Finite Group Databasebeta

← C3×#61826 #10680433

Subgroups — C3×#61826 #10680433

Order 768.

1726 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3502C211yesmaximal, characteristic, complemented
2C2×C3×#554C211yesmaximal, characteristic
3C3×#3385C218yesmaximal, complemented
4C3×#3372C218yesmaximal, complemented
5C3×#3236C214yesmaximal, complemented
6C3×#3230C212yesmaximal, complemented
7C3×#3110C214yesmaximal
8C3×#3101C212yesmaximal
9C3×#2983C211yesmaximal, characteristic, complemented
10#61826C311yesmaximal, characteristic, direct product factor, Sylow
11C3×Q82C2214yescomplemented
12C3×#557—24no—
13C3×#557C2214yescomplemented
14C3×#554—24no—
15C3×#554C2218yes—
16C3×#544—21no—
17C3×#544—22no—
18C3×#533C2212yes—
19C3×#533C2214yes—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#125C2212yes—
22C2×C3×C22⋊Q8—21no—
23C2×C3×C22⋊Q8—22no—
24C3×#519—21no—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×C4×D4C2212yes—
28C22×C3×C4⋊C4—21no—
29C3×#501C2218yescomplemented
30C3×#500C2214yescomplemented
31C3×#491C2218yescomplemented
32C3×#487C2214yescomplemented
33C3×#484C2218yes—
34C3×#483C2214yes—
35C3×#479C22116yescomplemented
36C3×#477C2218yescomplemented
37C3×#476C2214yes—
38C3×#474C2218yes—
39C3×#461C2214yes—
40C3×#461C2218yes—
41C3×#450C2214yes—
42C3×#446C2218yescomplemented
43C3×#440C22116yes—
44C3×C4×SD16C2218yes—
45C3×#432C2214yes—
46C3×#431C2212yes—
47#3502C2×C311yescharacteristic, complemented
48C2×#554C2×C311yescharacteristic
49#3385C2×C318yescomplemented
50#3372C2×C318yescomplemented
51#3236C2×C314yescomplemented
52#3230C2×C312yescomplemented
53#3110C2×C314yes—
54#3101C2×C312yes—
55#2983C2×C311yescharacteristic, complemented
56C22×C3×Q8—21no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4—22no—
61C2×C3×SD16C2318yes—
62C3×C4⋊Q8—24no—
63C3×C4⋊Q8—28no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8C2312yes—
66C3×C4⋊Q8C2314yes—
67C3×C4⋊Q8C2314yes—
68C3×C4⋊Q8C2312yes—
69C3×C4⋊Q8C2314yes—
70C3×#125—24no—
71C3×#125—28no—
72C3×#125—24no—
73C3×#125—24no—
74C3×#125—22no—
75C3×#125C2318yes—
76C3×C22⋊Q8—28no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—216no—
80C3×C22⋊Q8C2314yes—
81C3×C4×Q8—24no—
82C3×C4×Q8—24no—
83C3×C4×Q8—24no—
84C3×C4×Q8—22no—
85C3×C4×Q8D414yes—
86C3×C4×Q8C2314yes—
87C3×C4×Q8C2318yes—
88C3×C4×Q8C2314yes—
89C3×C4×D4—24no—
90C3×C4×D4—22no—
91C3×C4×D4C2314yes—
92C3×C4×D4C2318yes—
93C3×#117—24no—
94C3×#117—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—28no—
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4D412yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2312yes—
107C3×#107C23116yes—
108C3×#106C2318yes—
109C3×C4⋊C8C2314yes—
110C3×C4⋊C8C2318yes—
111C3×#103C2318yes—
112C3×#103C23116yes—
113C3×#102C2318yes—
114C3×C22⋊C8C2314yes—
115C3×#97C2318yes—
116C3×C4×C8C2314yes—
117Q82C22×C314yescomplemented
118#557—24no—
119#557C22×C314yescomplemented
120#554—24no—
121#554C22×C318yes—
122#544—21no—
123#544—22no—
124#533C22×C312yes—
125#533C22×C314yes—
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#125C22×C312yes—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—22no—
130#519—21no—
131C2×C4×Q8C22×C311yescharacteristic
132C2×C4×D4C22×C311yescharacteristic
133C2×C4×D4C22×C312yes—
134C22×C4⋊C4—21no—
135#501C22×C318yescomplemented
136#500C22×C314yescomplemented
137#491C22×C318yescomplemented
138#487C22×C314yescomplemented
139#484C22×C318yes—
140#483C22×C314yes—
141#479C22×C3116yescomplemented
142#477C22×C318yescomplemented
143#476C22×C314yes—
144#474C22×C318yes—
145#461C22×C314yes—
146#461C22×C318yes—
147#450C22×C314yes—
148#446C22×C318yescomplemented
149#440C22×C3116yes—
150C4×SD16C22×C318yes—
151#432C22×C314yes—
152#431C22×C312yes—
153C24×C3—21no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—24no—
157C2×C3×Q8—24no—
158C2×C3×Q8—82no—
159C2×C3×Q8C2×D414yes—
160C2×C3×Q8C2414yes—
161C2×C3×D4—24no—
162C2×C3×D4—21no—
163C2×C3×D4C2412yes—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—28no—
168C22×C3×C4—44no—
169C22×C3×C4—84no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C3×SD16—216no—
176C2×C3×C8C2418yes—
177C3×C4⋊C4—28no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—28no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—28no—
191C3×C4⋊C4C2×D414yes—
192C3×C4⋊C4C2×D418yes—
193C3×C4⋊C4C2414yes—
194C3×C4⋊C4C2412yes—
195C3×C4⋊C4C2414yes—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—28no—
199C3×C42—28no—
200C3×C42—24no—
201C3×C42C2×D414yes—
202C3×C42C2×D412yes—
203C3×C42C2412yes—
204C3×C42C2414yes—
205C22×Q8—21no—
206C22×Q8C3×D411yescharacteristic
207C22×D4C23×C311yescharacteristic
208C23×C4—21no—
209C23×C4—22no—
210C2×SD16C23×C318yes—
211C4⋊Q8—24no—
212C4⋊Q8—28no—
213C4⋊Q8—22no—
214C4⋊Q8C23×C312yes—
215C4⋊Q8C23×C314yes—
216C4⋊Q8C23×C314yes—
217C4⋊Q8C23×C314yes—
218C4⋊Q8C23×C312yes—
219#125—24no—
220#125—28no—
221#125—24no—
222#125—24no—
223#125—22no—
224#125C23×C318yes—
225C22⋊Q8—28no—
226C22⋊Q8—22no—
227C22⋊Q8—24no—
228C22⋊Q8—216no—
229C22⋊Q8C23×C314yes—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—22no—
234C4×Q8C3×D414yes—
235C4×Q8C23×C314yes—
236C4×Q8C23×C318yes—
237C4×Q8C23×C314yes—
238C4×D4—24no—
239C4×D4—22no—
240C4×D4C23×C314yes—
241C4×D4C23×C318yes—
242#117—24no—
243#117—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4C3×D411yescharacteristic
249C2×C4⋊C4C3×D412yes—
250C2×C4⋊C4C23×C311yescharacteristic
251C2×C4⋊C4C23×C312yes—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C42C23×C312yes—
255C2×C42C23×C311yescharacteristic
256#107C23×C3116yes—
257#106C23×C318yes—
258C4⋊C8C23×C314yes—
259C4⋊C8C23×C318yes—
260#103C23×C318yes—
261#103C23×C3116yes—
262#102C23×C318yes—
263C22⋊C8C23×C314yes—
264#97C23×C318yes—
265C4×C8C23×C314yes—
266C23×C3—24no—
267C23×C3—22no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3C22×D411yescharacteristic
271C3×Q8—28no—
272C3×Q8—28no—
273C3×Q8—84no—
274C3×Q8—82no—
275C3×Q8—84no—
276C3×D4—24no—
277C3×D4—22no—
278C3×D42−1+414yes—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—44no—
288C2×C3×C4—42no—
289C2×C3×C4—48no—
290C2×C3×C4—88no—
291C2×C3×C4#13714yes—
292C2×C3×C4C22×D414yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×D412yes—
295C2×C3×C4C22×D414yes—
296C2×C3×C4C22×D412yes—
297C2×C3×C4C2511yescharacteristic
298C3×C8—28no—
299C24—21no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—82no—
305C2×Q8C2×C3×D414yes—
306C2×Q8C24×C314yes—
307C2×D4—24no—
308C2×D4—21no—
309C2×D4C24×C312yes—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—44no—
315C22×C4—84no—
316C22×C4C2×C3×D412yes—
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C2×C3×D412yes—
319C22×C4C2×C3×D411yescharacteristic
320C22×C4C24×C311yescharacteristic
321SD16—216no—
322C2×C8C24×C318yes—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4C2×C3×D414yes—
338C4⋊C4C2×C3×D418yes—
339C4⋊C4C24×C314yes—
340C4⋊C4C24×C312yes—
341C4⋊C4C24×C314yes—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—28no—
345C42—28no—
346C42—24no—
347C42C2×C3×D414yes—
348C42C2×C3×D412yes—
349C42C24×C312yes—
350C42C24×C314yes—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—24no—
354C22×C3—48no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3C2×#13712yes—
358C22×C3C23×D411yescharacteristic
359C3×C4—24no—
360C3×C4—22no—
361C3×C4—21no—
362C3×C4—24no—
363C3×C4—44no—
364C3×C4—44no—
365C3×C4—44no—
366C3×C4—42no—
367C3×C4C2×#13714yes—
368C3×C4C2×2−1+412yes—
369C23—24no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23C22×C3×D411yescharacteristic
374Q8—28no—
375Q8—28no—
376Q8—84no—
377Q8—84no—
378Q8—82no—
379D4—24no—
380D4—22no—
381D4C3×2−1+414yes—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—48no—
393C2×C4—88no—
394C2×C4C3×#13714yes—
395C2×C4C22×C3×D414yes—
396C2×C4C22×C3×D411yescharacteristic
397C2×C4C22×C3×D412yes—
398C2×C4C22×C3×D414yes—
399C2×C4C22×C3×D412yes—
400C2×C4C25×C311yescharacteristic
401C8—28no—
402C2×C3—22no—
403C2×C3—44no—
404C2×C3#348611yescharacteristic
405C2×C3C22×#13711yescharacteristic
406C2×C3#358511yescharacteristic
407C22—21no—
408C22—21no—
409C22—24no—
410C22—48no—
411C22—42no—
412C22—42no—
413C22C2×C3×#13712yes—
414C22C23×C3×D411yescharacteristic
415C4—21no—
416C4—24no—
417C4—24no—
418C4—22no—
419C4—44no—
420C4—42no—
421C4—44no—
422C4—44no—
423C4C2×C3×#13714yes—
424C4C2×C3×2−1+412yes—
425C3#6182611yescharacteristic, direct product factor, Sylow
426C2—22no—
427C2—44no—
428C2C3×#348611yescharacteristic
429C2C22×C3×#13711yescharacteristic
430C2C3×#358511yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC22×C3×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3502C2semidirect
3. C3×#3385C2semidirect
4. C3×#3372C2semidirect
5. C3×#3236C2semidirect
6. C3×#3230C2semidirect
9. C3×#2983C2semidirect
10. #61826C3direct (N × H)
11. C3×Q82C22semidirect
13. C3×#557C22semidirect
29. C3×#501C22semidirect
30. C3×#500C22semidirect
31. C3×#491C22semidirect
32. C3×#487C22semidirect
35. C3×#479C22semidirect
36. C3×#477C22semidirect
42. C3×#446C22semidirect
47. #3502C2×C3semidirect
49. #3385C2×C3semidirect
50. #3372C2×C3semidirect
51. #3236C2×C3semidirect
52. #3230C2×C3semidirect
55. #2983C2×C3semidirect
117. Q82C22×C3semidirect
119. #557C22×C3semidirect
135. #501C22×C3semidirect
136. #500C22×C3semidirect
137. #491C22×C3semidirect
138. #487C22×C3semidirect
141. #479C22×C3semidirect
142. #477C22×C3semidirect
148. #446C22×C3semidirect
425. C3#61826direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35022
C2×C3×#5542
C3×#33852
C3×#33722
C3×#32362
C3×#32302
C3×#31102
C3×#31012
C3×#29832
#618263