Finite Group Databasebeta

← #10698549

Subgroups — #10698549

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2541C211yesmaximal, characteristic
2#71764C211yesmaximal, characteristic, complemented
3C2×#4544C211yesmaximal, characteristic, complemented
4#71604C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#69467C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#10409—31nomaximal, Sylow
9C2×C3×#127—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×#121—21no—
12C2×C3×#121—21no—
13C2×C3×C4×D4C2211yescharacteristic
14C3×#392—22no—
15C3×#390—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17C2×#832—22no—
18C2×#832—21no—
19C2×#808—21no—
20#4935C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4544—22no—
25D4⋊(C3⋊C8)—22no—
26#4529—21no—
27#4529—21no—
28#2541D311yescharacteristic
29#1957—31no—
30#1948—31no—
31#1802—31no—
32C2×#331—31no—
33C2×D4⋊C8—31no—
34#1279—31no—
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4D411yescharacteristic
40C22×C3×D4D412yes—
41C23×C3×C4—21no—
42C3×#127—21no—
43C3×#127—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—42no—
47C3×#127—42no—
48C3×#121—42no—
49C3×#121—42no—
50C3×#121—42no—
51C3×#121—42no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C42—21no—
63C2×C3×C42C2311yescharacteristic
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70C3×#95—21no—
71#832—44no—
72#832—42no—
73#808—42no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yescomplemented
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C2×#127—21no—
81C2×#127C2×D311yescharacteristic
82C2×#121—21no—
83C2×#121—21no—
84C2×C4×D4C2×D311yescharacteristic
85C2×#106—61no—
86C2×C4⋊C8—31no—
87C2×C4⋊C8—31no—
88C2×#102—61no—
89C2×#102—62no—
90C2×C4×C8—31no—
91#392—22no—
92#390—21no—
93#384—31no—
94C4×C4⋊C4C2×D311yescharacteristic
95#340—61no—
96#340—61no—
97#331—62no—
98D4⋊C8—62no—
99C24×C3—21no—
100C24×C3—22no—
101C2×C3×D4—21no—
102C2×C3×D4—22no—
103C2×C3×D4—21no—
104C2×C3×D4—22no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—42no—
109C2×C3×D4—42no—
110C2×C3×D4—44no—
111C2×C3×D4—44no—
112C2×C3×D4—44no—
113C2×C3×D4—44no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—82no—
127C2×C3×D4—82no—
128C2×C3×D4SD1612yes—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×Dic3—41no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4D812yes—
147C3×C4⋊C4SD1612yes—
148C3×C22⋊C4—44no—
149C3×C22⋊C4—44no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156C3×C42#4111yescharacteristic
157C3×C42#4111yescharacteristic
158C4⋊Dic3—42no—
159C4⋊Dic3—41no—
160C4⋊Dic3—41no—
161C4⋊Dic3—41no—
162C4⋊Dic3—41no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C2×C3⋊C8—41no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4#6711yescharacteristic
177C22×D4#6712yes—
178C23×C4—21no—
179C22×C8—61no—
180C22×C8—61no—
181C22×C8—61no—
182#127—21no—
183#127—21no—
184#127—21no—
185#127—21no—
186#127—42no—
187#127—42no—
188#121—42no—
189#121—42no—
190#121—42no—
191#121—42no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4#6711yescharacteristic
200C2×C4⋊C4#6711yescharacteristic
201C2×C4⋊C4#6711yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C42—21no—
206C2×C42C22×D311yescharacteristic
207#106—122no—
208C4⋊C8—32no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211C4⋊C8—61no—
212#102—124no—
213#102—122no—
214C4×C8—61no—
215C4×C8—61no—
216#95—21no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3#12011yescharacteristic
232C3×D4—22no—
233C3×D4—22no—
234C3×D4—44no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—82no—
238C3×D4—88no—
239C3×D4—88no—
240C3×D4—81no—
241C3×D4—81no—
242C3×D4—81no—
243C3×D4—81no—
244C3×D4—81no—
245C3×D4—81no—
246C3×D4—81no—
247C3×D4—81no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—81no—
260C2×C3×C4—81no—
261C2×C3×C4—81no—
262C2×C3×C4—81no—
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12311yescharacteristic
267C2×C3×C4#12311yescharacteristic
268C2×C3×C4#12411yescharacteristic
269C2×C3×C4C2×D811yescharacteristic
270C2×C3×C4C2×SD1611yescharacteristic
271C2×C3×C4C2×SD1611yescharacteristic
272C2×C3×C4#13511yescharacteristic
273C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
274C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
275C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
276C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
277C2×Dic3—81no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280C3⋊C8—42no—
281C3⋊C8—42no—
282C3⋊C8—81no—
283C24—21no—
284C24—22no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—82no—
311C2×D4—82no—
312C2×D4#21412yes—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—121no—
323C22×C4C2×#6711yescharacteristic
324C22×C4C2×#6711yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C2×C8—62no—
327C2×C8—62no—
328C2×C8—121no—
329C2×C8—121no—
330C2×C8—121no—
331C2×C8—121no—
332C2×C8—121no—
333C2×C8—121no—
334C2×C8—121no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—122no—
340C4⋊C4—121no—
341C4⋊C4—121no—
342C4⋊C4—121no—
343C4⋊C4—121no—
344C4⋊C4#21312yes—
345C4⋊C4#21412yes—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C42—41no—
351C42—41no—
352C42D3×D411yescharacteristic
353C42#23711yescharacteristic
354C42#23711yescharacteristic
355C42#23711yescharacteristic
356C22×C3—42no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3#39411yescharacteristic
371C22×C3#44711yescharacteristic
372C22×C3#44911yescharacteristic
373C22×C3#45011yescharacteristic
374C22×C3#45011yescharacteristic
375C22×C3#45311yescharacteristic
376C22×C3#45511yescharacteristic
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—81no—
382C3×C4#45911yescharacteristic
383C3×C4#46011yescharacteristic
384C3×C4#46511yescharacteristic
385C3×C4#46911yescharacteristic
386C3×C4#48111yescharacteristic
387C3×C4#48211yescharacteristic
388C3×C4#48611yescharacteristic
389C3×C4#49011yescharacteristic
390Dic3—161no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23#95311yescharacteristic
406D4—22no—
407D4—22no—
408D4—44no—
409D4—41no—
410D4—41no—
411D4—82no—
412D4—88no—
413D4—88no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—81no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—241no—
438C2×C4—241no—
439C2×C4—241no—
440C2×C4C2×#21311yescharacteristic
441C2×C4#93211yescharacteristic
442C2×C4#93211yescharacteristic
443C2×C4#93211yescharacteristic
444C2×C4C2×#21411yescharacteristic
445C2×C4C2×#21411yescharacteristic
446C2×C4#93511yescharacteristic
447C2×C4#93511yescharacteristic
448C2×C4#93611yescharacteristic
449C2×C4#93811yescharacteristic
450C2×C4#93911yescharacteristic
451C2×C4#93911yescharacteristic
452C2×C4#94911yescharacteristic
453C2×C4#95011yescharacteristic
454C8—122no—
455C8—122no—
456C8—241no—
457C2×C3—82no—
458C2×C3—162no—
459C2×C3#162111yescharacteristic
460C2×C3#162911yescharacteristic
461C2×C3#163911yescharacteristic
462C2×C3#164211yescharacteristic
463C2×C3#203411yescharacteristic
464C2×C3#203711yescharacteristic
465C2×C3#204111yescharacteristic
466C22—42no—
467C22—42no—
468C22—42no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22#521711yescharacteristic
481C22#521811yescharacteristic
482C22#521811yescharacteristic
483C22#522211yescharacteristic
484C22#523811yescharacteristic
485C22#524111yescharacteristic
486C22#524211yescharacteristic
487C4—21no—
488C4—21no—
489C4—42no—
490C4—42no—
491C4—81no—
492C4—481no—
493C4#503411yescharacteristic
494C4#503511yescharacteristic
495C4#503911yescharacteristic
496C4#504111yescharacteristic
497C4#507011yescharacteristic
498C4#507111yescharacteristic
499C4#507211yescharacteristic
500C4#507511yescharacteristic
501C3#1040911yescharacteristic, complemented, Sylow
502C2—82no—
503C2—162no—
504C2#7168011yescharacteristic
505C2#7176211yescharacteristic
506C2#7185211yescharacteristic
507C2#7190711yescharacteristic
508C2#7190811yescharacteristic
509C2#7191811yescharacteristic
510C2#7192111yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2541C2
socleC23×C3#120
hypercenter#2541D3
hypocenterC3#10409
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. #71764C2semidirect
3. C2×#4544C2semidirect
4. #71604C2semidirect
5. C2×D4⋊(C3⋊C8)C2semidirect
6. #69467C2semidirect
7. #67619C2semidirect
77. C4⋊(C3⋊C8)D4semidirect
501. C3#10409semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
#717642
C2×#45442
#716042
C2×D4⋊(C3⋊C8)2
#694672
#676192
#104093