Finite Group Databasebeta

← #10823946

Subgroups — #10823946

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5113C211yesmaximal, characteristic
2#82228C211yesmaximal, characteristic
3#82224C211yesmaximal, characteristic, complemented
4C2×#5002C211yesmaximal, characteristic
5C3×#1735C211yesmaximal, characteristic, complemented
6C2×#4550C211yesmaximal, characteristic, complemented
7#71580C211yesmaximal, characteristic, complemented
8#68915C212yesmaximal, complemented
9#68914C212yesmaximal, complemented
10#68903C212yesmaximal, complemented
11#68902C212yesmaximal, complemented
12#23627—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#926C2211yescharacteristic
18C22×C4×Dic3C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20#5113—24no—
21C2×#821C2212yes—
22C2×#816C2212yes—
23#5002—24no—
24#4988C2211yescharacteristic
25#4988C2211yescharacteristic
26#4977C2212yes—
27#4975C2212yes—
28#4973C2211yescharacteristic
29#4973C2211yescharacteristic
30C2×#810C2212yes—
31C2×#809C2212yes—
32#4958—22no—
33#4942—22no—
34C4×(C2×D3)⋊C4—22no—
35C4×Dic3⋊C4—22no—
36C3×#340C2212yes—
37C3×#340C2212yes—
38#4550C2212yes—
39#4550C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42C2×#416—31no—
43C2×#416—31no—
44#2297—31no—
45#2285—31no—
46#2074—32no—
47#2073—32no—
48#2072—32no—
49#2071—32no—
50#1735—31no—
51#1735D311yescharacteristic, complemented
52C2×#336—31no—
53C23×C3×C4C2311yescharacteristic
54C22×#67—21no—
55C23×Dic3D412yes—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235D414yes—
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C22×Dic6C2311yescharacteristic
65C22×C3×C8—22no—
66C22×C3×C8—22no—
67C22×C3×C8—22no—
68C22×C3×C8C2312yes—
69C3×#117—22no—
70C3×#117—22no—
71C3×#117—22no—
72C3×#117—22no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79C2×C3×C42—21no—
80C2×#217—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83#926—22no—
84#926—22no—
85C2×C4⋊Dic3C2312yes—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3D412yes—
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C22×C3⋊C8—22no—
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8C2312yes—
98C3⋊#95—24no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#821—24no—
102#816—24no—
103#810—24no—
104#809—24no—
105C22×#41—31no—
106C23×C8—31no—
107C23×C8C2×D311yescharacteristic
108C2×#117—31no—
109C2×#117C2×D311yescharacteristic
110C2×#117C2×D311yescharacteristic
111C22×C42—31no—
112#416—64no—
113#416—64no—
114C2×#103—32no—
115C2×#103—32no—
116C2×#102—32no—
117C2×#102—32no—
118#391—31no—
119#391—31no—
120#390—31no—
121#390—31no—
122#386—62no—
123#384—32no—
124#382—32no—
125#380—62no—
126C4×C4⋊C4—62no—
127C4×C22⋊C4—62no—
128#340—32no—
129#340—32no—
130#340C2×D312yes—
131#340C2×D312yes—
132#336—32no—
133#336—32no—
134C24×C3C2×D411yescharacteristic
135C23×D3—21no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4#4112yes—
142C22×C3×C4#4112yes—
143C22×C3×C4#4112yes—
144C22×C3×C4#4112yes—
145C22×C3×C4#4112yes—
146C22×C3×C4C2411yescharacteristic
147C2×#67—24no—
148C2×#67—42no—
149C2×#67—42no—
150C22×Dic3—24no—
151C22×Dic3—21no—
152C22×Dic3—24no—
153C22×Dic3—22no—
154C22×Dic3—42no—
155C22×Dic3C2×D412yes—
156C22×Dic3C2×D412yes—
157#235—28no—
158#235—28no—
159#235—44no—
160#235—44no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12C2×D412yes—
167C2×C4×D3—24no—
168C2×C4×D3—42no—
169C2×C4×D3—42no—
170C2×Dic6—21no—
171C2×Dic6—21no—
172C2×Dic6—21no—
173C2×Dic6—21no—
174C2×Dic6—21no—
175C2×Dic6C2×D412yes—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—44no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—42no—
188C3×C42—42no—
189C3×C42—42no—
190#217—42no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—44no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195Dic3⋊C4—44no—
196Dic3⋊C4—44no—
197C4×Dic3—24no—
198C4×Dic3—21no—
199C4×Dic3—21no—
200C4×Dic3—21no—
201C4×Dic3—21no—
202C4×Dic3—42no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—44no—
207C2×#41—34no—
208C2×#41—62no—
209C2×#41—62no—
210C2×#41—62no—
211C2×#41—62no—
212C2×#41—62no—
213C22×Q8—31no—
214C22×D4—31no—
215C22×D4—61no—
216C23×C4—32no—
217C23×C4—61no—
218C23×C4C22×D311yescharacteristic
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—32no—
223C22×C8—62no—
224C22×C8—62no—
225C22×C8—62no—
226C22×C8C22×D312yes—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—62no—
232#117—62no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D312yes—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—61no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—32no—
247C2×C42—31no—
248C2×C42—31no—
249C2×C42—62no—
250C2×C42—62no—
251C2×C42—61no—
252#103—64no—
253#103—64no—
254#102—64no—
255#102—64no—
256#95—62no—
257#95—62no—
258#95—64no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3#12112yes—
262C23×C3#12312yes—
263C23×C3#12412yes—
264C23×C3C22×D411yescharacteristic
265C22×D3—22no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—42no—
269C22×D3—42no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4#12112yes—
278C2×C3×C4#12314yes—
279C2×C3×C4#12314yes—
280C2×C3×C4#12312yes—
281C2×C3×C4#12412yes—
282C2×C3×C4#12614yes—
283C2×C3×C4#12614yes—
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290#67—48no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—48no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3#13514yes—
304D12—22no—
305D12—22no—
306D12—41no—
307D12—41no—
308C4×D3—48no—
309Dic6—22no—
310Dic6—22no—
311Dic6—41no—
312Dic6—41no—
313C3×C8—42no—
314C3⋊C8—42no—
315C24—61no—
316C24D3×D411yescharacteristic
317#41—68no—
318#41—68no—
319#41—124no—
320#41—124no—
321C2×Q8—32no—
322C2×Q8—61no—
323C2×Q8—61no—
324C2×Q8—61no—
325C2×Q8—61no—
326C2×Q8—61no—
327C2×D4—32no—
328C2×D4—64no—
329C2×D4—61no—
330C2×D4—61no—
331C2×D4—61no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—122no—
335C2×D4—122no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—32no—
340C22×C4—32no—
341C22×C4—64no—
342C22×C4—64no—
343C22×C4—61no—
344C22×C4—64no—
345C22×C4—62no—
346C22×C4—122no—
347C22×C4—122no—
348C22×C4—122no—
349C22×C4#23512yes—
350C22×C4#23512yes—
351C22×C4#23512yes—
352C22×C4D3×D412yes—
353C22×C4D3×D411yescharacteristic
354C22×C4#23712yes—
355C22×C4#23912yes—
356C22×C4C23×D311yescharacteristic
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—44no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—124no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—62no—
372C4⋊C4—62no—
373C4⋊C4—124no—
374C4⋊C4—124no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—124no—
378C22⋊C4—124no—
379C22⋊C4—122no—
380C42—42no—
381C42—42no—
382C42—64no—
383C42—61no—
384C42—61no—
385C42—61no—
386C42—61no—
387C42—122no—
388C22×C3—24no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—24no—
392C22×C3#39714yes—
393C22×C3C2×#12111yescharacteristic
394C22×C3C2×#12311yescharacteristic
395C22×C3C2×#12411yescharacteristic
396C22×C3C2×#13512yes—
397C22×C3#57512yes—
398C2×D3—42no—
399C2×D3—44no—
400C2×D3—82no—
401C2×D3—81no—
402C2×D3—81no—
403C3×C4—81no—
404C3×C4—81no—
405C3×C4—82no—
406C3×C4#39714yes—
407C3×C4C2×#12311yescharacteristic
408C3×C4C2×#12311yescharacteristic
409C3×C4C2×#12611yescharacteristic
410C3×C4C2×#12611yescharacteristic
411Dic3—24no—
412Dic3—42no—
413Dic3—42no—
414Dic3—82no—
415C23—22no—
416C23—22no—
417C23—62no—
418C23—121no—
419C23—121no—
420C23—122no—
421C23—122no—
422C23#88412yes—
423C23#88512yes—
424C23#89412yes—
425C23C2×D3×D411yescharacteristic
426Q8—62no—
427Q8—62no—
428Q8—121no—
429Q8—121no—
430D4—62no—
431D4—62no—
432D4—128no—
433D4—121no—
434D4—121no—
435C2×C4—34no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—128no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—128no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4#87614yes—
457C2×C4#88314yes—
458C2×C4#88412yes—
459C2×C4#88512yes—
460C2×C4#89412yes—
461C2×C4#89414yes—
462C2×C4#89814yes—
463C2×C4C2×#23511yescharacteristic
464C2×C4C2×#23511yescharacteristic
465C2×C4C2×#23511yescharacteristic
466C2×C4C2×D3×D411yescharacteristic
467C2×C4C2×#23711yescharacteristic
468C2×C4C2×#23911yescharacteristic
469C8—42no—
470C8—122no—
471C2×C3—24no—
472C2×C3C2×#39711yescharacteristic
473C2×C3#303512yes—
474C2×C3#309112yes—
475C2×C3#313111yescharacteristic
476C2×C3#313511yescharacteristic
477D3—82no—
478C22—24no—
479C22—24no—
480C22—22no—
481C22—22no—
482C22—124no—
483C22—122no—
484C22—242no—
485C22—241no—
486C22—241no—
487C22#467514yes—
488C22C2×#88411yescharacteristic
489C22C2×#88511yescharacteristic
490C22C2×#89411yescharacteristic
491C22D3×#13512yes—
492C22#576812yes—
493C4—64no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4—122no—
498C4—122no—
499C4—242no—
500C4#467514yes—
501C4C2×#87611yescharacteristic
502C4C2×#88311yescharacteristic
503C4C2×#89411yescharacteristic
504C4C2×#89811yescharacteristic
505C3#2362711yescharacteristic, complemented, Sylow
506C2—24no—
507C2—242no—
508C2C2×#467511yescharacteristic
509C2#7882511yescharacteristic
510C2#7882611yescharacteristic
511C2#7886112yes—
512C2#7918512yes—
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1735C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1735D3
hypocenterC3#23627
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
3. #82224C2semidirect
5. C3×#1735C2semidirect
6. C2×#4550C2semidirect
7. #71580C2semidirect
8. #68915C2semidirect
9. #68914C2semidirect
10. #68903C2semidirect
11. #68902C2semidirect
51. #1735D3semidirect
505. C3#23627semidirect

Maximal subgroups

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

subgroupindex
C2×#51132
#822282
#822242
C2×#50022
C3×#17352
C2×#45502
#715802
#689152
#689142
#689032
#689022
#236273