Finite Group Databasebeta

← #10690535

Subgroups — #10690535

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#946C211yesmaximal, characteristic
2C2×#4685C211yesmaximal, characteristic
3C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
4C2×#4535C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic
6#67704C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9075—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#946C411yescharacteristic
11C2×C42×D3C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5229C411yescharacteristic
15C2×(C2×D3)⋊C8—21no—
16C2×(C2×D3)⋊C8—21no—
17C2×Dic3⋊C8—21no—
18#4989—21no—
19#4987C411yescharacteristic
20#4978—21no—
21#4972C411yescharacteristic
22C4×(C2×D3)⋊C4—21no—
23C4×C4⋊Dic3C2211yescharacteristic
24C4×Dic3⋊C4—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4685—22no—
28#4685—22no—
29C3×Q8⋊C8—22no—
30#4550—21no—
31#4535—22no—
32C2×#431—31no—
33C4×C22⋊Q8—31no—
34#1797—31no—
35C2×Q8⋊C8—31no—
36C2×Q8⋊C8D311yescharacteristic, complemented
37#1279—31no—
38#1279—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C4×D3C2×C411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C22×C4×D3D411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53#946—22no—
54#946—22no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58C2×C4⋊Dic3C2×C411yescharacteristic
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3C2×C411yescharacteristic
63C2×C4×Dic3D411yescharacteristic
64C2×C4×Dic3D411yescharacteristic
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67C42×D3—21no—
68C42×D3—21no—
69C42×D3—21no—
70C42×D3—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4×C8—21no—
74C3×C4×C8D412yes—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80(C2×D3)⋊C8—24no—
81(C2×D3)⋊C8—42no—
82Dic3⋊C8—24no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4×C3⋊C8—21no—
86C4×C3⋊C8D412yes—
87C2×C22⋊Q8—31no—
88C2×C4×Q8C2×D311yescharacteristic
89C22×C42—31no—
90#431—62no—
91#431—62no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8—61no—
94C2×C4⋊C8C2×D311yescharacteristic
95C2×C22⋊C8—61no—
96C2×C22⋊C8—61no—
97C2×C4×C8—31no—
98C2×C4×C8C2×D311yescharacteristic
99#391—31no—
100#389—31no—
101#387—61no—
102#385—61no—
103#380—31no—
104C4×C4⋊C4—31no—
105C4×C4⋊C4—61no—
106C4×C22⋊C4—61no—
107#336—61no—
108Q8⋊C8—22no—
109Q8⋊C8—62no—
110C23×D3C22⋊C411yescharacteristic
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8M1612yes—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3C22⋊C411yescharacteristic
125C22×Dic3C22⋊C411yescharacteristic
126C22×Dic3C22⋊C411yescharacteristic
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×C4×D3—21no—
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—41no—
140C2×C4×D3—41no—
141C2×C4×D3SD1612yes—
142C2×C4×D3Q1612yes—
143C2×C3×C8—22no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158C3×C42#4111yescharacteristic
159(C2×D3)⋊C4—24no—
160(C2×D3)⋊C4—42no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3M1612yes—
165Dic3⋊C4—24no—
166Dic3⋊C4—42no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C4×Dic3—21no—
172C4×Dic3—21no—
173C4×Dic3—41no—
174C4×Dic3—41no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181C22×Q8C4×D311yescharacteristic
182C23×C4—31no—
183C23×C4—31no—
184C23×C4—31no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—61no—
188C22×C8—61no—
189C22⋊Q8—62no—
190C22⋊Q8—62no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4C4×D311yescharacteristic
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C42—21no—
204C2×C42—31no—
205C2×C42—31no—
206C2×C42—31no—
207C2×C42—61no—
208C2×C42—61no—
209C2×C42—61no—
210C2×C42—61no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—64no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218C22⋊C8—64no—
219C22⋊C8—122no—
220C4×C8—21no—
221C4×C8—32no—
222C4×C8—61no—
223C4×C8#6712yes—
224#95—61no—
225#95—61no—
226#95—61no—
227#95—61no—
228#95—61no—
229C23×C3C2×C22⋊C411yescharacteristic
230C22×D3—21no—
231C22×D3—21no—
232C22×D3—21no—
233C22×D3—21no—
234C22×D3—21no—
235C22×D3—21no—
236C22×D3#10312yes—
237C3×Q8—22no—
238C3×Q8—22no—
239C3×Q8—41no—
240C3×Q8—41no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12311yescharacteristic
255C2×C3×C4C2×M1611yescharacteristic
256C2×C3×C4C2×M1611yescharacteristic
257C2×C3×C4#13111yescharacteristic
258C2×C3×C4#13111yescharacteristic
259C2×C3×C4C2×SD1611yescharacteristic
260C2×C3×C4C2×Q1611yescharacteristic
261C2×C3×C4#13511yescharacteristic
262C2×C3×C4#13511yescharacteristic
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—22no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—21no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3#10312yes—
278C4×D3—22no—
279C4×D3—22no—
280C4×D3—22no—
281C4×D3—22no—
282C4×D3—42no—
283C4×D3—42no—
284C4×D3—41no—
285C4×D3—41no—
286C4×D3—41no—
287C4×D3—41no—
288C3×C8—42no—
289C3×C8—81no—
290C3⋊C8—42no—
291C3⋊C8—81no—
292C24—31no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8ΓL(1,𝔽25)12yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—31no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
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—61no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4C2×C4×D311yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4D3×D411yescharacteristic
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—62no—
332C2×C8—121no—
333C2×C8—121no—
334C2×C8—121no—
335C2×C8—121no—
336C2×C8—121no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—32no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—64no—
343C4⋊C4—61no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C4⋊C4—122no—
347C22⋊C4—64no—
348C22⋊C4—122no—
349C42—41no—
350C42—41no—
351C42—61no—
352C42—61no—
353C42—61no—
354C42—61no—
355C42—61no—
356C42—61no—
357C42—121no—
358C42—121no—
359C42#23511yescharacteristic
360C42D3×D411yescharacteristic
361C42#23711yescharacteristic
362C42C2×#6711yescharacteristic
363C22×C3#38611yescharacteristic
364C22×C3#40711yescharacteristic
365C22×C3#40811yescharacteristic
366C22×C3C2×#10311yescharacteristic
367C22×C3#41611yescharacteristic
368C22×C3#42111yescharacteristic
369C22×C3#42111yescharacteristic
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—41no—
374C2×D3—41no—
375C2×D3—41no—
376C2×D3—41no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—42no—
380C3×C4—81no—
381C3×C4#43511yescharacteristic
382C3×C4#43511yescharacteristic
383C3×C4#43611yescharacteristic
384C3×C4#43611yescharacteristic
385C3×C4#46511yescharacteristic
386C3×C4#46511yescharacteristic
387C3×C4#46711yescharacteristic
388C3×C4#46711yescharacteristic
389Dic3—22no—
390Dic3—41no—
391Dic3—41no—
392Dic3—42no—
393Dic3—81no—
394C23—32no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—61no—
399C23—61no—
400C23—61no—
401C23D3×C22⋊C411yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—41no—
405Q8—41no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—32no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—62no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4#67⋊C411yescharacteristic
440C2×C4#88311yescharacteristic
441C2×C4#88411yescharacteristic
442C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
443C2×C4#90111yescharacteristic
444C2×C4D3×M1611yescharacteristic
445C2×C4#90711yescharacteristic
446C2×C4D3×SD1611yescharacteristic
447C2×C4#91611yescharacteristic
448C2×C4D3×Q1611yescharacteristic
449C2×C4#91911yescharacteristic
450C2×C4C4×#6711yescharacteristic
451C2×C4#93711yescharacteristic
452C2×C4#93811yescharacteristic
453C8—42no—
454C8—81no—
455C8—122no—
456C8—241no—
457C2×C3#148111yescharacteristic
458C2×C3#148211yescharacteristic
459C2×C3#148711yescharacteristic
460C2×C3#148811yescharacteristic
461C2×C3#187011yescharacteristic
462C2×C3#189211yescharacteristic
463C2×C3#189711yescharacteristic
464D3—42no—
465C22—62no—
466C22—62no—
467C22—62no—
468C22—121no—
469C22—121no—
470C22—121no—
471C22—121no—
472C22#472611yescharacteristic
473C22#472711yescharacteristic
474C22D3×#10311yescharacteristic
475C22#480411yescharacteristic
476C22#482111yescharacteristic
477C22#482111yescharacteristic
478C22#495611yescharacteristic
479C4—21no—
480C4—21no—
481C4—42no—
482C4—62no—
483C4—81no—
484C4—121no—
485C4—122no—
486C4—121no—
487C4—241no—
488C4#472911yescharacteristic
489C4#473011yescharacteristic
490C4#480711yescharacteristic
491C4#481311yescharacteristic
492C4#511111yescharacteristic
493C4#512811yescharacteristic
494C4#517111yescharacteristic
495C4#518811yescharacteristic
496C3#907511yescharacteristic, complemented, Sylow
497C2—122no—
498C2#6823211yescharacteristic
499C2#6823311yescharacteristic
500C2#6823611yescharacteristic
501C2#6823711yescharacteristic
502C2#6995811yescharacteristic
503C2#7027011yescharacteristic
504C2#7038311yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenterC2×Q8⋊C8D3
hypocenterC3#9075
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×C3×Q8⋊C8C2semidirect
4. C2×#4535C2semidirect
6. #67704C2semidirect
7. #67655C2semidirect
36. C2×Q8⋊C8D3semidirect
496. C3#9075semidirect

Maximal subgroups

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

subgroupindex
C4×#9462
C2×#46852
C2×C3×Q8⋊C82
C2×#45352
#704752
#677042
#676552
#90753