Finite Group Databasebeta

← #10690804

Subgroups — #10690804

Order 768.

600 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83008C211yesmaximal, characteristic
2C2×#4705C211yesmaximal, characteristic, complemented
3C2×C3×#329C211yesmaximal, characteristic, complemented
4C2×#4540C211yesmaximal, characteristic, complemented
5#69807C211yesmaximal, characteristic, complemented
6#67659C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#9124—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×#946C411yescharacteristic
11C2×#938C411yescharacteristic
12C2×C42×D3C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15#5229C411yescharacteristic
16#5221C411yescharacteristic
17C2×(C2×D3)⋊C8—21no—
18C2×(C2×D3)⋊C8—21no—
19C2×Dic3⋊C8—21no—
20#4956—21no—
21#4956—21no—
22#4944—21no—
23#4944—21no—
24#4936C2211yescharacteristic
25C2×#803C2211yescharacteristic
26C2×#803C2211yescharacteristic
27#4705—22no—
28#4705—22no—
29C3×#329—22no—
30#4550—21no—
31#4540—22no—
32C2×#432—31no—
33#2850—31no—
34#1749—31no—
35C2×#329—31no—
36C2×#329D311yescharacteristic, complemented
37#1295—31no—
38#1295—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×#67—21no—
42C22×C4×D3C2×C411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×C4×D3D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57#946—42no—
58#938—42no—
59C2×(C2×D3)⋊C4—21no—
60C2×C4⋊Dic3C2×C411yescharacteristic
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×Dic3⋊C4—21no—
63C2×C4×Dic3C2×C411yescharacteristic
64C2×C4×Dic3D411yescharacteristic
65C2×C4×Dic3D411yescharacteristic
66C22×C3⋊C8—21no—
67C22×C3⋊C8—21no—
68C42×D3—21no—
69C42×D3—21no—
70C42×D3—21no—
71C42×D3—21no—
72C3×#97—21no—
73C3×#97—21no—
74C3×#97—21no—
75C3×#97D412yescomplemented
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80(C2×D3)⋊C8—24no—
81(C2×D3)⋊C8—24no—
82Dic3⋊C8—24no—
83#803—21no—
84#803—21no—
85#803—21no—
86#803D412yescomplemented
87C2×#124C2×D311yescharacteristic
88C2×C22⋊Q8—31no—
89C2×#121—31no—
90C22×C42—31no—
91#432—62no—
92#432—62no—
93C2×C4⋊C8—61no—
94C2×C22⋊C8—61no—
95C2×C22⋊C8—61no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97C2×D311yescharacteristic
99C2×#97C2×D311yescharacteristic
100#391—31no—
101#390—31no—
102#388—61no—
103#388—61no—
104#386—61no—
105#386—61no—
106#382—31no—
107#336—61no—
108#329—22no—
109#329—62no—
110C24×C3—21no—
111C23×D3C22⋊C411yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C22×C3×C4—21no—
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C2×#67—42no—
125C2×#67—42no—
126C2×#67—42no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3C22⋊C411yescharacteristic
130C22×Dic3C22⋊C411yescharacteristic
131C22×Dic3C22⋊C411yescharacteristic
132C2×C4×D3—22no—
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3—22no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C4×D3—41no—
143C2×C4×D3—41no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C2×C3×C8—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157C3×C42#4111yescharacteristic
158#217—42no—
159#217—42no—
160#217—42no—
161(C2×D3)⋊C4—42no—
162C4⋊Dic3—41no—
163C4⋊Dic3—41no—
164Dic3⋊C4—42no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C22×Q8C4×D311yescharacteristic
178C22×D4—61no—
179C22×D4C4×D311yescharacteristic
180C23×C4—31no—
181C23×C4—31no—
182C23×C4—31no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—61no—
186C22×C8—61no—
187#124—21no—
188#124—21no—
189#124—21no—
190#124—21no—
191C22⋊Q8—122no—
192#121—122no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C42—31no—
203C2×C42—31no—
204C2×C42—31no—
205C2×C42—61no—
206C2×C42—61no—
207C2×C42—61no—
208C2×C42—61no—
209C2×C42C22×D311yescharacteristic
210C4⋊C8—64no—
211C22⋊C8—64no—
212C22⋊C8—64no—
213#97—21no—
214#97—21no—
215#97—21no—
216#97—32no—
217#97—61no—
218#97—61no—
219#97—61no—
220#97#6712yescomplemented
221#95—61no—
222#95—61no—
223#95—61no—
224#95—61no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3C2×C22⋊C411yescharacteristic
233C22×D3—21no—
234C22×D3—21no—
235C22×D3—21no—
236C22×D3—21no—
237C22×D3—21no—
238C22×D3—21no—
239C22×D3#10012yes—
240C3×Q8—81no—
241C3×Q8—81no—
242C3×D4—81no—
243C3×D4—81no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4C4×D411yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12311yescharacteristic
259#67—84no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3#10412yes—
275C4×D3—42no—
276C4×D3—42no—
277C4×D3—42no—
278C4×D3—42no—
279C4×D3—42no—
280C4×D3—42no—
281C3×C8—42no—
282C3×C8—42no—
283C3⋊C8—42no—
284C3⋊C8—42no—
285C24—21no—
286C24—31no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—122no—
298C22×C4—21no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4—121no—
316C22×C4C2×C4×D311yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4D3×D411yescharacteristic
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—62no—
326C2×C8—62no—
327C2×C8—121no—
328C2×C8—121no—
329C2×C8—121no—
330C2×C8—121no—
331C4⋊C4—121no—
332C4⋊C4—121no—
333C4⋊C4—122no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—122no—
339C22⋊C4—122no—
340C22⋊C4—122no—
341C22⋊C4—122no—
342C42—61no—
343C42—61no—
344C42—61no—
345C42—61no—
346C42—61no—
347C42—61no—
348C42#23511yescharacteristic
349C42D3×D411yescharacteristic
350C42#23711yescharacteristic
351C42C2×#6711yescharacteristic
352C22×C3—81no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3#38611yescharacteristic
360C22×C3C2×#10011yescharacteristic
361C22×C3#41311yescharacteristic
362C22×C3C2×#10411yescharacteristic
363C22×C3#42111yescharacteristic
364C22×C3#42111yescharacteristic
365C22×C3#42111yescharacteristic
366C2×D3—22no—
367C2×D3—22no—
368C2×D3—22no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—41no—
372C2×D3—41no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—81no—
380Dic3—22no—
381Dic3—41no—
382Dic3—41no—
383Dic3—81no—
384Dic3—81no—
385C23—32no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—61no—
394C23—61no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—61no—
399C23D3×C22⋊C411yescharacteristic
400Q8—81no—
401Q8—81no—
402D4—81no—
403D4—81no—
404D4—244no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—32no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4#67⋊C411yescharacteristic
436C2×C4#88311yescharacteristic
437C2×C4#88411yescharacteristic
438C2×C4C4×#6711yescharacteristic
439C2×C4#93711yescharacteristic
440C2×C4#93811yescharacteristic
441C8—42no—
442C8—42no—
443C8—122no—
444C8—122no—
445C2×C3—161no—
446C2×C3#152911yescharacteristic
447C2×C3#153011yescharacteristic
448C2×C3#153311yescharacteristic
449C2×C3#153511yescharacteristic
450C2×C3#187111yescharacteristic
451C2×C3#189711yescharacteristic
452C2×C3#190011yescharacteristic
453D3—42no—
454C22—62no—
455C22—62no—
456C22—62no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—121no—
465C22—121no—
466C22—121no—
467C22—121no—
468C22D3×#10011yescharacteristic
469C22#474511yescharacteristic
470C22D3×#10411yescharacteristic
471C22#482111yescharacteristic
472C22#482111yescharacteristic
473C22#482111yescharacteristic
474C22#495611yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—62no—
482C4—81no—
483C4—121no—
484C4—121no—
485C4—241no—
486C4—241no—
487C3#912411yescharacteristic, complemented, Sylow
488C2—122no—
489C2—161no—
490C2#6834411yescharacteristic
491C2#6834511yescharacteristic
492C2#6834711yescharacteristic
493C2#6834811yescharacteristic
494C2#7008111yescharacteristic
495C2#7038111yescharacteristic
496C2#7038311yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#329C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#938C4
cosocleC2×C3×C42C23
hypercenterC2×#329D3
hypocenterC3#9124
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
2. C2×#4705C2semidirect
3. C2×C3×#329C2semidirect
4. C2×#4540C2semidirect
5. #69807C2semidirect
6. #67659C2semidirect
7. #67659C2semidirect
36. C2×#329D3semidirect
75. C3×#97D4semidirect
86. #803D4semidirect
220. #97#67semidirect
487. C3#9124semidirect

Maximal subgroups

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

subgroupindex
#830082
C2×#47052
C2×C3×#3292
C2×#45402
#698072
#676592
#91243