q8—44no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8C3×C22⋊C414yes—
286C2×Q8C3×C22⋊C412yes—
287C2×Q8C22×C3×C412yes—
288C2×Q8C22×C3×C412yes—
289C2×Q8C22×C3×C411yescharacteristic
290C2×Q8C2×C3×D414yes—
291C2×Q8C2×C3×D412yes—
292C2×Q8C24×C311yescharacteristic
293C2×D4—24no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4C22×C3×C412yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C3×C22⋊C412yes—
312C22×C4C3×C22⋊C411yescharacteristic
313C22×C4C3×C22⋊C412yes—
314C22×C4C3×C22⋊C414yes—
315C22×C4C22×C3×C411yescharacteristic
316C22×C4C22×C3×C412yes—
317C22×C4C2×C3×D412yes—
318C22×C4C2×C3×D414yes—
319M16—28no—
320M16—44no—
321M16—48no—
322C2×C8—44no—
323C2×C8—48no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—28no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C42—21no—
333C42—21no—
334C42C22×C3×C412yes—
335C42C22×C3×C412yes—
336C22×C3—22no—
337C22×C3—24no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—44no—
341C22×C3—42no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3C2×#9911yescharacteristic
345C22×C3C2×#9914yes—
346C22×C3C2×#9911yescharacteristic
347C22×C3C22×C22⋊C411yescharacteristic
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—24no—
352C3×C4—22no—
353C3×C4—21no—
354C3×C4—42no—
355C3×C4—42no—
356C3×C4—42no—
357C23—22no—
358C23—42no—
359C23—44no—
360C23—41no—
361C23—41no—
362C23C3×#9912yes—
363C23C3×#9912yes—
364C23C2×C3×C22⋊C411yescharacteristic
365C23C2×C3×C22⋊C412yes—
366Q8—44no—
367Q8—42no—
368Q8—42no—
369Q8—41no—
370Q8—44no—
371Q8—42no—
372Q8—41no—
373Q8—42no—
374Q8—42no—
375D4—42no—
376D4—42no—
377D4—48no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4C2×C3×C22⋊C411yescharacteristic
401C2×C4C2×C3×C22⋊C414yes—
402C2×C4C2×C3×C22⋊C412yes—
403C2×C4C2×C3×C22⋊C412yes—
404C2×C4C23×C3×C411yescharacteristic
405C2×C4C22×C3×D412yes—
406C8—48no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—24no—
410C2×C3—82no—
411C2×C3#213312yesdirect product factor
412C2×C3C22×#9911yescharacteristic
413C22—22no—
414C22—24no—
415C22—42no—
416C22—42no—
417C22—44no—
418C22—42no—
419C22—82no—
420C22—82no—
421C22C2×C3×#9911yescharacteristic
422C22C2×C3×#9914yes—
423C22C2×C3×#9911yescharacteristic
424C22C22×C3×C22⋊C411yescharacteristic
425C4—22no—
426C4—22no—
427C4—21no—
428C4—24no—
429C4—22no—
430C4—21no—
431C4—42no—
432C4—42no—
433C4—42no—
434C3C2×#213311yescharacteristic, direct product factor, Sylow
435C2—21no—
436C2—21no—
437C2—24no—
438C2—82no—
439C2C3×#213312yesdirect product factor
440C2C22×C3×#9911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#99
commutatorC2×C4C23×C3×C4
Frattini, cosocleC2×Q8C24×C3
prime radicalC22×C3×#41C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×#101C2semidirect
3. C3×#2133C2direct (N × H)
4. C2×C3×#356C2semidirect
5. C2×C3×#355C2semidirect
6. C2×#2133C3direct (N × H)
23. C2×C3×#101C22semidirect
24. C3×#356C22semidirect
25. C3×#355C22semidirect
27. C22×#101C2×C3semidirect
28. #2133C2×C3direct (N × H)
29. C2×#356C2×C3semidirect
30. C2×#355C2×C3semidirect
97. C2×#101C22×C3semidirect
98. #356C22×C3semidirect
99. #355C22×C3semidirect
411. C2×C3#2133direct (N × H)
434. C3C2×#2133direct (N × H)
439. C2C3×#2133direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C22×C3×#1012
C3×#21332
C2×C3×#3562
C2×C3×#3552
C2×#21333