Finite Group Databasebeta

← C3×#25816 #10644423

Subgroups — C3×#25816 #10644423

Order 768.

1250 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#501C212yesmaximal, complemented
3C3×#2851C211yesmaximal, characteristic
4C3×#2058C212yesmaximal, complemented
5C3×#2057C212yesmaximal, complemented
6C3×#1993C212yesmaximal, complemented
7C3×#1853C212yesmaximal
8C3×#1852C212yesmaximal
9C3×#1749C211yesmaximal, characteristic
10#25816C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8—21no—
12C2×C3×C4⋊Q8—28no—
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8—21no—
15C2×C3×C4⋊Q8C2212yes—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×#125C2211yescharacteristic
18C22×C3×C4⋊C4—21no—
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C42C2211yescharacteristic
21C3×#501—28no—
22C2×C3×#107C2214yes—
23C2×C3×#106C2214yes—
24C2×C3×C22⋊C8C2212yes—
25C2×C3×#97C2212yes—
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#387—21no—
29C3×#387—21no—
30C3×#385—21no—
31C3×#385—21no—
32C3×#382C2211yescharacteristic
33C3×#380C2211yescharacteristic
34C3×#380C2211yescharacteristic
35C3×#340C2214yes—
36C3×#340C2214yes—
37C3×#336C2212yes—
38C22×C4⋊Q8C2×C311yescharacteristic
39C2×#501C2×C312yescomplemented
40#2851C2×C311yescharacteristic
41#2058C2×C312yescomplemented
42#2057C2×C312yescomplemented
43#1993C2×C312yescomplemented
44#1853C2×C312yes—
45#1852C2×C312yes—
46#1749C2×C311yescharacteristic
47C22×C3×Q8—28no—
48C22×C3×Q8—22no—
49C22×C3×Q8—22no—
50C22×C3×Q8—44no—
51C22×C3×Q8—44no—
52C23×C3×C4—22no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×C3×C8C2314yes—
57C3×C4⋊Q8—28no—
58C3×C4⋊Q8—28no—
59C3×C4⋊Q8—28no—
60C3×C4⋊Q8—24no—
61C3×C4⋊Q8D418yes—
62C3×#125—24no—
63C2×C3×C4⋊C4—28no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—28no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2314yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C42—24no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C3×#107—28no—
85C3×#106—28no—
86C3×C22⋊C8Q818yescomplemented
87C3×#97—24no—
88C3×#95—21no—
89C3×#95—21no—
90C3×#95—21no—
91C3×#95—21no—
92C3×#95—21no—
93C3×#95—21no—
94C23×Q8—21no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C22×C312yes—
99C2×C4⋊Q8C22×C312yes—
100C2×#125C22×C311yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4C22×C312yes—
103C22×C42C22×C311yescharacteristic
104#501—28no—
105C2×#107C22×C314yes—
106C2×#106C22×C314yes—
107C2×C22⋊C8C22×C312yes—
108C2×#97C22×C312yes—
109#389C22×C311yescharacteristic
110#389C22×C311yescharacteristic
111#387—21no—
112#387—21no—
113#385—21no—
114#385—21no—
115#382C22×C311yescharacteristic
116#380C22×C311yescharacteristic
117#380C22×C311yescharacteristic
118#340C22×C314yes—
119#340C22×C314yes—
120#336C22×C312yes—
121C24×C3C2×D411yescharacteristic
122C2×C3×Q8—28no—
123C2×C3×Q8—28no—
124C2×C3×Q8—432no—
125C2×C3×Q8—44no—
126C2×C3×Q8—44no—
127C2×C3×Q8—44no—
128C2×C3×Q8—44no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—28no—
138C22×C3×C4—24no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—44no—
144C22×C3×C4—44no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×Q814yes—
151C22×C3×C4C2411yescharacteristic
152C2×C3×C8—24no—
153C2×C3×C8—24no—
154C2×C3×C8C2×Q818yescomplemented
155C3×C4⋊C4—28no—
156C3×C4⋊C4—28no—
157C3×C4⋊C4—28no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—42no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4C2×D418yes—
168C3×C22⋊C4—42no—
169C3×C22⋊C4—42no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42C2×D414yes—
174C22×Q8—28no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—44no—
178C22×Q8—44no—
179C23×C4—22no—
180C23×C4C3×D411yescharacteristic
181C23×C4C3×D411yescharacteristic
182C23×C4C23×C311yescharacteristic
183C22×C8C23×C314yes—
184C4⋊Q8—28no—
185C4⋊Q8—28no—
186C4⋊Q8—28no—
187C4⋊Q8—24no—
188C4⋊Q8C3×D418yes—
189#125—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C23×C311yescharacteristic
200C2×C4⋊C4C23×C311yescharacteristic
201C2×C4⋊C4C23×C311yescharacteristic
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C4⋊C4C23×C314yes—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—24no—
207C2×C42C3×D411yescharacteristic
208C2×C42C3×D411yescharacteristic
209C2×C42C23×C311yescharacteristic
210C2×C42C23×C311yescharacteristic
211#107—28no—
212#106—28no—
213C22⋊C8C3×Q818yescomplemented
214#97—24no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3C4⋊Q812yes—
226C23×C3AGL(1,ℤ/8)12yes—
227C23×C3#13712yes—
228C23×C3C22×D411yescharacteristic
229C3×Q8—416no—
230C3×Q8—416no—
231C2×C3×C4—28no—
232C2×C3×C4—28no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—28no—
244C2×C3×C4—28no—
245C2×C3×C4—416no—
246C2×C3×C4—44no—
247C2×C3×C4—44no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4#12012yes—
261C2×C3×C4#12012yes—
262C2×C3×C4C4⋊Q812yes—
263C2×C3×C4AGL(1,ℤ/8)12yes—
264C2×C3×C4#13712yes—
265C2×C3×C4C22×D412yes—
266C2×C3×C4C22×Q812yes—
267C2×C3×C42−1+412yes—
268C3×C8—28no—
269C24C2×C3×D411yescharacteristic
270C2×Q8—28no—
271C2×Q8—28no—
272C2×Q8—432no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—44no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C2×C3×D411yescharacteristic
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×Q814yes—
299C22×C4C24×C311yescharacteristic
300C2×C8—24no—
301C2×C8—24no—
302C2×C8C2×C3×Q818yescomplemented
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4C2×C3×D418yes—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42C2×C3×D414yes—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3#50114yescomplemented
329C22×C3C2×#12011yescharacteristic
330C22×C3C2×C4⋊Q811yescharacteristic
331C22×C3#53611yescharacteristic
332C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
333C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
334C22×C3C2×#13711yescharacteristic
335C22×C3C2×#13711yescharacteristic
336C3×C4—22no—
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—48no—
342C3×C4—81no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4—81no—
346C3×C4D4×Q814yes—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23C3×C4⋊Q812yes—
352C23C3×AGL(1,ℤ/8)12yes—
353C23C3×#13712yes—
354C23C22×C3×D411yescharacteristic
355Q8—416no—
356Q8—416no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—28no—
370C2×C4—28no—
371C2×C4—416no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4C3×#12012yes—
387C2×C4C3×#12012yes—
388C2×C4C3×C4⋊Q812yes—
389C2×C4C3×AGL(1,ℤ/8)12yes—
390C2×C4C3×#13712yes—
391C2×C4C22×C3×D412yes—
392C2×C4C22×C3×Q812yes—
393C2×C4C3×2−1+412yes—
394C8—28no—
395C2×C3—24no—
396C2×C3#243411yescharacteristic
397C2×C3#300311yescharacteristic
398C2×C3#300411yescharacteristic
399C2×C3C2×#50111yescharacteristic
400C2×C3#316511yescharacteristic
401C2×C3#323811yescharacteristic
402C2×C3#323911yescharacteristic
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22C3×#50114yescomplemented
410C22C2×C3×#12011yescharacteristic
411C22C2×C3×C4⋊Q811yescharacteristic
412C22C3×#53611yescharacteristic
413C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
414C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
415C22C2×C3×#13711yescharacteristic
416C22C2×C3×#13711yescharacteristic
417C4—22no—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—48no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4—81no—
427C4C3×D4×Q814yes—
428C3#2581611yescharacteristic, direct product factor, Sylow
429C2—24no—
430C2C3×#243411yescharacteristic
431C2C3×#300311yescharacteristic
432C2C3×#300411yescharacteristic
433C2C2×C3×#50111yescharacteristic
434C2C3×#316511yescharacteristic
435C2C3×#323811yescharacteristic
436C2C3×#323911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
4. C3×#2058C2semidirect
5. C3×#2057C2semidirect
6. C3×#1993C2semidirect
10. #25816C3direct (N × H)
39. C2×#501C2×C3semidirect
41. #2058C2×C3semidirect
42. #2057C2×C3semidirect
43. #1993C2×C3semidirect
86. C3×C22⋊C8Q8semidirect
154. C2×C3×C8C2×Q8semidirect
213. C22⋊C8C3×Q8semidirect
302. C2×C8C2×C3×Q8semidirect
328. C22×C3#501semidirect
409. C22C3×#501semidirect
428. C3#25816direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#5012
C3×#28512
C3×#20582
C3×#20572
C3×#19932
C3×#18532
C3×#18522
C3×#17492
#258163