Finite Group Databasebeta

← #10844702

Subgroups — #10844702

Order 768.

850 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
1C3×#2568C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic
3#82500C211yesmaximal, characteristic
4C2×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
5#71785C211yesmaximal, characteristic, complemented
6#69682C212yesmaximal, complemented
7#69680C212yesmaximal
8#69676C212yesmaximal
9#69674C212yesmaximal, complemented
10C4×#809C211yesmaximal, characteristic
11#69467C211yesmaximal, characteristic, complemented
12#25293—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#946—21no—
16C22×#215C2212yes—
17C2×C4×D12C2211yescharacteristic
18C3×#400—22no—
19C3×#391—21no—
20C3×#391C2212yes—
21C3×#391C2212yes—
22C3×#382—21no—
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#835C2212yes—
25C2×#835C2212yes—
26C2×#835C2212yes—
27#5083—24no—
28Q8×C3⋊C8—24no—
29#4994—22no—
30#4987—21no—
31C2×#809C2211yescharacteristic
32C2×#809C2211yescharacteristic
33C2×#808C2212yes—
34#4939C2212yes—
35#4935C2211yescharacteristic
36#4928C2212yes—
37C2×C4⋊(C3⋊C8)C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C2×C4×C3⋊C8C2212yes—
40C2×C4×C3⋊C8C2211yescharacteristic
41#4529C2211yescharacteristic
42#4529C2212yes—
43#4529C2211yescharacteristic
44C2×#461—31no—
45C2×C8×Q8—31no—
46#2568D311yescharacteristic, complemented
47#2538—31no—
48#2066—32no—
49#2055—32no—
50#1968—32no—
51#1956—31no—
52#1948—31no—
53#1933—32no—
54C4×#102—31no—
55C22×C3×Q8D411yescharacteristic
56C22×C3×Q8C2312yes—
57C22×D12C2311yescharacteristic
58C22×C4×D3—21no—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4D412yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C42—21no—
70C2×C3×C42C2312yes—
71C2×C3×C42C2311yescharacteristic
72#946—44no—
73C2×#215—24no—
74C2×#215—24no—
75C2×#215—42no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×Dic3⋊C4—22no—
84C22×C3⋊C8C2311yescharacteristic
85C22×C3⋊C8C2312yes—
86C22×C3⋊C8C2311yescharacteristic
87C4×D12—22no—
88C4×D12—22no—
89C3×#95—21no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94#835—24no—
95#835—24no—
96#835—42no—
97C3⋊#95—22no—
98#809—22no—
99#809—22no—
100#808—42no—
101C4⋊(C3⋊C8)—24no—
102C4⋊(C3⋊C8)D414yes—
103C4×C3⋊C8—22no—
104C4×C3⋊C8—24no—
105C22×SD16—32no—
106C2×C4⋊Q8C2×D311yescharacteristic
107C2×C22⋊Q8—61no—
108C2×C4×Q8C2×D311yescharacteristic
109C2×C4×D4—31no—
110#461—64no—
111C8×Q8—64no—
112C2×#106—32no—
113C2×C4⋊C8—31no—
114C2×C4⋊C8—32no—
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118C2×#102—31no—
119C2×#102—31no—
120C2×C4×C8—32no—
121C2×C4×C8—31no—
122#400—22no—
123#396—62no—
124#391—21no—
125#391C2×D312yes—
126#391C2×D312yes—
127#390—32no—
128#389—32no—
129#384—31no—
130#382—21no—
131#380—61no—
132C4×C4⋊C4C2×D311yescharacteristic
133#340—32no—
134#340—31no—
135#340—31no—
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—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8SD1614yescomplemented
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—21no—
155C22×Dic3—22no—
156C2×D12—21no—
157C2×D12—21no—
158C2×D12—21no—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12#4112yes—
162C2×C4×D3—42no—
163C2×C4×D3—42no—
164C2×C4×D3—42no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C42—22no—
174C3×C42—22no—
175C3×C42—42no—
176C3×C42C2×D412yes—
177C3×C42#4112yes—
178#215—48no—
179(C2×D3)⋊C4—42no—
180(C2×D3)⋊C4—44no—
181(C2×D3)⋊C4—82no—
182C4⋊Dic3—21no—
183C4⋊Dic3—22no—
184C4⋊Dic3—21no—
185C4⋊Dic3—22no—
186C4⋊Dic3—42no—
187C4⋊Dic3—81no—
188Dic3⋊C4—44no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8C2×D414yes—
197C2×C3⋊C8#4112yes—
198C22×Q8#6711yescharacteristic
199C22×Q8C22×D312yes—
200C22×D4—31no—
201C23×C4—61no—
202C2×SD16—64no—
203C2×SD16—64no—
204C2×SD16—122no—
205C22×C8—32no—
206C22×C8—31no—
207C22×C8—31no—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C22⋊Q8—124no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×D4—62no—
214C4×D4—62no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4#6711yescharacteristic
223C2×C4⋊C4#6712yes—
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C4⋊C4C22×D311yescharacteristic
226C2×C22⋊C4—61no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C42—21no—
230C2×C42C22×D312yes—
231C2×C42C22×D311yescharacteristic
232#106—122no—
233C4⋊C8—34no—
234C4⋊C8—64no—
235#103—64no—
236#103—64no—
237#103—122no—
238#102—62no—
239#102—62no—
240C4×C8—64no—
241C4×C8—62no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—62no—
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—28no—
257C3×Q8—42no—
258C3×Q8—42no—
259C3×Q8—42no—
260C3×Q8—42no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4#12112yes—
278C2×C3×C4#12112yes—
279C2×C3×C4C2×SD1614yes—
280C2×C3×C4C2×SD1612yes—
281C2×C3×C4#13512yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4H(5,𝔽2)11yescharacteristic
287C2×C3×C42−1+411yescharacteristic
288C2×Dic3—41no—
289C2×Dic3—42no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—42no—
294D12—22no—
295D12—22no—
296D12—41no—
297D12—41no—
298C4×D3—84no—
299C3⋊C8—24no—
300C3⋊C8—22no—
301C3⋊C8—41no—
302C24—61no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8#21514yescomplemented
312C2×D4—32no—
313C2×D4—61no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—61no—
322C22×C4—62no—
323C22×C4—122no—
324C22×C4—122no—
325C22×C4—122no—
326C22×C4C2×#6712yes—
327C22×C4C2×#6711yescharacteristic
328C22×C4C2×#6712yes—
329C22×C4C2×#6711yescharacteristic
330C22×C4C23×D311yescharacteristic
331SD16—128no—
332C2×C8—34no—
333C2×C8—32no—
334C2×C8—61no—
335C2×C8—62no—
336C2×C8—62no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4—62no—
352C4⋊C4—62no—
353C4⋊C4—124no—
354C4⋊C4—122no—
355C4⋊C4—241no—
356C22⋊C4—122no—
357C22⋊C4—124no—
358C22⋊C4—242no—
359C42—22no—
360C42—22no—
361C42—42no—
362C42D3×D412yes—
363C42#23712yes—
364C22×C3C2×#12111yescharacteristic
365C22×C3#54811yescharacteristic
366C22×C3#55011yescharacteristic
367C22×C3C22×SD1611yescharacteristic
368C22×C3C2×#13511yescharacteristic
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—24no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—81no—
382C3×C4—81no—
383C3×C4—82no—
384C3×C4#46514yes—
385C3×C4#52911yescharacteristic
386C3×C4#54211yescharacteristic
387C3×C4#54611yescharacteristic
388C3×C4#54811yescharacteristic
389Dic3—82no—
390Dic3—81no—
391C23—62no—
392C23—121no—
393C23—121no—
394C23—121no—
395C23—121no—
396C23—121no—
397C23—121no—
398C23C22×#6711yescharacteristic
399Q8—28no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404D4—62no—
405D4—62no—
406D4—121no—
407D4—121no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—121no—
425C2×C4—122no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—244no—
431C2×C4#93912yes—
432C2×C4#93912yes—
433C2×C4C2×#21514yes—
434C2×C4C2×#21512yes—
435C2×C4#95012yes—
436C2×C4C2×D3×D411yescharacteristic
437C2×C4C2×#23711yescharacteristic
438C2×C4D3×#4111yescharacteristic
439C2×C4D3×#4111yescharacteristic
440C2×C4#100911yescharacteristic
441C2×C4#101011yescharacteristic
442C8—64no—
443C8—62no—
444C8—121no—
445C2×C3#242611yescharacteristic
446C2×C3C2×#46511yescharacteristic
447C2×C3#306311yescharacteristic
448C2×C3#335911yescharacteristic
449C2×C3#336011yescharacteristic
450C2×C3#339211yescharacteristic
451C2×C3#339311yescharacteristic
452D3—82no—
453C22—122no—
454C22—122no—
455C22—122no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22C2×#93911yescharacteristic
461C22C22×#21511yescharacteristic
462C22C2×#95011yescharacteristic
463C22#583111yescharacteristic
464C22#583411yescharacteristic
465C22#583711yescharacteristic
466C22#583811yescharacteristic
467C4—24no—
468C4—22no—
469C4—42no—
470C4—82no—
471C4—81no—
472C4—81no—
473C4—241no—
474C4—242no—
475C4#504814yes—
476C4#567611yescharacteristic
477C4#567711yescharacteristic
478C4#568011yescharacteristic
479C4#568211yescharacteristic
480C3#2529311yescharacteristic, complemented, Sylow
481C2—242no—
482C2C2×#504811yescharacteristic
483C2#8299511yescharacteristic
484C2#8340011yescharacteristic
485C2#8366711yescharacteristic
486C2#8367011yescharacteristic
487C2#8368511yescharacteristic
488C2#8368811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2568C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2568D3
hypocenterC3#25293
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
1. C3×#2568C2semidirect
4. C2×Q8×C3⋊C8C2semidirect
5. #71785C2semidirect
6. #69682C2semidirect
9. #69674C2semidirect
11. #69467C2semidirect
46. #2568D3semidirect
145. C2×C3×Q8SD16semidirect
311. C2×Q8#215semidirect
480. C3#25293semidirect

Maximal subgroups

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

subgroupindex
C3×#25682
C2×#50832
#825002
C2×Q8×C3⋊C82
#717852
#696822
#696802
#696762
#696742
C4×#8092
#694672
#252933