Finite Group Databasebeta

← #10690786

Subgroups — #10690786

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
1C2×#4707C211yesmaximal, characteristic, complemented
2#74256C211yesmaximal, characteristic
3#69807C211yesmaximal, characteristic, complemented
4C2×#4464C211yesmaximal, characteristic, complemented
5C2×#4464C211yesmaximal, characteristic, complemented
6C3×#1295C211yesmaximal, characteristic, complemented
7#67630C211yesmaximal, characteristic, complemented
8#9120—31nomaximal, Sylow
9C22×ΓL(1,𝔽25)—21no—
10C2×#897C411yescharacteristic
11C2×#895C411yescharacteristic
12C2×#875C2211yescharacteristic
13C2×C42×D3C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15C2×C3×#97C2211yescharacteristic
16C3×#382C2211yescharacteristic
17C2×(C2×D3)⋊C8—21no—
18C2×C8×Dic3—21no—
19#4988C411yescharacteristic
20#4973C411yescharacteristic
21C2×#803C2211yescharacteristic
22C2×#803C2211yescharacteristic
23#4707—22no—
24#4707—22no—
25#4670—21no—
26#4670—21no—
27#4665—21no—
28#4665—21no—
29#4550—21no—
30#4464—22no—
31#4464—22no—
32C2×C4×M16—31no—
33#2850—31no—
34#1749—31no—
35C2×#329—31no—
36C2×#329—31no—
37#1295—31no—
38#1295D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12C2×C411yescharacteristic
41C22×C4×D3C2×C411yescharacteristic
42C22×C4×D3D411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×Dic6C2×C411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—21no—
57C2×C4×Dic3C2×C411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C2×C4×Dic3D411yescharacteristic
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62C2×ΓL(1,𝔽25)—24no—
63C2×ΓL(1,𝔽25)—24no—
64C2×ΓL(1,𝔽25)—24no—
65#897—42no—
66#895—42no—
67#875—21no—
68#875—21no—
69#875—21no—
70#875—21no—
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×#124—31no—
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#382C2×D311yescharacteristic
109#336—61no—
110#329—62no—
111#329—62no—
112C23×D3—21no—
113C23×D3C22⋊C411yescharacteristic
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—21no—
120C22×Dic3C22⋊C411yescharacteristic
121C22×Dic3C22⋊C411yescharacteristic
122C22×Dic3C22⋊C411yescharacteristic
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—42no—
128C2×D12—42no—
129C2×D12—42no—
130C2×C4×D3—22no—
131C2×C4×D3—21no—
132C2×C4×D3—21no—
133C2×C4×D3—22no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—22no—
139C2×C4×D3—22no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×Dic6—21no—
143C2×Dic6—21no—
144C2×Dic6—41no—
145C2×Dic6—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×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157C3×C42#4111yescharacteristic
158(C2×D3)⋊C4—42no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—42no—
164(C2×D3)⋊C4—42no—
165(C2×D3)⋊C4—42no—
166C4⋊Dic3—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×Q8—31no—
181C22×D4—31no—
182C22×D4—61no—
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—61no—
194#124—61no—
195#124—61no—
196#124—61no—
197C22⋊Q8—122no—
198#121—122no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4C4×D311yescharacteristic
201C2×C4⋊C4C4×D311yescharacteristic
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
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#97D1212yescomplemented
225C4×C8—64no—
226#95—21no—
227#95—21no—
228#95—61no—
229#95—61no—
230C23×C3C2×C22⋊C411yescharacteristic
231C22×D3—21no—
232C22×D3—21no—
233C22×D3—21no—
234C22×D3—21no—
235C22×D3—21no—
236C22×D3—21no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3—41no—
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3#10412yes—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4C4×D411yescharacteristic
258C2×C3×C4C4×D411yescharacteristic
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12311yescharacteristic
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3#10012yes—
273C2×Dic3#10412yes—
274C2×Dic3#10412yes—
275D12—81no—
276D12—81no—
277D12—84no—
278C4×D3—24no—
279C4×D3—24no—
280C4×D3—42no—
281C4×D3—42no—
282C4×D3—42no—
283C4×D3—42no—
284Dic6—81no—
285Dic6—81no—
286C3×C8—24no—
287C3×C8—42no—
288C3⋊C8—24no—
289C3⋊C8—42no—
290C24—31no—
291C24—61no—
292C2×Q8—61no—
293C2×Q8—61no—
294C2×Q8—121no—
295C2×Q8—121no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—122no—
301C2×D4—122no—
302C2×D4—122no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—62no—
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—41no—
338C4⋊C4—41no—
339C4⋊C4—122no—
340C22⋊C4—122no—
341C22⋊C4—122no—
342C22⋊C4—122no—
343C22⋊C4—122no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C22⋊C4—121no—
348C42—61no—
349C42—61no—
350C42—61no—
351C42—61no—
352C42—61no—
353C42—61no—
354C42C2×D1211yescharacteristic
355C42#23511yescharacteristic
356C42D3×D411yescharacteristic
357C42#23911yescharacteristic
358C22×C3#38611yescharacteristic
359C22×C3C2×#10011yescharacteristic
360C22×C3#41311yescharacteristic
361C22×C3C2×#10411yescharacteristic
362C22×C3C2×#10411yescharacteristic
363C22×C3C2×#10411yescharacteristic
364C22×C3#42111yescharacteristic
365C2×D3—22no—
366C2×D3—22no—
367C2×D3—22no—
368C2×D3—41no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—41no—
372C2×D3—81no—
373C2×D3—81no—
374C2×D3—81no—
375C2×D3—81no—
376C2×D3—81no—
377C2×D3—81no—
378C2×D3—81no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—81no—
386C3×C4—81no—
387Dic3—22no—
388Dic3—22no—
389Dic3—22no—
390Dic3—81no—
391C23—32no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—121no—
399C23—121no—
400C23—121no—
401C23—121no—
402C23—121no—
403C23—121no—
404C23—121no—
405C23D3×C22⋊C411yescharacteristic
406Q8—241no—
407Q8—241no—
408D4—241no—
409D4—241no—
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—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—64no—
427C2×C4—64no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4C4×D1211yescharacteristic
442C2×C4#88211yescharacteristic
443C2×C4#88411yescharacteristic
444C2×C4#89311yescharacteristic
445C2×C4#89411yescharacteristic
446C2×C4#89511yescharacteristic
447C8—24no—
448C8—42no—
449C8—64no—
450C8—122no—
451C2×C3#152411yescharacteristic
452C2×C3#152511yescharacteristic
453C2×C3#152711yescharacteristic
454C2×C3#152911yescharacteristic
455C2×C3#187111yescharacteristic
456C2×C3#189611yescharacteristic
457C2×C3#190011yescharacteristic
458D3—42no—
459D3—161no—
460C22—62no—
461C22—62no—
462C22—62no—
463C22—121no—
464C22—121no—
465C22—121no—
466C22—121no—
467C22—241no—
468C22—241no—
469C22—241no—
470C22—241no—
471C22—241no—
472C22—241no—
473C22—241no—
474C22#466511yescharacteristic
475C22D3×#10011yescharacteristic
476C22#475111yescharacteristic
477C22D3×#10411yescharacteristic
478C22D3×#10411yescharacteristic
479C22D3×#10411yescharacteristic
480C22#482111yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—62no—
488C4—62no—
489C4—62no—
490C4—81no—
491C4—81no—
492C4—241no—
493C3#912011yescharacteristic, complemented, Sylow
494C2—122no—
495C2—481no—
496C2#6834211yescharacteristic
497C2#6834311yescharacteristic
498C2#6836111yescharacteristic
499C2#6836411yescharacteristic
500C2#6875511yescharacteristic
501C2#6875611yescharacteristic
502C2#6899211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#895C4
cosocleC2×C3×C42C23
hypercenter#1295D3
hypocenterC3#9120
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
1. C2×#4707C2semidirect
3. #69807C2semidirect
4. C2×#4464C2semidirect
5. C2×#4464C2semidirect
6. C3×#1295C2semidirect
7. #67630C2semidirect
38. #1295D3semidirect
78. C3×#97D4semidirect
88. #803D4semidirect
224. #97D12semidirect
493. C3#9120semidirect

Maximal subgroups

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

subgroupindex
C2×#47072
#742562
#698072
C2×#44642
C3×#12952
#676302
#91203