Finite Group Databasebeta

← #10901349

Subgroups — #10901349

Order 768.

1348 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×#5007C212yesmaximal, complemented
3#82076C211yesmaximal, characteristic, complemented
4#71782C212yesmaximal, complemented
5#71762C212yesmaximal, complemented
6#71679C214yesmaximal, complemented
7#71604C212yesmaximal, complemented
8#71530C211yesmaximal, characteristic, complemented
9#29687—31nomaximal, Sylow
10C22×C3×#41—21no—
11C3×#532—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#127C2211yescharacteristic
14C2×C3×#124—21no—
15C2×C3×#121—21no—
16C2×C3×#121C2212yes—
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C2×#924—21no—
21C2×#869C2211yescharacteristic
22C2×#832C2214yes—
23C2×#832C2214yes—
24#5212—21no—
25#5212C2212yes—
26C2×#830C2212yes—
27#5007—28no—
28C2×#807C2214yes—
29#4935C2211yescharacteristic
30#4929—21no—
31#4928—21no—
32#4928C2212yes—
33C2×C4⋊(C3⋊C8)C2212yes—
34#4529C2214yes—
35#4524C2212yes—
36C2×#532D311yescharacteristic
37C2×#474—32no—
38#2857—31no—
39#2057—32no—
40#2041—32no—
41#2033—34no—
42#1802—32no—
43#1797—31no—
44C2×C3×#41—28no—
45C2×C3×#41—44no—
46C2×C3×#41—44no—
47C22×C3×Q8—21no—
48C22×C3×D4—22no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4C2312yes—
52C23×C3×C4—22no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×Dic6—21no—
57C3×C4⋊Q8—24no—
58C3×#127—24no—
59C3×#124—28no—
60C3×#121—28no—
61C3×#121—24no—
62C3×#121—24no—
63C3×C4×D4—28no—
64C3×C4×D4—24no—
65C3×C4×D4D418yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C42—24no—
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D411yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2312yes—
81#924—44no—
82C2×Dic3⋊C4—22no—
83C22×C3⋊C8C2314yes—
84#869—24no—
85#832—28no—
86#832—44no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89#830—42no—
90#807—28no—
91C4⋊(C3⋊C8)—24no—
92C22×#41—21no—
93#532—28no—
94C2×C4⋊Q8—31no—
95C2×C4⋊Q8C2×D311yescharacteristic
96C2×#127C2×D311yescharacteristic
97C2×#124—21no—
98C2×C22⋊Q8—61no—
99C2×#121—21no—
100C2×#121C2×D312yes—
101C2×C4×D4—21no—
102C2×C4×D4C2×D312yes—
103C22×C42C2×D311yescharacteristic
104#474—68no—
105C2×#107—34no—
106C2×C4⋊C8—32no—
107C2×#102—34no—
108C2×#102—34no—
109C2×C22⋊C8—32no—
110#391—61no—
111#389—32no—
112#389—61no—
113#384—31no—
114#380—32no—
115#380—61no—
116#340—34no—
117#336—32no—
118C24×C3—22no—
119C24×C3C2×D411yescharacteristic
120C3×#41—432no—
121C2×C3×Q8—24no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×D4—24no—
125C2×C3×D4—28no—
126C2×C3×D4—24no—
127C2×C3×D4—22no—
128C2×C3×D4—22no—
129C2×C3×D4—22no—
130C2×C3×D4—22no—
131C2×C3×D4—22no—
132C2×C3×D4—42no—
133C2×C3×D4—42no—
134C2×C3×D4—44no—
135C2×C3×D4—44no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4C2×D414yes—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—28no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—41no—
150C22×C3×C4—44no—
151C22×C3×C4—44no—
152C22×C3×C4D814yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—22no—
161C2×Dic6—42no—
162C2×Dic6—42no—
163C2×Dic6—42no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4C2×D414yes—
167C3×C22⋊C4—28no—
168C3×C22⋊C4—28no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42C2×D414yes—
173#217—44no—
174C4⋊Dic3—24no—
175C4⋊Dic3—22no—
176C4⋊Dic3—22no—
177C4⋊Dic3—42no—
178C4⋊Dic3—42no—
179C4⋊Dic3—42no—
180Dic3⋊C4—44no—
181C2×C3⋊C8—24no—
182C2×C3⋊C8—24no—
183C2×C3⋊C8—24no—
184C2×#41—28no—
185C2×#41—44no—
186C2×#41—44no—
187C22×Q8—21no—
188C22×Q8—61no—
189C22×D4—22no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4C22×D312yes—
193C23×C4—22no—
194C23×C4#6711yescharacteristic
195C23×C4#6711yescharacteristic
196C23×C4C22×D311yescharacteristic
197C22×C8—34no—
198C4⋊Q8—24no—
199C4⋊Q8—64no—
200#127—24no—
201#124—28no—
202C22⋊Q8—124no—
203#121—28no—
204#121—24no—
205#121—24no—
206C4×D4—28no—
207C4×D4—24no—
208C4×D4#6718yes—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—32no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4—61no—
215C2×C4⋊C4—61no—
216C2×C4⋊C4C22×D312yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—62no—
220C2×C42—24no—
221C2×C42#6711yescharacteristic
222C2×C42#6711yescharacteristic
223C2×C42C22×D311yescharacteristic
224C2×C42C22×D311yescharacteristic
225#107—68no—
226C4⋊C8—64no—
227#102—68no—
228#102—124no—
229C22⋊C8—122no—
230#95—62no—
231#95—62no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—24no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3C2×D812yes—
246C23×C3C22×D411yescharacteristic
247C3×Q8—48no—
248C3×D4—24no—
249C3×D4—24no—
250C3×D4—48no—
251C3×D4—48no—
252C3×D4—416no—
253C3×D4—42no—
254C3×D4—42no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—28no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—44no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—416no—
272C2×C3×C4#12012yes—
273C2×C3×C4#12012yes—
274C2×C3×C4C2×D812yes—
275C2×C3×C4C2×D812yes—
276C2×C3×C4AGL(1,ℤ/8)12yes—
277C2×C3×C4#13712yes—
278C2×C3×C4C22×D412yes—
279C2×C3×C4H(5,𝔽2)11yescharacteristic
280C2×C3×C42−1+411yescharacteristic
281C2×C3×C42−1+412yes—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288Dic6—84no—
289C3⋊C8—44no—
290C24—22no—
291C24C2×#6711yescharacteristic
292#41—432no—
293C2×Q8—24no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—122no—
297C2×Q8—122no—
298C2×Q8—122no—
299C2×D4—24no—
300C2×D4—28no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4C2×#6714yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—28no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4#21314yes—
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332C22×C4C2×#6711yescharacteristic
333C22×C4C2×#6711yescharacteristic
334C22×C4C2×#6711yescharacteristic
335C22×C4C23×D311yescharacteristic
336C2×C8—64no—
337C2×C8—64no—
338C2×C8—64no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—64no—
344C4⋊C4—122no—
345C4⋊C4—124no—
346C4⋊C4—122no—
347C4⋊C4—122no—
348C4⋊C4C2×#6714yes—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—124no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42C2×#6714yes—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3—44no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3C2×#12011yescharacteristic
371C22×C3#53611yescharacteristic
372C22×C3#53711yescharacteristic
373C22×C3C22×D811yescharacteristic
374C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
375C22×C3C2×#13711yescharacteristic
376C22×C3#57511yescharacteristic
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—41no—
381C3×C4—44no—
382C3×C4#44714yes—
383C3×C4#53612yes—
384C3×C4#53712yes—
385Dic3—82no—
386Dic3—82no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—24no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23C2×#21312yes—
401C23C22×#6711yescharacteristic
402Q8—48no—
403Q8—244no—
404D4—24no—
405D4—24no—
406D4—48no—
407D4—48no—
408D4—416no—
409D4—42no—
410D4—42no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—28no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—416no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4C2×#21312yes—
435C2×C4C2×#21312yes—
436C2×C4#93212yes—
437C2×C4#95112yes—
438C2×C4#95312yes—
439C2×C4#95312yes—
440C2×C4#100911yescharacteristic
441C2×C4#101011yescharacteristic
442C2×C4#101012yes—
443C2×C4C22×#6712yes—
444C8—124no—
445C2×C3—42no—
446C2×C3—84no—
447C2×C3#261611yescharacteristic
448C2×C3C2×#44711yescharacteristic
449C2×C3#300711yescharacteristic
450C2×C3#322611yescharacteristic
451C2×C3#323511yescharacteristic
452C2×C3#324511yescharacteristic
453C2×C3#325411yescharacteristic
454C22—22no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22C22×#21311yescharacteristic
469C22C2×#93211yescharacteristic
470C22#582211yescharacteristic
471C22C2×#95111yescharacteristic
472C22#582511yescharacteristic
473C22#583311yescharacteristic
474C22C2×#95311yescharacteristic
475C4—22no—
476C4—22no—
477C4—22no—
478C4—44no—
479C4—41no—
480C4—242no—
481C4—242no—
482C4#521714yes—
483C4#582512yes—
484C4#583312yes—
485C3#2968711yescharacteristic, complemented, Sylow
486C2—42no—
487C2—84no—
488C2#8327011yescharacteristic
489C2#8327411yescharacteristic
490C2#8331011yescharacteristic
491C2#8332111yescharacteristic
492C2C2×#521711yescharacteristic
493C2#8408411yescharacteristic
494C2#8409411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#29687
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#5007C2semidirect
3. #82076C2semidirect
4. #71782C2semidirect
5. #71762C2semidirect
6. #71679C2semidirect
7. #71604C2semidirect
8. #71530C2semidirect
485. C3#29687semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50072
#820762
#717822
#717622
#716792
#716042
#715302
#296873