Finite Group Databasebeta

← #10814404

Subgroups — #10814404

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2349C211yesmaximal, characteristic, complemented
2C2×C4×#213C211yesmaximal, characteristic
3#73645C211yesmaximal, characteristic, complemented
4#71613C212yesmaximal
5#71609C212yesmaximal, complemented
6#71605C211yesmaximal, characteristic, complemented
7#69617C212yesmaximal, complemented
8#69600C212yesmaximal, complemented
9#69556C211yesmaximal, characteristic, complemented
10#69547C211yesmaximal, characteristic, complemented
11#69477C211yesmaximal, characteristic, complemented
12#22764—31nomaximal, Sylow
13C2×C3×C4×D4C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C22×#213C2211yescharacteristic
16C2×C4×D12C2211yescharacteristic
17C3×#389C2212yes—
18C3×#385—22no—
19C3×#382C2211yescharacteristic
20C3×#380—21no—
21C3×C4×C4⋊C4C2211yescharacteristic
22C2×C3×#95—21no—
23C2×#832C2212yes—
24C2×#832C2211yescharacteristic
25C4×#213C418yescomplemented
26#4990—22no—
27C2×#809C2212yes—
28C2×#809C2211yescharacteristic
29C2×#807C2211yescharacteristic
30#4939C2212yes—
31#4936C2211yescharacteristic
32C4×C4⋊Dic3C2212yes—
33C2×C4⋊(C3⋊C8)C2211yescharacteristic
34C2×#803C2212yes—
35C2×C4×C3⋊C8C2211yescharacteristic
36#4666—21no—
37#4664—21no—
38#4529C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4524C2212yes—
43C2×C4×D8—31no—
44#2357—31no—
45#2349D311yescharacteristic, complemented
46#1936—32no—
47#1918—32no—
48#1879—32no—
49#1835—31no—
50#1804—31no—
51#1804—31no—
52#1770—31no—
53#1762—32no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C23×C3×C4—22no—
57C22×D12C2311yescharacteristic
58C22×C4×D3—21no—
59C3×C4×D4—24no—
60C3×C4×D4—22no—
61C3×C4×D4—42no—
62C3×C4×D4C2×C414yes—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C42C2312yes—
70C2×C3×C42C2311yescharacteristic
71C2×#213—22no—
72C2×#213—22no—
73C2×#213C2×C414yes—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4×Dic3—22no—
80C22×C3⋊C8C2312yes—
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C4×D12—22no—
84C4×D12C2×C414yescomplemented
85C3×#95—21no—
86C3×#95—22no—
87C3×#95—21no—
88C3×#95D411yescharacteristic
89C3×#95D411yescharacteristic
90#832—24no—
91#832C2×C414yes—
92C3⋊#95—21no—
93C3⋊#95—22no—
94C3⋊#95—21no—
95#809—42no—
96#809C2×C414yescomplemented
97#807C2×C414yescomplemented
98C4⋊(C3⋊C8)—22no—
99#803—42no—
100C4×C3⋊C8C2×C414yescomplemented
101C22×D8—31no—
102C2×C4×D4—31no—
103C2×C4×D4C2×D312yes—
104C2×C4×D4C2×D311yescharacteristic
105C4×D8—38no—
106C2×#107—31no—
107C2×C4⋊C8—31no—
108C2×#102—31no—
109C2×#102—31no—
110C2×#102—32no—
111C2×#102—32no—
112C2×#97—32no—
113C2×C4×C8—31no—
114#390—32no—
115#389C2×D312yes—
116#388—62no—
117#385—22no—
118#382—31no—
119#382C2×D311yescharacteristic
120#381—61no—
121#380—21no—
122C4×C4⋊C4—32no—
123C4×C4⋊C4C2×D311yescharacteristic
124C4×C22⋊C4—61no—
125C2×#95—21no—
126#340—32no—
127#340—32no—
128#340—32no—
129#340—32no—
130#336—32no—
131C24×C3—21no—
132C23×D3—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4C22×C412yes—
137C2×C3×D4#4112yes—
138C2×C3×D4#4112yes—
139C22×C3×C4—21no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—44no—
143C22×C3×C4—44no—
144C22×C3×C4—82no—
145C22×C3×C4—82no—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—22no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—21no—
156C2×D12—21no—
157C2×D12—21no—
158C2×D12C22×C412yes—
159C2×C4×D3—24no—
160C2×C4×D3—82no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—82no—
166C3×C4⋊C4C22×C412yes—
167C3×C22⋊C4—24no—
168C3×C22⋊C4—44no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42C22×C412yes—
172C3×C42#4112yes—
173#213—28no—
174(C2×D3)⋊C4—24no—
175(C2×D3)⋊C4—44no—
176(C2×D3)⋊C4—82no—
177C4⋊Dic3—21no—
178C4⋊Dic3—22no—
179C4⋊Dic3—22no—
180C4⋊Dic3C22×C412yes—
181C4×Dic3—82no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8C22×C412yes—
190C2×C3⋊C8C22×C412yes—
191C22×D4—31no—
192C22×D4C22×D311yescharacteristic
193C23×C4—21no—
194C23×C4—22no—
195C23×C4—61no—
196C2×D8—34no—
197C2×D8—62no—
198C2×D8—62no—
199C22×C8—31no—
200C22×C8—31no—
201C22×C8—32no—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—34no—
205C4×D4—42no—
206C4×D4—62no—
207C4×D4C4×D314yes—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4#6712yes—
212C2×C4⋊C4C22×D312yes—
213C2×C4⋊C4C22×D311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C42—62no—
220C2×C42C22×D312yes—
221C2×C42C22×D311yescharacteristic
222#107—34no—
223C4⋊C8—62no—
224#102—34no—
225#102—34no—
226#102—64no—
227#102—122no—
228#97—122no—
229C4×C8—34no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—61no—
234#95—62no—
235#95—61no—
236#95#6711yescharacteristic
237#95#6711yescharacteristic
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3C22×D411yescharacteristic
244C22×D3—22no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—42no—
248C22×D3—42no—
249C3×D4—22no—
250C3×D4—22no—
251C3×D4—22no—
252C3×D4#11714yes—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—84no—
267C2×C3×C4—88no—
268C2×C3×C4C4×D412yes—
269C2×C3×C4C4×D412yes—
270C2×C3×C4C23×C411yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4H(5,𝔽2)11yescharacteristic
275C2×C3×C42−1+411yescharacteristic
276C2×Dic3—22no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281D12—22no—
282D12—22no—
283D12—41no—
284D12—41no—
285C4×D3—84no—
286C3⋊C8—22no—
287C3⋊C8—22no—
288C3⋊C8—42no—
289C24—21no—
290C24—61no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—32no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C2×D4C2×C4×D312yes—
301C2×D4#23712yes—
302C2×D4#23912yes—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—61no—
309C22×C4—64no—
310C22×C4—62no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—242no—
314C22×C4C2×#6712yes—
315C22×C4C2×#6712yes—
316C22×C4C2×#6711yescharacteristic
317C22×C4C2×#6711yescharacteristic
318C22×C4C23×D311yescharacteristic
319D8—68no—
320C2×C8—32no—
321C2×C8—32no—
322C2×C8—61no—
323C2×C8—61no—
324C2×C8—61no—
325C2×C8—61no—
326C2×C8—62no—
327C2×C8—62no—
328C2×C8—62no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—32no—
333C4⋊C4—42no—
334C4⋊C4—61no—
335C4⋊C4—62no—
336C4⋊C4—62no—
337C4⋊C4—82no—
338C4⋊C4C2×C4×D312yes—
339C22⋊C4—24no—
340C22⋊C4—44no—
341C22⋊C4—64no—
342C22⋊C4—124no—
343C22⋊C4—242no—
344C42—22no—
345C42—22no—
346C42—242no—
347C42C2×C4×D312yes—
348C42#23512yes—
349C22×C3—24no—
350C22×C3—44no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—44no—
354C22×C3C2×C4×D411yescharacteristic
355C22×C3#54611yescharacteristic
356C22×C3#54811yescharacteristic
357C22×C3#57512yes—
358C22×C3#57612yes—
359C2×D3—42no—
360C2×D3—42no—
361C2×D3—42no—
362C2×D3—82no—
363C2×D3—82no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—42no—
367C3×C4—42no—
368C3×C4—42no—
369C3×C4—82no—
370C3×C4C2×#11711yescharacteristic
371C3×C4#52011yescharacteristic
372C3×C4#55211yescharacteristic
373C3×C4#55311yescharacteristic
374Dic3—42no—
375Dic3—82no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—42no—
380C23—42no—
381C23—62no—
382C23—121no—
383C23—121no—
384C23—122no—
385C23—122no—
386C23C22×#6711yescharacteristic
387D4—22no—
388D4—22no—
389D4—22no—
390D4—62no—
391D4—62no—
392D4—121no—
393D4—121no—
394D4#89214yes—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—62no—
409C2×C4—84no—
410C2×C4—88no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—244no—
416C2×C4C4×#6712yes—
417C2×C4C4×#6712yes—
418C2×C4C22×C4×D311yescharacteristic
419C2×C4C2×#23511yescharacteristic
420C2×C4C2×#23711yescharacteristic
421C2×C4#100411yescharacteristic
422C2×C4C2×#23911yescharacteristic
423C2×C4#100711yescharacteristic
424C8—62no—
425C8—62no—
426C8—122no—
427C2×C3—44no—
428C2×C3#235211yescharacteristic
429C2×C3#294311yescharacteristic
430C2×C3#294811yescharacteristic
431C2×C3#332212yes—
432C2×C3#337012yes—
433D3—82no—
434C22—24no—
435C22—44no—
436C22—42no—
437C22—42no—
438C22—44no—
439C22—122no—
440C22—122no—
441C22—122no—
442C22—242no—
443C22—242no—
444C22C2×C4×#6711yescharacteristic
445C22#580611yescharacteristic
446C22#581711yescharacteristic
447C22#582212yes—
448C22#583512yes—
449C4—22no—
450C4—22no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—122no—
456C4—242no—
457C4C2×#89211yescharacteristic
458C4#550411yescharacteristic
459C4#551311yescharacteristic
460C4#551411yescharacteristic
461C3#2276411yescharacteristic, complemented, Sylow
462C2—44no—
463C2—242no—
464C2#8216111yescharacteristic
465C2#8250611yescharacteristic
466C2#8253811yescharacteristic
467C2#8350012yes—
468C2#8363912yes—
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2349C2
socleC23×C3C22×D4
prime radicalC22×#213C22
cosocleC22×C3×C4C24
hypercenter#2349D3
hypocenterC3#22764
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
1. C3×#2349C2semidirect
3. #73645C2semidirect
5. #71609C2semidirect
6. #71605C2semidirect
7. #69617C2semidirect
8. #69600C2semidirect
9. #69556C2semidirect
10. #69547C2semidirect
11. #69477C2semidirect
25. C4×#213C4semidirect
45. #2349D3semidirect
84. C4×D12C2×C4semidirect
96. #809C2×C4semidirect
97. #807C2×C4semidirect
100. C4×C3⋊C8C2×C4semidirect
461. C3#22764semidirect

Maximal subgroups

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

subgroupindex
C3×#23492
C2×C4×#2132
#736452
#716132
#716092
#716052
#696172
#696002
#695562
#695472
#694772
#227643