Finite Group Databasebeta

← #10844701

Subgroups — #10844701

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×#2538C211yesmaximal, characteristic
2#82479C211yesmaximal, characteristic, complemented
3C2×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×#5019C211yesmaximal, characteristic
5C4×#832C211yesmaximal, characteristic
6#69651C212yesmaximal
7#69646C212yesmaximal
8#69634C212yesmaximal, complemented
9#69614C212yesmaximal, complemented
10#69579C211yesmaximal, characteristic, complemented
11#69568C211yesmaximal, characteristic, complemented
12#25293—31nomaximal, Sylow
13C2×C3×C22⋊Q8—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C22×#214C2212yes—
17C2×#869C2211yescharacteristic
18C3×#396—22no—
19C3×#390C2212yes—
20C3×#389C2212yes—
21C3×#384C2211yescharacteristic
22C3×#380—21no—
23#5229—21no—
24C2×#832C2211yescharacteristic
25C2×#832C2211yescharacteristic
26Q8×C3⋊C8—24no—
27#5019—24no—
28#4986—22no—
29#4977—21no—
30C2×#810C2212yes—
31C2×#810C2212yes—
32C2×#810C2212yes—
33C2×#808C2212yes—
34C4×C4⋊Dic3C2211yescharacteristic
35#4929C2212yes—
36#4929C2212yes—
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#2568—31no—
47#2538D311yescharacteristic
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×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C22×Dic6C2312yes—
59C3×C22⋊Q8—44no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76C2×#214—24no—
77C2×#214—24no—
78C2×#214—42no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4×Dic3—21no—
84C22×C3⋊C8C2311yescharacteristic
85C22×C3⋊C8C2312yes—
86C22×C3⋊C8C2311yescharacteristic
87#869—22no—
88#869—22no—
89C3×#95—22no—
90#832—22no—
91#832—22no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—21no—
97#810—24no—
98#810—24no—
99#810—42no—
100#808—42no—
101C4⋊(C3⋊C8)—24no—
102C4⋊(C3⋊C8)D414yes—
103C4×C3⋊C8—22no—
104C4×C3⋊C8—24no—
105C22×SD16—32no—
106C2×C4⋊Q8—31no—
107C2×C22⋊Q8—21no—
108C2×C4×Q8C2×D311yescharacteristic
109C2×C4×D4C2×D311yescharacteristic
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—62no—
123#396—22no—
124#391—32no—
125#391—32no—
126#391—61no—
127#390C2×D312yes—
128#389C2×D312yes—
129#384C2×D311yescharacteristic
130#382—61no—
131#380—21no—
132C4×C4⋊C4—31no—
133#340—32no—
134#340—31no—
135#340—31no—
136C24×C3—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8SD1614yescomplemented
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4#4112yes—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—22no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—44no—
173C3×C4⋊C4—81no—
174C3×C22⋊C4—42no—
175C3×C22⋊C4—44no—
176C3×C22⋊C4—82no—
177C3×C42—22no—
178C3×C42—22no—
179C3×C42C2×D412yes—
180C3×C42#4112yes—
181#214—48no—
182C4⋊Dic3—21no—
183C4⋊Dic3—21no—
184C4⋊Dic3—21no—
185C4⋊Dic3—21no—
186C4⋊Dic3—42no—
187C4⋊Dic3—42no—
188C4×Dic3—42no—
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—32no—
199C22×Q8#6711yescharacteristic
200C22×D4C22×D311yescharacteristic
201C23×C4—21no—
202C2×SD16—64no—
203C2×SD16—64no—
204C2×SD16—122no—
205C22×C8—32no—
206C22×C8—31no—
207C22×C8—31no—
208C4⋊Q8—62no—
209C4⋊Q8—62no—
210C22⋊Q8—44no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×D4—22no—
214C4×D4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4#6711yescharacteristic
223C2×C4⋊C4#6712yes—
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C4⋊C4C22×D312yes—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C42—61no—
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—22no—
243#95—61no—
244#95—62no—
245#95—62no—
246#95—62no—
247#95—62no—
248C23×C3—22no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3C22×D411yescharacteristic
256C3×Q8—28no—
257C3×D4—22no—
258C3×D4—22no—
259C3×D4—41no—
260C3×D4—41no—
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—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—84no—
275C2×C3×C4#12112yes—
276C2×C3×C4#12112yes—
277C2×C3×C4C2×SD1614yes—
278C2×C3×C4C2×SD1612yes—
279C2×C3×C4#13512yes—
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4H(5,𝔽2)11yescharacteristic
285C2×C3×C42−1+411yescharacteristic
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295Dic6—42no—
296Dic6—42no—
297Dic6—42no—
298Dic6—42no—
299C3⋊C8—24no—
300C3⋊C8—22no—
301C3⋊C8—41no—
302C24—21no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—62no—
306C2×Q8—62no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8#21414yescomplemented
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4#23912yes—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—62no—
326C22×C4C2×#6712yes—
327C22×C4C2×#6712yes—
328C22×C4C2×#6711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330C22×C4C23×D311yescharacteristic
331SD16—128no—
332C2×C8—34no—
333C2×C8—32no—
334C2×C8—61no—
335C2×C8—62no—
336C2×C8—61no—
337C2×C8—62no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—81no—
354C4⋊C4—122no—
355C4⋊C4—122no—
356C22⋊C4—44no—
357C22⋊C4—42no—
358C22⋊C4—82no—
359C42—22no—
360C42—22no—
361C42—122no—
362C42D3×D412yes—
363C42#23712yes—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—81no—
368C22×C3—81no—
369C22×C3—81no—
370C22×C3—81no—
371C22×C3C2×#12111yescharacteristic
372C22×C3#54811yescharacteristic
373C22×C3#55011yescharacteristic
374C22×C3C22×SD1611yescharacteristic
375C22×C3C2×#13511yescharacteristic
376C22×C3#57711yescharacteristic
377C22×C3#57811yescharacteristic
378C3×C4—24no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—82no—
382C3×C4—81no—
383C3×C4#46514yes—
384C3×C4#52911yescharacteristic
385C3×C4#54211yescharacteristic
386C3×C4#54611yescharacteristic
387C3×C4#54811yescharacteristic
388Dic3—81no—
389Dic3—81no—
390Dic3—82no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23C22×#6711yescharacteristic
399Q8—28no—
400Q8—122no—
401Q8—122no—
402Q8—122no—
403Q8—122no—
404D4—22no—
405D4—22no—
406D4—41no—
407D4—41no—
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—42no—
421C2×C4—84no—
422C2×C4—122no—
423C2×C4—121no—
424C2×C4—122no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—122no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4C2×#21414yes—
432C2×C4C2×#21412yes—
433C2×C4#93912yes—
434C2×C4#93912yes—
435C2×C4#95012yes—
436C2×C4C2×D3×D411yescharacteristic
437C2×C4C2×#23711yescharacteristic
438C2×C4C2×#23911yescharacteristic
439C2×C4#100711yescharacteristic
440C2×C4D3×#4111yescharacteristic
441C2×C4#100911yescharacteristic
442C8—62no—
443C8—64no—
444C8—121no—
445C2×C3—82no—
446C2×C3#242611yescharacteristic
447C2×C3C2×#46511yescharacteristic
448C2×C3#306311yescharacteristic
449C2×C3#335911yescharacteristic
450C2×C3#336011yescharacteristic
451C2×C3#339211yescharacteristic
452C2×C3#339311yescharacteristic
453C22—42no—
454C22—42no—
455C22—42no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22C22×#21411yescharacteristic
461C22C2×#93911yescharacteristic
462C22#581711yescharacteristic
463C22C2×#95011yescharacteristic
464C22#583411yescharacteristic
465C22#583611yescharacteristic
466C22#583811yescharacteristic
467C4—24no—
468C4—22no—
469C4—42no—
470C4—82no—
471C4—81no—
472C4—242no—
473C4—241no—
474C4—241no—
475C4#504114yes—
476C4#562411yescharacteristic
477C4#563011yescharacteristic
478C4#563311yescharacteristic
479C4#564311yescharacteristic
480C3#2529311yescharacteristic, complemented, Sylow
481C2—82no—
482C2C2×#504111yescharacteristic
483C2#8275511yescharacteristic
484C2#8342311yescharacteristic
485C2#8356811yescharacteristic
486C2#8358811yescharacteristic
487C2#8363711yescharacteristic
488C2#8367011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2538D3
hypocenterC3#25293
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. #82479C2semidirect
3. C2×Q8×C3⋊C8C2semidirect
8. #69634C2semidirect
9. #69614C2semidirect
10. #69579C2semidirect
11. #69568C2semidirect
139. C2×C3×Q8SD16semidirect
311. C2×Q8#214semidirect
480. C3#25293semidirect

Maximal subgroups

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

subgroupindex
C3×#25382
#824792
C2×Q8×C3⋊C82
C2×#50192
C4×#8322
#696512
#696462
#696342
#696142
#695792
#695682
#252933