Finite Group Databasebeta

← #10877255

Subgroups — #10877255

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#930C211yesmaximal, characteristic
2#82062C211yesmaximal, characteristic, complemented
3C3×#1797C211yesmaximal, characteristic, complemented
4#70480C212yesmaximal, complemented
5#70455C212yesmaximal, complemented
6#69933C212yesmaximal
7#69929C212yesmaximal, complemented
8#68682C211yesmaximal, characteristic, complemented
9#68681C211yesmaximal, characteristic, complemented
10#68643C211yesmaximal, characteristic, complemented
11#68642C211yesmaximal, characteristic, complemented
12#27683—31nomaximal, Sylow
13C22×C3×C42C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C2×C4×#67—21no—
16C2×#926C2212yes—
17C2×C4×D12C2211yescharacteristic
18C2×C3×C4⋊C8C2212yes—
19C2×C3×C22⋊C8C2212yes—
20#5213—21no—
21#5212—21no—
22#5212—21no—
23#5212C2211yescharacteristic
24C2×#821C2212yes—
25C2×#821C2212yes—
26C2×#818C2212yes—
27C2×#817C2212yes—
28#4940—21no—
29#4939C2211yescharacteristic
30C4×(C2×D3)⋊C4—21no—
31#4936C2212yes—
32#4935C2211yescharacteristic
33#4935C2211yescharacteristic
34C4×C4⋊Dic3—21no—
35C4×C4⋊Dic3—21no—
36C4×C4⋊Dic3C2211yescharacteristic
37#4928C2211yescharacteristic
38C3×#336C2211yescharacteristic
39C3×#336C2211yescharacteristic
40#4551C2212yes—
41#4551C2212yes—
42#4551C2212yes—
43#4551C2212yes—
44C4×#121—31no—
45#2285—31no—
46#2100—31no—
47#2099—31no—
48#2094—31no—
49#2093—31no—
50#1949—32no—
51#1875—32no—
52#1854—32no—
53#1804—32no—
54#1797D311yescharacteristic, complemented
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×C3×C8C2312yes—
62C22×C3×C8C2312yes—
63C2×C3×C42—24no—
64C2×C3×C42D411yescharacteristic
65C2×C3×C42D411yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C2×#217—22no—
69C2×#217—21no—
70#930—22no—
71#930Q814yes—
72C4×#67—44no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77#926—42no—
78#926—42no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×Dic3⋊C4—21no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—21no—
91C4×D12—24no—
92C3×C4⋊C8—24no—
93C3×C22⋊C8—24no—
94C3⋊#95—21no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99#821—42no—
100#821—42no—
101#818—42no—
102#817—42no—
103C2×#121—31no—
104C2×C4×D4—31no—
105C2×C4×D4—61no—
106C2×#117—32no—
107C22×C42C2×D311yescharacteristic
108C2×#107—32no—
109C2×#106—32no—
110C2×C4⋊C8C2×D312yes—
111C2×#102—32no—
112C2×#102—32no—
113C2×C22⋊C8C2×D312yes—
114#390—31no—
115#389—31no—
116#388—61no—
117#386—61no—
118#384—31no—
119#384—31no—
120#382—32no—
121#380—31no—
122#380—61no—
123#380—61no—
124C4×C4⋊C4—31no—
125C4×C4⋊C4—61no—
126C4×C4⋊C4—61no—
127C4×C22⋊C4—61no—
128#340—32no—
129#340—32no—
130#340—32no—
131#340—32no—
132#336C2×D311yescharacteristic
133#336C2×D311yescharacteristic
134C24×C3C2×D411yescharacteristic
135C23×D3—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×Q812yes—
150C22×C3×C4#4112yes—
151C22×C3×C4C2411yescharacteristic
152C2×#67—24no—
153C2×#67—42no—
154C2×#67—42no—
155C22×Dic3—21no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—21no—
163C2×D12C2×Q812yes—
164C2×C4×D3—42no—
165C2×C4×D3—42no—
166C2×C4×D3—42no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C3×C42—22no—
174C3×C42—22no—
175C3×C42—22no—
176C3×C42#4114yes—
177#217—44no—
178#217—42no—
179(C2×D3)⋊C4—24no—
180(C2×D3)⋊C4—42no—
181(C2×D3)⋊C4—42no—
182(C2×D3)⋊C4—42no—
183C4⋊Dic3—21no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—42no—
191C4⋊Dic3—42no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4⋊Dic3C2×Q812yes—
195Dic3⋊C4—42no—
196C4×Dic3—44no—
197C4×Dic3—42no—
198C22×D4—31no—
199C22×D4—61no—
200C23×C4—61no—
201C23×C4D1211yescharacteristic
202C23×C4D1211yescharacteristic
203C23×C4C22×D311yescharacteristic
204C22×C8C22×D312yes—
205C22×C8C22×D312yes—
206#121—34no—
207#121—62no—
208C4×D4—64no—
209C4×D4—124no—
210#117—122no—
211#117—122no—
212C2×C4⋊C4—31no—
213C2×C4⋊C4—32no—
214C2×C4⋊C4—32no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C22⋊C4—61no—
223C2×C22⋊C4—62no—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C42—24no—
229C2×C42—62no—
230C2×C42—61no—
231C2×C42D1211yescharacteristic
232C2×C42D1211yescharacteristic
233C2×C42C22×D311yescharacteristic
234C2×C42C22×D311yescharacteristic
235#107—122no—
236#106—122no—
237C4⋊C8—24no—
238#102—122no—
239#102—122no—
240C22⋊C8—24no—
241#95—61no—
242#95—61no—
243#95—61no—
244#95—62no—
245#95—62no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3C22⋊Q812yes—
253C23×C3C22×D411yescharacteristic
254C22×D3—22no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—41no—
260C22×D3—41no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—44no—
274C2×C3×C4—44no—
275C2×C3×C4C22⋊Q812yes—
276C2×C3×C4#12312yes—
277C2×C3×C4#12312yes—
278C2×C3×C4#13512yes—
279C2×C3×C4AGL(1,ℤ/8)12yes—
280C2×C3×C4C22×Q811yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4112yes—
283C2×C3×C4H(5,𝔽2)12yes—
284#67—48no—
285C2×Dic3—22no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300D12—24no—
301D12—42no—
302C4×D3—84no—
303C3×C8—42no—
304C3×C8—42no—
305C24—61no—
306C24C2×D1211yescharacteristic
307C2×D4—32no—
308C2×D4—64no—
309C2×D4—62no—
310C2×D4—61no—
311C2×D4—62no—
312C2×D4—122no—
313C2×D4—122no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—61no—
327C22×C4—122no—
328C22×C4—122no—
329C22×C4—122no—
330C22×C4C2×D1211yescharacteristic
331C22×C4C2×D1211yescharacteristic
332C22×C4C2×D1211yescharacteristic
333C22×C4C2×D1211yescharacteristic
334C22×C4C2×D1211yescharacteristic
335C22×C4#23712yes—
336C22×C4D3×Q812yes—
337C22×C4C23×D311yescharacteristic
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C4⋊C4—32no—
345C4⋊C4—61no—
346C4⋊C4—62no—
347C4⋊C4—62no—
348C4⋊C4—62no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—122no—
353C4⋊C4—122no—
354C4⋊C4—122no—
355C4⋊C4—122no—
356C4⋊C4—122no—
357C22⋊C4—64no—
358C22⋊C4—124no—
359C22⋊C4—122no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22⋊C4—122no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—124no—
367C42—122no—
368C42#23714yes—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3C2×C22⋊Q811yescharacteristic
377C22×C3C2×#12311yescharacteristic
378C22×C3#53511yescharacteristic
379C22×C3C2×#13511yescharacteristic
380C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
381C22×C3#57511yescharacteristic
382C22×C3#57511yescharacteristic
383C2×D3—42no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—81no—
389C2×D3—81no—
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4#55312yes—
396C3×C4#55412yes—
397Dic3—42no—
398Dic3—81no—
399Dic3—82no—
400Dic3—81no—
401Dic3—82no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—62no—
409C23—121no—
410C23—121no—
411C23—121no—
412C23—121no—
413C23—121no—
414C23—121no—
415C23#89712yes—
416C23C22×D1211yescharacteristic
417D4—64no—
418D4—128no—
419D4—122no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—62no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—122no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—122no—
442C2×C4—121no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—121no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—121no—
449C2×C4—244no—
450C2×C4#88612yes—
451C2×C4#88612yes—
452C2×C4#89712yes—
453C2×C4#90412yes—
454C2×C4#90812yes—
455C2×C4C2×#23712yes—
456C2×C4C2×#23711yescharacteristic
457C2×C4C2×D3×Q811yescharacteristic
458C2×C4#100912yes—
459C8—42no—
460C8—42no—
461C2×C3—42no—
462C2×C3#248711yescharacteristic
463C2×C3#307611yescharacteristic
464C2×C3#307711yescharacteristic
465C2×C3#309111yescharacteristic
466C2×C3#309411yescharacteristic
467C2×C3#320711yescharacteristic
468C2×C3#321911yescharacteristic
469D3—82no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—122no—
478C22—122no—
479C22—122no—
480C22—241no—
481C22—241no—
482C22—241no—
483C22—241no—
484C22C2×#88611yescharacteristic
485C22C2×#89711yescharacteristic
486C22#552511yescharacteristic
487C22C2×#90411yescharacteristic
488C22C2×#90811yescharacteristic
489C22#574811yescharacteristic
490C22#574811yescharacteristic
491C4—22no—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—122no—
497C4—241no—
498C4—241no—
499C4—242no—
500C4—242no—
501C4#561812yes—
502C4#563512yes—
503C3#2768311yescharacteristic, complemented, Sylow
504C2—42no—
505C2—242no—
506C2#7402811yescharacteristic
507C2#7859111yescharacteristic
508C2#7862411yescharacteristic
509C2#7902511yescharacteristic
510C2#7905911yescharacteristic
511C2#7930411yescharacteristic
512C2#7935411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1797C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1797D3
hypocenterC3#27683
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #82062C2semidirect
3. C3×#1797C2semidirect
4. #70480C2semidirect
5. #70455C2semidirect
7. #69929C2semidirect
8. #68682C2semidirect
9. #68681C2semidirect
10. #68643C2semidirect
11. #68642C2semidirect
54. #1797D3semidirect
503. C3#27683semidirect

Maximal subgroups

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

subgroupindex
C4×#9302
#820622
C3×#17972
#704802
#704552
#699332
#699292
#686822
#686812
#686432
#686422
#276833