Finite Group Databasebeta

← #10844804

Subgroups — #10844804

Order 768.

830 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83244C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic
3C2×#4679C211yesmaximal, characteristic, complemented
4#74369C211yesmaximal, characteristic
5C3×#1956C211yesmaximal, characteristic, complemented
6#70280C212yesmaximal
7#70262C212yesmaximal, complemented
8#70260C212yesmaximal
9#70255C212yesmaximal, complemented
10C4×#821C211yesmaximal, characteristic
11#69567C211yesmaximal, characteristic, complemented
12#25301—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C22×#215C2212yes—
15C2×#897—21no—
16C2×C4×D12C2211yescharacteristic
17C2×C4×Dic6C2211yescharacteristic
18C2×C3×#103C2212yes—
19C2×C3×#103C2212yes—
20C2×C3×C4×C8C2211yescharacteristic
21C3×#384C2211yescharacteristic
22#5229C2212yes—
23#5229C2212yes—
24C2×#835C2212yes—
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#815C2212yes—
28C2×Dic3⋊C8C2212yes—
29#5083—24no—
30#4994—22no—
31#4988C2212yes—
32#4987—21no—
33#4985—22no—
34#4984C2212yes—
35#4977—21no—
36#4973—21no—
37C2×#807C2212yes—
38C4×C4⋊Dic3C2211yescharacteristic
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4679—24no—
41#4551C2211yescharacteristic
42#4551C2211yescharacteristic
43#4529C2212yes—
44C2×#461—31no—
45C2×#446—31no—
46#2568—31no—
47#2538—31no—
48#2068—32no—
49#2055—32no—
50#1973—32no—
51#1956D311yescharacteristic, complemented
52#1947—31no—
53#1935—32no—
54C4×#102—31no—
55C22×C3×Q8C2312yes—
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×Dic6D411yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42C2311yescharacteristic
69C2×#215—24no—
70C2×#215—24no—
71C2×#215—42no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3D411yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4D412yes—
81C2×C4×Dic3—21no—
82C2×C4×Dic3C2312yes—
83C22×C3⋊C8C2312yes—
84#897—44no—
85C4×D12—22no—
86C4×D12—22no—
87C4×Dic6—22no—
88C4×Dic6—22no—
89C3×#103—24no—
90C3×#103—42no—
91C3×C4×C8—22no—
92#835—24no—
93C3⋊#95—21no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#821—22no—
100#821—22no—
101#815—24no—
102Dic3⋊C8—24no—
103#807—42no—
104C4⋊(C3⋊C8)D414yes—
105C22×SD16—32no—
106C2×C4⋊Q8C2×D311yescharacteristic
107C2×C22⋊Q8—61no—
108C2×C4×Q8—31no—
109C2×C4×D4—31no—
110#461—64no—
111#446—64no—
112C2×#107—32no—
113C2×C4⋊C8—31no—
114C2×C4⋊C8—32no—
115C2×#103—32no—
116C2×#103C2×D312yes—
117C2×#103C2×D312yes—
118C2×#102—31no—
119C2×#102—31no—
120C2×#97—32no—
121C2×C4×C8C2×D311yescharacteristic
122#400—62no—
123#396—62no—
124#391—32no—
125#391—32no—
126#391—61no—
127#390—32no—
128#389—32no—
129#384C2×D311yescharacteristic
130#382—61no—
131#380—61no—
132C4×C4⋊C4—31no—
133#340—31no—
134#340—31no—
135#340—32no—
136C23×D3—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2411yescharacteristic
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3C2×D412yes—
152C22×Dic3C2×D412yes—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—21no—
156C2×D12—21no—
157C2×D12—21no—
158C2×D12#4112yes—
159C2×C4×D3—42no—
160C2×C4×D3—42no—
161C2×C4×D3—42no—
162C2×Dic6—22no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8#4112yes—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—81no—
177C3×C42C2×D412yes—
178C3×C42#4112yes—
179#215—48no—
180(C2×D3)⋊C4—44no—
181(C2×D3)⋊C4—42no—
182(C2×D3)⋊C4—82no—
183C4⋊Dic3—22no—
184C4⋊Dic3—21no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—44no—
191Dic3⋊C4—24no—
192C4×Dic3—22no—
193C4×Dic3—22no—
194C4×Dic3—42no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8C2×D414yes—
198C22×Q8—31no—
199C22×Q8C22×D312yes—
200C22×D4—31no—
201C23×C4—61no—
202C2×SD16—64no—
203C2×SD16—64no—
204C2×SD16—122no—
205C22×C8—32no—
206C22×C8C22×D311yescharacteristic
207C22×C8C22×D311yescharacteristic
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C22⋊Q8—124no—
211C4×Q8—62no—
212C4×Q8—62no—
213C4×D4—62no—
214C4×D4—62no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4C22×D312yes—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—62no—
229C2×C42—32no—
230C2×C42—61no—
231C2×C42C22×D311yescharacteristic
232#107—122no—
233C4⋊C8—34no—
234C4⋊C8—64no—
235#103—24no—
236#103—42no—
237#103—64no—
238#102—62no—
239#102—62no—
240#97—64no—
241C4×C8—22no—
242#95—62no—
243#95—62no—
244#95—62no—
245#95—62no—
246#95—62no—
247#95—61no—
248C23×C3C22×D411yescharacteristic
249C22×D3—22no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C3×Q8—42no—
257C3×Q8—42no—
258C3×Q8—42no—
259C3×Q8—42no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4#12112yes—
269C2×C3×C4#12112yes—
270C2×C3×C4AGL(1,ℤ/8)12yes—
271C2×C3×C4#13712yes—
272C2×C3×C4C22×D411yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4H(5,𝔽2)11yescharacteristic
277C2×C3×C42−1+411yescharacteristic
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293D12—22no—
294D12—22no—
295D12—41no—
296D12—41no—
297C4×D3—84no—
298Dic6—44no—
299C3×C8—22no—
300C3×C8—41no—
301C3⋊C8—24no—
302C24—61no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×D4—32no—
313C2×D4—61no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—32no—
321C22×C4—32no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—122no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C23×D311yescharacteristic
331SD16—128no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—34no—
338C2×C8—62no—
339C2×C8—62no—
340C2×C8#23712yes—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—64no—
347C4⋊C4—62no—
348C4⋊C4—61no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—81no—
355C4⋊C4—124no—
356C22⋊C4—124no—
357C22⋊C4—122no—
358C22⋊C4—242no—
359C42—62no—
360C42—62no—
361C42—122no—
362C42#23512yes—
363C42D3×D412yes—
364C22×C3C2×#12111yescharacteristic
365C22×C3#54811yescharacteristic
366C22×C3#55011yescharacteristic
367C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
368C22×C3C2×#13711yescharacteristic
369C22×C3#57711yescharacteristic
370C22×C3#57811yescharacteristic
371C2×D3—42no—
372C2×D3—42no—
373C2×D3—42no—
374C2×D3—81no—
375C2×D3—81no—
376C2×D3—81no—
377C2×D3—81no—
378C3×C4—22no—
379C3×C4—82no—
380C3×C4—82no—
381C3×C4#46814yes—
382C3×C4#52911yescharacteristic
383C3×C4#54211yescharacteristic
384C3×C4#54611yescharacteristic
385C3×C4#54811yescharacteristic
386Dic3—42no—
387Dic3—42no—
388Dic3—81no—
389Dic3—81no—
390Dic3—81no—
391C23—62no—
392C23—121no—
393C23—121no—
394C23—121no—
395C23—121no—
396C23—121no—
397C23—121no—
398C23C2×D3×D411yescharacteristic
399Q8—42no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403Q8—124no—
404D4—62no—
405D4—62no—
406D4—121no—
407D4—121no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—244no—
432C2×C4#88412yes—
433C2×C4#88412yes—
434C2×C4#91412yes—
435C2×C4#91812yes—
436C2×C4C2×#23511yescharacteristic
437C2×C4C2×D3×D411yescharacteristic
438C2×C4C2×#23711yescharacteristic
439C2×C4#100411yescharacteristic
440C2×C4D3×#4111yescharacteristic
441C2×C4#101011yescharacteristic
442C8—22no—
443C8—41no—
444C8—64no—
445C2×C3#242611yescharacteristic
446C2×C3C2×#46811yescharacteristic
447C2×C3#306311yescharacteristic
448C2×C3#336711yescharacteristic
449C2×C3#337211yescharacteristic
450C2×C3#339711yescharacteristic
451C2×C3#339811yescharacteristic
452D3—82no—
453C22—122no—
454C22—122no—
455C22—122no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22C2×#88411yescharacteristic
461C22#567711yescharacteristic
462C22#571811yescharacteristic
463C22C2×#91411yescharacteristic
464C22#576211yescharacteristic
465C22C2×#91811yescharacteristic
466C22#576611yescharacteristic
467C4—22no—
468C4—82no—
469C4—82no—
470C4—122no—
471C4—122no—
472C4—241no—
473C4—241no—
474C4—241no—
475C4#481214yes—
476C4#555711yescharacteristic
477C4#556111yescharacteristic
478C4#559811yescharacteristic
479C4#561211yescharacteristic
480C3#2530111yescharacteristic, complemented, Sylow
481C2—242no—
482C2#7644311yescharacteristic
483C2C2×#481211yescharacteristic
484C2#7880011yescharacteristic
485C2#8162211yescharacteristic
486C2#8162311yescharacteristic
487C2#8175711yescharacteristic
488C2#8179711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1956C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1956D3
hypocenterC3#25301
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. #83244C2semidirect
3. C2×#4679C2semidirect
5. C3×#1956C2semidirect
7. #70262C2semidirect
9. #70255C2semidirect
11. #69567C2semidirect
51. #1956D3semidirect
480. C3#25301semidirect

Maximal subgroups

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

subgroupindex
#832442
C2×#50832
C2×#46792
#743692
C3×#19562
#702802
#702622
#702602
#702552
C4×#8212
#695672
#253013