Finite Group Databasebeta

← #10801921

Subgroups — #10801921

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#442C211yesmaximal, characteristic, complemented
2C2×#5159C211yesmaximal, characteristic
3#82396C212yesmaximal, complemented
4C2×#4708C211yesmaximal, characteristic, complemented
5#72038C211yesmaximal, characteristic, complemented
6C4×#832C212yesmaximal, complemented
7#70578C211yesmaximal, characteristic, complemented
8#70559C211yesmaximal, characteristic, complemented
9#70202C212yesmaximal, complemented
10#70188C212yesmaximal, complemented
11#69814C211yesmaximal, characteristic, complemented
12#21541—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2212yes—
15C2×#938C2212yes—
16C22×ΓL(1,𝔽25)C2211yescharacteristic
17C22×D3×C8C2211yescharacteristic
18C2×#872C2211yescharacteristic
19C3×#442—24no—
20C2×C3×#106C2211yescharacteristic
21C2×C3×#102C2212yes—
22C2×C3×#97C2211yescharacteristic
23#5221C2212yes—
24C2×#832C2212yes—
25C2×#832C2212yes—
26#5213—22no—
27#5210—22no—
28#5159C418yescomplemented
29C2×#818C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#815C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33#4987C2212yes—
34#4972C2212yes—
35#4957—22no—
36#4945—22no—
37C4×C4⋊Dic3C2212yes—
38C2×C4×C3⋊C8C2211yescharacteristic
39#4708—24no—
40#4551C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#469—31no—
44C2×#442D311yescharacteristic, complemented
45C2×#405—31no—
46#2324—32no—
47#1970—31no—
48#1941—31no—
49#1936—32no—
50#1893—32no—
51#1809—31no—
52#1776—31no—
53C4×#102—32no—
54C22×C3×D4C2312yes—
55C23×C3×C4—22no—
56C22×#67—22no—
57C22×C4×D3C2311yescharacteristic
58C2×C3×D8—22no—
59C2×C3×D8—22no—
60C2×C3×D8C2×C414yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4×D4—24no—
64C3×C4×D4—24no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42C2311yescharacteristic
68C2×#217—22no—
69C2×#217—22no—
70C2×#217—22no—
71#938—24no—
72#938C2×C418yescomplemented
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4D412yes—
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2312yes—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4D412yes—
79C2×C4×Dic3—22no—
80C2×C4×Dic3C2311yescharacteristic
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C2×ΓL(1,𝔽25)—22no—
84C2×ΓL(1,𝔽25)—22no—
85C2×ΓL(1,𝔽25)C2×C414yescomplemented
86C2×D3×C8—22no—
87C2×D3×C8—22no—
88C2×D3×C8—22no—
89#872—22no—
90#872—22no—
91C3×#106—22no—
92C3×#102—24no—
93C3×#97D414yescomplemented
94#832—24no—
95#832C2×C418yescomplemented
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#818—22no—
100#817C2×C414yescomplemented
101#815—22no—
102C8×Dic3—22no—
103C4×C3⋊C8—22no—
104C22×D8C2×D311yescharacteristic
105C22×M16—31no—
106C23×C8—31no—
107C2×#121—32no—
108C2×C4×D4C2×D312yes—
109C2×#117—31no—
110#469—38no—
111#442—24no—
112C2×#107—31no—
113C2×#106—31no—
114C2×#106C2×D311yescharacteristic
115C2×#102—32no—
116C2×#102—32no—
117C2×#102C2×D312yes—
118#405—64no—
119C2×#97—31no—
120C2×#97C2×D311yescharacteristic
121C2×C4×C8—31no—
122C2×C4×C8—31no—
123#390—32no—
124#390—62no—
125#389—32no—
126#388—62no—
127#386—62no—
128#385—62no—
129#380—32no—
130C4×C4⋊C4—32no—
131#340—32no—
132#340—32no—
133#340—32no—
134C24×C3—22no—
135C23×D3C2×D411yescharacteristic
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4C22×C414yes—
142C22×C3×C4—22no—
143C22×C3×C4—44no—
144C22×C3×C4—82no—
145C22×C3×C4—82no—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2411yescharacteristic
148C2×#67—28no—
149C2×#67—82no—
150C2×#67—82no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3C2×D412yes—
154C22×Dic3C2×D411yescharacteristic
155C2×C4×D3—21no—
156C2×C4×D3—21no—
157C2×C4×D3—21no—
158C2×C4×D3—21no—
159C2×C4×D3C22×C412yes—
160C2×C4×D3#4112yes—
161C3×D8—28no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8C22×C412yes—
165C2×C3×C8C2×D412yes—
166C2×C3×C8C2×D412yes—
167C2×C3×C8#4112yes—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C22⋊C4—44no—
171C3×C42C2×D412yes—
172C3×C42#4112yes—
173#217—28no—
174#217—82no—
175#217—82no—
176(C2×D3)⋊C4—24no—
177(C2×D3)⋊C4—82no—
178C4⋊Dic3—22no—
179C4⋊Dic3—22no—
180C4⋊Dic3—22no—
181C4⋊Dic3C22×C414yes—
182Dic3⋊C4—24no—
183Dic3⋊C4—82no—
184C4×Dic3—21no—
185C4×Dic3—21no—
186C4×Dic3—82no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8C22×C412yes—
192C2×C3⋊C8#4112yes—
193ΓL(1,𝔽25)—28no—
194D3×C8—28no—
195C22×D4—62no—
196C22×D4C22×D312yes—
197C23×C4—22no—
198C23×C4—31no—
199C2×D8—22no—
200C2×D8—22no—
201C2×D8C4×D314yes—
202C2×M16—34no—
203C2×M16—62no—
204C2×M16—62no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8—62no—
208C22×C8—62no—
209C22×C8—62no—
210C22×C8C22×D311yescharacteristic
211C22×C8C22×D311yescharacteristic
212#121—38no—
213#121—64no—
214C4×D4—24no—
215C4×D4—24no—
216#117—62no—
217#117—62no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4C22×D312yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—32no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C42—31no—
230C2×C42—62no—
231C2×C42C22×D311yescharacteristic
232#107—62no—
233#106—22no—
234#106—34no—
235#102—24no—
236#102—38no—
237#102—64no—
238#97—62no—
239#97#6714yescomplemented
240C4×C8—62no—
241C4×C8—62no—
242#95—62no—
243#95—62no—
244#95—62no—
245C23×C3—24no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3C22×D411yescharacteristic
253C22×D3—21no—
254C22×D3—21no—
255C22×D3—22no—
256C22×D3—22no—
257C22×D3C4×D412yes—
258C3×D4—24no—
259C3×D4—24no—
260C3×D4—42no—
261C3×D4—42no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—88no—
268C2×C3×C4#12114yes—
269C2×C3×C4C23×C411yescharacteristic
270C2×C3×C4C22×D411yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275#67—88no—
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—24no—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3C4×D412yes—
288C4×D3—22no—
289C4×D3—22no—
290C4×D3—22no—
291C4×D3—22no—
292C3×C8—22no—
293C3×C8C4×D414yes—
294C3⋊C8—22no—
295C3⋊C8—22no—
296C24—22no—
297C24—31no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—68no—
304C2×D4—242no—
305C2×D4—242no—
306C2×D4C2×C4×D314yes—
307C22×C4—22no—
308C22×C4—31no—
309C22×C4—32no—
310C22×C4—32no—
311C22×C4—32no—
312C22×C4—44no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4D3×D412yes—
322C22×C4C23×D311yescharacteristic
323D8—28no—
324M16—68no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—32no—
328C2×C8—32no—
329C2×C8—68no—
330C2×C8—61no—
331C2×C8—61no—
332C2×C8—61no—
333C2×C8—61no—
334C2×C8C2×C4×D312yes—
335C2×C8#23512yes—
336C2×C8C2×#6712yes—
337C2×C8C2×#6712yes—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—34no—
341C4⋊C4—64no—
342C4⋊C4—62no—
343C4⋊C4—62no—
344C4⋊C4—62no—
345C4⋊C4—242no—
346C22⋊C4—44no—
347C22⋊C4—68no—
348C22⋊C4—64no—
349C22⋊C4—242no—
350C22⋊C4—242no—
351C22⋊C4—242no—
352C42—61no—
353C42—61no—
354C42—242no—
355C42#23712yes—
356C42C2×#6712yes—
357C22×C3—44no—
358C22×C3—44no—
359C22×C3—44no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3C2×C4×D411yescharacteristic
365C22×C3C2×#12111yescharacteristic
366C22×C3#53211yescharacteristic
367C22×C3#57612yes—
368C22×C3#57712yes—
369C2×D3—24no—
370C2×D3—22no—
371C2×D3—41no—
372C2×D3—41no—
373C2×D3—42no—
374C3×C4—24no—
375C3×C4—82no—
376C3×C4C2×C4×D411yescharacteristic
377C3×C4C4×#4111yescharacteristic
378C3×C4#52511yescharacteristic
379C3×C4#52911yescharacteristic
380Dic3—22no—
381Dic3—44no—
382Dic3—42no—
383Dic3—82no—
384C23—24no—
385C23—32no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—62no—
393C23—62no—
394C23—61no—
395C23—61no—
396C23C2×D3×D411yescharacteristic
397D4—24no—
398D4—24no—
399D4—42no—
400D4—42no—
401D4—248no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—32no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—64no—
417C2×C4—62no—
418C2×C4—88no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4#93814yes—
425C2×C4C22×C4×D311yescharacteristic
426C2×C4C2×#23511yescharacteristic
427C2×C4C2×#23711yescharacteristic
428C2×C4D3×#4111yescharacteristic
429C2×C4D3×#4111yescharacteristic
430C2×C4C22×#6711yescharacteristic
431C8—22no—
432C8—62no—
433C8—62no—
434C8C4×#6714yes—
435C2×C3—84no—
436C2×C3C4×#12111yescharacteristic
437C2×C3#294612yes—
438C2×C3#306212yes—
439C2×C3#317311yescharacteristic
440C2×C3#317611yescharacteristic
441D3—42no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—62no—
446C22—64no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—122no—
452C22—121no—
453C22—121no—
454C22C4×D3×D411yescharacteristic
455C22#554811yescharacteristic
456C22#577412yes—
457C22#577512yes—
458C22C2×#93811yescharacteristic
459C4—24no—
460C4—62no—
461C4—82no—
462C4—122no—
463C4—124no—
464C4—242no—
465C4C4×#23711yescharacteristic
466C4#555711yescharacteristic
467C4C2×C4×#6711yescharacteristic
468C4#582811yescharacteristic
469C3#2154111yescharacteristic, complemented, Sylow
470C2—84no—
471C2—122no—
472C2#7985911yescharacteristic
473C2#7986011yescharacteristic
474C2#7986612yes—
475C2C4×#93811yescharacteristic
476C2#8395112yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
prime radicalC2×#5159C2
cosocleC22×C3×C4C24
hypercenterC2×#442D3
hypocenterC3#21541
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×C3×#442C2semidirect
3. #82396C2semidirect
4. C2×#4708C2semidirect
5. #72038C2semidirect
6. C4×#832C2semidirect
7. #70578C2semidirect
8. #70559C2semidirect
9. #70202C2semidirect
10. #70188C2semidirect
11. #69814C2semidirect
28. #5159C4semidirect
44. C2×#442D3semidirect
72. #938C2×C4semidirect
85. C2×ΓL(1,𝔽25)C2×C4semidirect
93. C3×#97D4semidirect
95. #832C2×C4semidirect
100. #817C2×C4semidirect
239. #97#67semidirect
469. C3#21541semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4422
C2×#51592
#823962
C2×#47082
#720382
C4×#8322
#705782
#705592
#702022
#701882
#698142
#215413