Finite Group Databasebeta

← C3×#34388 #10652995

Subgroups — C3×#34388 #10652995

Order 768.

2438 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C3×#2381C212yesmaximal, complemented
3C3×#2380C212yesmaximal, complemented
4C3×#2364C214yesmaximal, complemented
5C3×#2353C214yesmaximal, complemented
6C3×#2339C214yesmaximal, complemented
7C3×#2337C214yesmaximal, complemented
8C3×#2325C214yesmaximal, complemented
9C3×#2318C214yesmaximal, complemented
10C3×#2283C211yesmaximal, characteristic, complemented
11C2×C3×#383C211yesmaximal, characteristic, complemented
12#34388C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C3×#532C4116yescomplemented
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124C2212yes—
18C2×C3×#121C2212yes—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4C2218yes—
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2212yes—
23C22×C3×C42C2211yescharacteristic
24C3×#391C2218yes—
25C3×#388C2218yes—
26C3×#388C2218yes—
27C3×#387C2218yes—
28C3×#387C2218yes—
29C3×#386C2218yes—
30C3×#386C2218yes—
31C3×#386C2218yes—
32C3×#386C2218yes—
33C3×#385C2218yes—
34C3×#383C2212yes—
35C3×#383C2212yes—
36C3×#383C2212yes—
37C3×#383C2212yes—
38C3×#383C2212yes—
39C3×#382C2212yes—
40C3×#381C2212yes—
41C3×#381C2214yes—
42C3×#380C2212yes—
43C3×#380C2214yes—
44C3×C4×C4⋊C4C2212yes—
45C3×C4×C22⋊C4C2212yes—
46C3×C4×C22⋊C4C2218yes—
47C3×C4×C22⋊C4C2214yes—
48C3×#376C2212yes—
49C2×C3×#95C2212yes—
50C2×C3×#95C2212yes—
51C2×C3×#95C2212yes—
52C2×C3×#95C2214yes—
53C2×#532C2×C311yescharacteristic
54#2381C2×C312yescomplemented
55#2380C2×C312yescomplemented
56#2364C2×C314yescomplemented
57#2353C2×C314yescomplemented
58#2339C2×C314yescomplemented
59#2337C2×C314yescomplemented
60#2325C2×C314yescomplemented
61#2318C2×C314yescomplemented
62#2283C2×C311yescharacteristic, complemented
63C2×#383C2×C311yescharacteristic, complemented
64C2×C3×#41—48no—
65C2×C3×#41C2×C4116yescomplemented
66C22×C3×Q8C2312yes—
67C22×C3×D4C2314yes—
68C22×C3×D4C2312yes—
69C22×C3×D4C2312yes—
70C22×C3×D4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2314yes—
73C23×C3×C4C2318yes—
74C23×C3×C4C2314yes—
75C23×C3×C4C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C3×C4⋊Q8C2×C418yescomplemented
78C3×#127C2×C418yescomplemented
79C3×#124C2×C4116yescomplemented
80C3×#121C2×C4116yescomplemented
81C3×#121C2×C4116yescomplemented
82C3×C4×D4—416no—
83C3×C4×D4C2×C4116yescomplemented
84C3×C4×D4C2×C4116yescomplemented
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2318yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4C2314yes—
90C2×C3×C22⋊C4C2318yes—
91C2×C3×C22⋊C4C2318yes—
92C2×C3×C22⋊C4C2318yes—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C22⋊C4C2314yes—
95C2×C3×C42C2×C418yescomplemented
96C2×C3×C42C2314yes—
97C2×C3×C42C2318yes—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C3×#95C2314yes—
103C3×#95C2314yes—
104C3×#95C2314yes—
105C3×#95C2314yes—
106C3×#95C2314yes—
107C3×#95C2314yes—
108C3×#95C2314yes—
109C3×#95C2318yes—
110C3×#95C2314yes—
111C3×#95C2318yes—
112C3×#95C2314yes—
113C3×#95C2314yes—
114C22×#41C22×C312yes—
115#532C3×C4116yescomplemented
116C2×C4⋊Q8C22×C311yescharacteristic
117C2×#127C22×C311yescharacteristic
118C2×#124C22×C312yes—
119C2×#121C22×C312yes—
120C2×#121C22×C312yes—
121C2×C4×D4C22×C318yes—
122C2×C4×D4C22×C312yes—
123C2×C4×D4C22×C312yes—
124C22×C42C22×C311yescharacteristic
125#391C22×C318yes—
126#388C22×C318yes—
127#388C22×C318yes—
128#387C22×C318yes—
129#387C22×C318yes—
130#386C22×C318yes—
131#386C22×C318yes—
132#386C22×C318yes—
133#386C22×C318yes—
134#385C22×C318yes—
135#383C22×C312yes—
136#383C22×C312yes—
137#383C22×C312yes—
138#383C22×C312yes—
139#383C22×C312yes—
140#382C22×C312yes—
141#381C22×C314yes—
142#381C22×C312yes—
143#380C22×C314yes—
144#380C22×C312yes—
145C4×C4⋊C4C22×C312yes—
146C4×C22⋊C4C22×C314yes—
147C4×C22⋊C4C22×C318yes—
148C4×C22⋊C4C22×C312yes—
149#376C22×C312yes—
150C2×#95C22×C314yes—
151C2×#95C22×C312yes—
152C2×#95C22×C312yes—
153C2×#95C22×C312yes—
154C24×C3C2414yes—
155C24×C3C2411yescharacteristic
156C3×#41—432no—
157C2×C3×Q8—44no—
158C2×C3×Q8C22×C418yes—
159C2×C3×D4—216no—
160C2×C3×D4—42no—
161C2×C3×D4—44no—
162C2×C3×D4—42no—
163C2×C3×D4—44no—
164C2×C3×D4C22×C4116yes—
165C2×C3×D4C22×C418yes—
166C2×C3×D4C22×C418yes—
167C2×C3×D4C22×C418yes—
168C22×C3×C4—24no—
169C22×C3×C4—28no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4—48no—
173C22×C3×C4—48no—
174C22×C3×C4—48no—
175C22×C3×C4—48no—
176C22×C3×C4—44no—
177C22×C3×C4—44no—
178C22×C3×C4C22×C414yes—
179C22×C3×C4C22×C414yes—
180C22×C3×C4C22×C414yes—
181C22×C3×C4C22×C4116yes—
182C22×C3×C4C2418yes—
183C22×C3×C4C2418yes—
184C22×C3×C4C2411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C22×C3×C4C2414yes—
191C3×C4⋊C4—44no—
192C3×C4⋊C4—48no—
193C3×C4⋊C4C22×C418yes—
194C3×C4⋊C4C22×C418yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—48no—
197C3×C22⋊C4—48no—
198C3×C22⋊C4—48no—
199C3×C22⋊C4C22×C4116yes—
200C3×C22⋊C4C22×C4116yes—
201C3×C42—44no—
202C3×C42—48no—
203C3×C42C22×C414yes—
204C3×C42C22×C414yes—
205C3×C42C22×C414yes—
206C3×C42C22×C414yes—
207C2×#41—48no—
208C2×#41C2×C3×C4116yescomplemented
209C22×Q8C23×C312yes—
210C22×D4C23×C314yes—
211C22×D4C23×C312yes—
212C22×D4C23×C312yes—
213C22×D4C23×C312yes—
214C23×C4C23×C311yescharacteristic
215C23×C4C23×C314yes—
216C23×C4C23×C318yes—
217C23×C4C23×C314yes—
218C23×C4C23×C311yescharacteristic
219C23×C4C23×C311yescharacteristic
220C4⋊Q8C2×C3×C418yescomplemented
221#127C2×C3×C418yescomplemented
222#124C2×C3×C4116yescomplemented
223#121C2×C3×C4116yescomplemented
224#121C2×C3×C4116yescomplemented
225C4×D4—416no—
226C4×D4C2×C3×C4116yescomplemented
227C4×D4C2×C3×C4116yescomplemented
228C2×C4⋊C4C23×C318yes—
229C2×C4⋊C4C23×C314yes—
230C2×C4⋊C4C23×C312yes—
231C2×C4⋊C4C23×C312yes—
232C2×C22⋊C4C23×C314yes—
233C2×C22⋊C4C23×C318yes—
234C2×C22⋊C4C23×C318yes—
235C2×C22⋊C4C23×C318yes—
236C2×C22⋊C4C23×C314yes—
237C2×C22⋊C4C23×C314yes—
238C2×C42C2×C3×C418yescomplemented
239C2×C42C23×C318yes—
240C2×C42C23×C314yes—
241C2×C42C23×C311yescharacteristic
242C2×C42C23×C311yescharacteristic
243C2×C42C23×C311yescharacteristic
244C2×C42C23×C311yescharacteristic
245#95C23×C314yes—
246#95C23×C314yes—
247#95C23×C314yes—
248#95C23×C314yes—
249#95C23×C314yes—
250#95C23×C314yes—
251#95C23×C314yes—
252#95C23×C318yes—
253#95C23×C314yes—
254#95C23×C318yes—
255#95C23×C314yes—
256#95C23×C314yes—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—28no—
260C23×C3—24no—
261C23×C3—28no—
262C23×C3—24no—
263C23×C3C23×C418yes—
264C23×C3C23×C412yes—
265C23×C3H(5,𝔽2)14yes—
266C23×C3C2511yescharacteristic
267C3×Q8—48no—
268C3×D4—232no—
269C3×D4—48no—
270C3×D4—48no—
271C3×D4—48no—
272C2×C3×C4—28no—
273C2×C3×C4—28no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—28no—
281C2×C3×C4—28no—
282C2×C3×C4—28no—
283C2×C3×C4—28no—
284C2×C3×C4—28no—
285C2×C3×C4—28no—
286C2×C3×C4—44no—
287C2×C3×C4—44no—
288C2×C3×C4—416no—
289C2×C3×C4—88no—
290C2×C3×C4—816no—
291C2×C3×C4C23×C412yes—
292C2×C3×C4C23×C412yes—
293C2×C3×C4C23×C412yes—
294C2×C3×C4C23×C412yes—
295C2×C3×C4C23×C412yes—
296C2×C3×C4C23×C412yes—
297C2×C3×C4C23×C418yes—
298C24C24×C314yes—
299C24C24×C311yescharacteristic
300#41—432no—
301C2×Q8—44no—
302C2×Q8C22×C3×C418yes—
303C2×D4—216no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—42no—
308C2×D4C22×C3×C4116yes—
309C2×D4C22×C3×C418yes—
310C2×D4C22×C3×C418yes—
311C2×D4C22×C3×C418yes—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—48no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4C22×C3×C414yes—
323C22×C4C22×C3×C414yes—
324C22×C4C22×C3×C414yes—
325C22×C4C22×C3×C4116yes—
326C22×C4C24×C318yes—
327C22×C4C24×C318yes—
328C22×C4C24×C311yescharacteristic
329C22×C4C24×C311yescharacteristic
330C22×C4C24×C311yescharacteristic
331C22×C4C24×C311yescharacteristic
332C22×C4C24×C311yescharacteristic
333C22×C4C24×C311yescharacteristic
334C22×C4C24×C314yes—
335C4⋊C4—48no—
336C4⋊C4—44no—
337C4⋊C4C22×C3×C418yes—
338C4⋊C4C22×C3×C418yes—
339C22⋊C4—28no—
340C22⋊C4—48no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4C22×C3×C4116yes—
344C22⋊C4C22×C3×C4116yes—
345C42—48no—
346C42—44no—
347C42C22×C3×C414yes—
348C42C22×C3×C414yes—
349C42C22×C3×C414yes—
350C42C22×C3×C414yes—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—28no—
354C22×C3—28no—
355C22×C3—28no—
356C22×C3—28no—
357C22×C3—48no—
358C22×C3—48no—
359C22×C3#51814yes—
360C22×C3C24×C411yescharacteristic
361C22×C3C2×H(5,𝔽2)12yes—
362C22×C3#58512yes—
363C22×C3#58512yes—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—28no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—22no—
371C3×C4—48no—
372C3×C4—48no—
373C23—22no—
374C23—22no—
375C23—28no—
376C23—24no—
377C23—28no—
378C23—24no—
379C23C23×C3×C418yes—
380C23C23×C3×C412yes—
381C23C3×H(5,𝔽2)14yes—
382C23C25×C311yescharacteristic
383Q8—48no—
384D4—232no—
385D4—48no—
386D4—48no—
387D4—48no—
388C2×C4—28no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—416no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—88no—
406C2×C4—816no—
407C2×C4C23×C3×C412yes—
408C2×C4C23×C3×C412yes—
409C2×C4C23×C3×C412yes—
410C2×C4C23×C3×C412yes—
411C2×C4C23×C3×C412yes—
412C2×C4C23×C3×C412yes—
413C2×C4C23×C3×C418yes—
414C2×C3—24no—
415C2×C3—48no—
416C2×C3C2×#51811yescharacteristic
417C2×C3#342811yescharacteristic
418C2×C3#342811yescharacteristic
419C2×C3#356014yes—
420C22—22no—
421C22—22no—
422C22—28no—
423C22—28no—
424C22—28no—
425C22—28no—
426C22—48no—
427C22—48no—
428C22C3×#51814yes—
429C22C24×C3×C411yescharacteristic
430C22C2×C3×H(5,𝔽2)12yes—
431C22C3×#58512yes—
432C22C3×#58512yes—
433C4—22no—
434C4—22no—
435C4—28no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—48no—
441C4—48no—
442C3#3438811yescharacteristic, direct product factor, Sylow
443C2—24no—
444C2—48no—
445C2C2×C3×#51811yescharacteristic
446C2C3×#342811yescharacteristic
447C2C3×#342811yescharacteristic
448C2C3×#356014yes—
subgroupG/H
center, socleC23×C3C25
commutatorC22C24×C3×C4
Frattini, cosocleC23C25×C3
prime radicalC2×C3×#532C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2381C2semidirect
3. C3×#2380C2semidirect
4. C3×#2364C2semidirect
5. C3×#2353C2semidirect
6. C3×#2339C2semidirect
7. C3×#2337C2semidirect
8. C3×#2325C2semidirect
9. C3×#2318C2semidirect
10. C3×#2283C2semidirect
11. C2×C3×#383C2semidirect
12. #34388C3direct (N × H)
14. C3×#532C4semidirect
54. #2381C2×C3semidirect
55. #2380C2×C3semidirect
56. #2364C2×C3semidirect
57. #2353C2×C3semidirect
58. #2339C2×C3semidirect
59. #2337C2×C3semidirect
60. #2325C2×C3semidirect
61. #2318C2×C3semidirect
62. #2283C2×C3semidirect
63. C2×#383C2×C3semidirect
65. C2×C3×#41C2×C4semidirect
77. C3×C4⋊Q8C2×C4semidirect
78. C3×#127C2×C4semidirect
79. C3×#124C2×C4semidirect
80. C3×#121C2×C4semidirect
81. C3×#121C2×C4semidirect
83. C3×C4×D4C2×C4semidirect
84. C3×C4×D4C2×C4semidirect
95. C2×C3×C42C2×C4semidirect
115. #532C3×C4semidirect
208. C2×#41C2×C3×C4semidirect
220. C4⋊Q8C2×C3×C4semidirect
221. #127C2×C3×C4semidirect
222. #124C2×C3×C4semidirect
223. #121C2×C3×C4semidirect
224. #121C2×C3×C4semidirect
226. C4×D4C2×C3×C4semidirect
227. C4×D4C2×C3×C4semidirect
238. C2×C42C2×C3×C4semidirect
442. C3#34388direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C3×#23812
C3×#23802
C3×#23642
C3×#23532
C3×#23392
C3×#23372
C3×#23252
C3×#23182
C3×#22832
C2×C3×#3832
#343883