Finite Group Databasebeta

← #10690784

Subgroups — #10690784

Order 768.

684 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83008C211yesmaximal, characteristic
2C2×#4707C211yesmaximal, characteristic, complemented
3C2×C3×#329C211yesmaximal, characteristic, complemented
4C2×#4540C211yesmaximal, characteristic, complemented
5#69807C211yesmaximal, characteristic, complemented
6#67659C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#9120—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×#946C411yescharacteristic
11C2×#938C411yescharacteristic
12C22×ΓL(1,𝔽25)—21no—
13C2×C42×D3C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15C2×C3×#97C2211yescharacteristic
16#5229C411yescharacteristic
17#5221C411yescharacteristic
18C2×(C2×D3)⋊C8—21no—
19C2×C8×Dic3—21no—
20#4956—21no—
21#4956—21no—
22#4944—21no—
23#4944—21no—
24#4936C2211yescharacteristic
25C2×#803C2211yescharacteristic
26C2×#803C2211yescharacteristic
27#4707—22no—
28#4707—22no—
29C3×#329—22no—
30#4550—21no—
31#4540—22no—
32C2×C4×M16—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—
68C2×ΓL(1,𝔽25)—24no—
69C2×ΓL(1,𝔽25)—24no—
70C2×ΓL(1,𝔽25)—24no—
71C42×D3—21no—
72C42×D3—21no—
73C42×D3—21no—
74C42×D3—21no—
75C3×#97—21no—
76C3×#97—21no—
77C3×#97—21no—
78C3×#97D412yescomplemented
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83(C2×D3)⋊C8—24no—
84C8×Dic3—24no—
85#803—21no—
86#803—21no—
87#803—21no—
88#803D412yescomplemented
89C22×M16—61no—
90C2×#124C2×D311yescharacteristic
91C2×C22⋊Q8—31no—
92C2×#121—31no—
93C22×C42—31no—
94C2×C22⋊C8—61no—
95C4×M16—62no—
96C4×M16—62no—
97C2×#97—31no—
98C2×#97—31no—
99C2×#97C2×D311yescharacteristic
100C2×#97C2×D311yescharacteristic
101C2×C4×C8—61no—
102#391—31no—
103#390—31no—
104#388—61no—
105#388—61no—
106#386—61no—
107#386—61no—
108#382—31no—
109#336—61no—
110#329—22no—
111#329—62no—
112C24×C3—21no—
113C23×D3C22⋊C411yescharacteristic
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C22×C3×C4—21no—
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×#67—42no—
127C2×#67—42no—
128C2×#67—42no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3C22⋊C411yescharacteristic
132C22×Dic3C22⋊C411yescharacteristic
133C22×Dic3C22⋊C411yescharacteristic
134C2×C4×D3—22no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—22no—
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—21no—
141C2×C4×D3—21no—
142C2×C4×D3—22no—
143C2×C4×D3—22no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—41no—
151C2×C3×C8—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159C3×C42#4111yescharacteristic
160#217—42no—
161#217—42no—
162#217—42no—
163(C2×D3)⋊C4—42no—
164C4⋊Dic3—41no—
165C4⋊Dic3—41no—
166Dic3⋊C4—42no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C4×Dic3—21no—
172C4×Dic3—21no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179ΓL(1,𝔽25)—216no—
180C22×Q8C4×D311yescharacteristic
181C22×D4—61no—
182C22×D4C4×D311yescharacteristic
183C23×C4—31no—
184C23×C4—31no—
185C23×C4—31no—
186C2×M16—64no—
187C2×M16—64no—
188C2×M16—64no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—61no—
192C22×C8—61no—
193#124—21no—
194#124—21no—
195#124—21no—
196#124—21no—
197C22⋊Q8—122no—
198#121—122no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—61no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C42—31no—
209C2×C42—31no—
210C2×C42—31no—
211C2×C42—61no—
212C2×C42—61no—
213C2×C42—61no—
214C2×C42—61no—
215C2×C42C22×D311yescharacteristic
216C22⋊C8—64no—
217#97—21no—
218#97—21no—
219#97—21no—
220#97—32no—
221#97—61no—
222#97—61no—
223#97—61no—
224#97#6712yescomplemented
225C4×C8—64no—
226#95—61no—
227#95—61no—
228#95—61no—
229#95—61no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3C2×C22⋊C411yescharacteristic
238C22×D3—21no—
239C22×D3—21no—
240C22×D3—21no—
241C22×D3—21no—
242C22×D3—21no—
243C22×D3—21no—
244C22×D3#10412yes—
245C3×Q8—81no—
246C3×Q8—81no—
247C3×D4—81no—
248C3×D4—81no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4C4×D411yescharacteristic
259C2×C3×C4C4×D411yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12311yescharacteristic
264#67—84no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3#10012yes—
278C2×Dic3#10412yes—
279C2×Dic3#10412yes—
280C4×D3—24no—
281C4×D3—24no—
282C4×D3—42no—
283C4×D3—42no—
284C4×D3—42no—
285C4×D3—42no—
286C3×C8—24no—
287C3×C8—42no—
288C3⋊C8—24no—
289C3⋊C8—42no—
290C24—21no—
291C24—31no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—122no—
301C2×D4—122no—
302C2×D4—122no—
303C22×C4—21no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4C2×C4×D311yescharacteristic
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324M16—616no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—62no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—121no—
336C2×C8—121no—
337C4⋊C4—121no—
338C4⋊C4—121no—
339C4⋊C4—122no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—122no—
345C22⋊C4—122no—
346C22⋊C4—122no—
347C22⋊C4—122no—
348C42—61no—
349C42—61no—
350C42—61no—
351C42—61no—
352C42—61no—
353C42—61no—
354C42#23511yescharacteristic
355C42D3×D411yescharacteristic
356C42#23711yescharacteristic
357C42C2×#6711yescharacteristic
358C22×C3—81no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3#38611yescharacteristic
366C22×C3C2×#10011yescharacteristic
367C22×C3#41311yescharacteristic
368C22×C3C2×#10411yescharacteristic
369C22×C3C2×#10411yescharacteristic
370C22×C3C2×#10411yescharacteristic
371C22×C3#42111yescharacteristic
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—22no—
375C2×D3—41no—
376C2×D3—41no—
377C2×D3—41no—
378C2×D3—41no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—81no—
386Dic3—22no—
387Dic3—22no—
388Dic3—22no—
389Dic3—81no—
390Dic3—81no—
391C23—32no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—61no—
400C23—61no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23D3×C22⋊C411yescharacteristic
406Q8—81no—
407Q8—81no—
408D4—81no—
409D4—81no—
410D4—244no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—64no—
424C2×C4—64no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4#67⋊C411yescharacteristic
442C2×C4#88311yescharacteristic
443C2×C4#88411yescharacteristic
444C2×C4C4×#6711yescharacteristic
445C2×C4#93711yescharacteristic
446C2×C4#93811yescharacteristic
447C8—24no—
448C8—42no—
449C8—64no—
450C8—122no—
451C2×C3—161no—
452C2×C3#152411yescharacteristic
453C2×C3#152511yescharacteristic
454C2×C3#152711yescharacteristic
455C2×C3#152911yescharacteristic
456C2×C3#187111yescharacteristic
457C2×C3#189611yescharacteristic
458C2×C3#190011yescharacteristic
459D3—42no—
460C22—62no—
461C22—62no—
462C22—62no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—121no—
471C22—121no—
472C22—121no—
473C22—121no—
474C22D3×#10011yescharacteristic
475C22#474511yescharacteristic
476C22D3×#10411yescharacteristic
477C22D3×#10411yescharacteristic
478C22D3×#10411yescharacteristic
479C22#482111yescharacteristic
480C22#495611yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—62no—
488C4—62no—
489C4—62no—
490C4—81no—
491C4—241no—
492C4—241no—
493C3#912011yescharacteristic, complemented, Sylow
494C2—122no—
495C2—161no—
496C2#6834111yescharacteristic
497C2#6834211yescharacteristic
498C2#6834311yescharacteristic
499C2#6834411yescharacteristic
500C2#7008111yescharacteristic
501C2#7038111yescharacteristic
502C2#7038211yescharacteristic
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#9120
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×#4707C2semidirect
3. C2×C3×#329C2semidirect
4. C2×#4540C2semidirect
5. #69807C2semidirect
6. #67659C2semidirect
7. #67659C2semidirect
36. C2×#329D3semidirect
78. C3×#97D4semidirect
88. #803D4semidirect
224. #97#67semidirect
493. C3#9120semidirect

Maximal subgroups

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

subgroupindex
#830082
C2×#47072
C2×C3×#3292
C2×#45402
#698072
#676592
#91203