Finite Group Databasebeta

← #10848844

Subgroups — #10848844

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5518C211yesmaximal, characteristic, complemented
2#82287C211yesmaximal, characteristic
3C2×#4715C211yesmaximal, characteristic, complemented
4C2×#4697C211yesmaximal, characteristic, complemented
5C3×#1750C211yesmaximal, characteristic, complemented
6#69964C212yesmaximal, complemented
7#69962C212yesmaximal
8#69827C211yesmaximal, characteristic, complemented
9#69771C211yesmaximal, characteristic, complemented
10#68686C212yesmaximal, complemented
11#68685C212yesmaximal, complemented
12#25612—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C2×#924C2212yes—
16C2×#924C2211yescharacteristic
17C22×Dic12C2212yes—
18C22×#204C2212yes—
19#5518—24no—
20#5518—24no—
21C2×#889—22no—
22C2×#879—22no—
23C2×#875C2211yescharacteristic
24C2×#870C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2212yes—
27C2×C3×C22⋊C8C2212yes—
28C2×C3×#97C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30C2×#821C2212yes—
31C2×#816C2212yes—
32C2×#816C2212yes—
33C2×#816C2212yes—
34#4994—22no—
35#4961—22no—
36#4939C2211yescharacteristic
37#4935C2211yescharacteristic
38#4929C2211yescharacteristic
39#4715—24no—
40#4697—24no—
41C3×#336C2212yes—
42#4551C2212yes—
43#4551C2212yes—
44C2×#541—31no—
45C2×#497—31no—
46C2×#488—31no—
47#2620—31no—
48#2001—32no—
49#2000—32no—
50#1992—31no—
51#1956—31no—
52#1880—32no—
53#1878—32no—
54#1750D311yescharacteristic, complemented
55C23×C3×C4C2311yescharacteristic
56C22×#67—21no—
57C22×D12C2311yescharacteristic
58C22×Dic6C2311yescharacteristic
59C22×Dic6C2312yes—
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×#117—22no—
63C3×#117—22no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C22⋊C4D412yes—
66C2×C3×C42C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C2×#217—22no—
69C2×#217—21no—
70#930—42no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2312yes—
79#924—24no—
80#924—24no—
81#924—42no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—21no—
86C2×C4×Dic3—22no—
87C2×Dic12—24no—
88C2×Dic12—24no—
89C2×Dic12—42no—
90C2×#204—24no—
91C2×#204—24no—
92C2×#204—42no—
93#889—216no—
94#879—216no—
95#875—22no—
96#875—22no—
97#870—22no—
98#870D414yescomplemented
99#869—22no—
100#869D414yescomplemented
101C4×Dic6—24no—
102C4×Dic6—24no—
103C3×C22⋊C8—24no—
104C3×#97—22no—
105C3×C4×C8—22no—
106C3⋊#95—22no—
107#821—24no—
108#816—24no—
109#816—42no—
110#816—42no—
111C22×Q16—32no—
112C22×SD16—32no—
113#541—64no—
114#541—64no—
115C2×C4⋊Q8—31no—
116C2×#126—62no—
117C2×#125—31no—
118C2×#125—62no—
119C2×#124—31no—
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8—31no—
122C2×#121—31no—
123C2×C4×Q8—32no—
124C2×#117C2×D311yescharacteristic
125#497—64no—
126#488—64no—
127C2×#103—32no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#102—32no—
131C2×C22⋊C8C2×D312yes—
132C2×#97C2×D311yescharacteristic
133C2×C4×C8C2×D311yescharacteristic
134#396—62no—
135#394—62no—
136#391—31no—
137#390—31no—
138#384—31no—
139#340—32no—
140#340—32no—
141#336C2×D312yes—
142C24×C3C2×D411yescharacteristic
143C23×D3—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2411yescharacteristic
152C2×#67—42no—
153C2×#67—82no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—22no—
162C2×D12—22no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6C2×D414yes—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8#4114yes—
179C3×C4⋊C4—24no—
180C3×C22⋊C4—24no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42C2×D412yes—
184C3×C42C2×D412yes—
185#217—28no—
186#217—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—82no—
190C4⋊Dic3—28no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3—82no—
195C4⋊Dic3C2×D414yes—
196Dic3⋊C4—28no—
197Dic3⋊C4—28no—
198Dic3⋊C4—28no—
199Dic3⋊C4—42no—
200C4×Dic3—28no—
201Dic12—48no—
202#204—48no—
203C22×Q8—32no—
204C22×Q8—31no—
205C22×D4—31no—
206C22×D4—61no—
207C23×C4C22×D311yescharacteristic
208C2×Q16—64no—
209C2×Q16—64no—
210C2×Q16—122no—
211C2×SD16—64no—
212C2×SD16—64no—
213C2×SD16—122no—
214C22×C8C22×D312yes—
215C22×C8C22×D312yes—
216C4⋊Q8—34no—
217C4⋊Q8—62no—
218#126—616no—
219#125—34no—
220#125—616no—
221#125—62no—
222#124—62no—
223#124—62no—
224C22⋊Q8—64no—
225C22⋊Q8—64no—
226C22⋊Q8—122no—
227#121—122no—
228C4×Q8—64no—
229C4×Q8—64no—
230#117—22no—
231#117—22no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4D1212yes—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4D1212yes—
248C2×C42—62no—
249C2×C42C22×D311yescharacteristic
250C2×C42C22×D311yescharacteristic
251#103—64no—
252#103—122no—
253#103—122no—
254#102—64no—
255C22⋊C8—24no—
256#97—22no—
257C4×C8—22no—
258#95—62no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3C22×D411yescharacteristic
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—42no—
269C22×D3—42no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4#12014yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4112yes—
284C2×C3×C4H(5,𝔽2)12yes—
285#67—84no—
286C2×Dic3—24no—
287C2×Dic3—24no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299D12—42no—
300D12—42no—
301Dic6—24no—
302Dic6—24no—
303Dic6—42no—
304Dic6—42no—
305Dic6—42no—
306Dic6—42no—
307C3×C8—24no—
308C3×C8—42no—
309C24—61no—
310C24C2×D1211yescharacteristic
311C2×Q8—34no—
312C2×Q8—62no—
313C2×Q8—62no—
314C2×Q8—62no—
315C2×Q8—62no—
316C2×Q8—61no—
317C2×Q8—61no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—62no—
321C2×D4—62no—
322C2×D4—61no—
323C2×D4—61no—
324C2×D4—62no—
325C2×D4—122no—
326C2×D4—242no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4C2×D1212yes—
337C22×C4C2×D1211yescharacteristic
338C22×C4C2×D1212yes—
339C22×C4C23×D311yescharacteristic
340Q16—128no—
341SD16—128no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8#23514yes—
348C4⋊C4—24no—
349C4⋊C4—34no—
350C4⋊C4—68no—
351C4⋊C4—68no—
352C4⋊C4—68no—
353C4⋊C4—68no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—62no—
357C4⋊C4—122no—
358C4⋊C4—242no—
359C22⋊C4—24no—
360C22⋊C4—68no—
361C22⋊C4—122no—
362C22⋊C4—122no—
363C22⋊C4—124no—
364C22⋊C4—242no—
365C42—21no—
366C42—21no—
367C42—68no—
368C42D3×D412yes—
369C42D3×D412yes—
370C22×C3—42no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—42no—
375C22×C3C2×#12011yescharacteristic
376C22×C3#53211yescharacteristic
377C22×C3#53511yescharacteristic
378C22×C3#57712yes—
379C22×C3#57812yes—
380C2×D3—82no—
381C2×D3—81no—
382C2×D3—81no—
383C2×D3—81no—
384C2×D3—82no—
385C3×C4—24no—
386C3×C4—41no—
387C3×C4—42no—
388C3×C4#54612yes—
389C3×C4#55012yes—
390Dic3—44no—
391Dic3—44no—
392Dic3—81no—
393Dic3—81no—
394Dic3—81no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—21no—
400C23—121no—
401C23—121no—
402C23—122no—
403C23—122no—
404C23—121no—
405C23C22×D1211yescharacteristic
406Q8—64no—
407Q8—64no—
408Q8—122no—
409Q8—122no—
410Q8—122no—
411Q8—122no—
412D4—122no—
413D4—122no—
414D4—244no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—64no—
426C2×C4—64no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4#88214yes—
439C2×C4C2×#23512yes—
440C2×C4C2×D3×D411yescharacteristic
441C2×C4C2×D3×D411yescharacteristic
442C2×C4#100912yes—
443C8—24no—
444C8—42no—
445C2×C3—81no—
446C2×C3#243111yescharacteristic
447C2×C3#301312yes—
448C2×C3#317812yes—
449C2×C3#322211yescharacteristic
450C2×C3#322311yescharacteristic
451D3—161no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—41no—
456C22—42no—
457C22—241no—
458C22—241no—
459C22—242no—
460C22—241no—
461C22—242no—
462C22#547711yescharacteristic
463C22C2×#88211yescharacteristic
464C22#552511yescharacteristic
465C22#575112yes—
466C22#575312yes—
467C4—24no—
468C4—42no—
469C4—41no—
470C4—124no—
471C4—124no—
472C4—241no—
473C4—241no—
474C4—241no—
475C4#555012yes—
476C4#555312yes—
477C3#2561211yescharacteristic, complemented, Sylow
478C2—81no—
479C2—481no—
480C2#7358611yescharacteristic
481C2#7857012yes—
482C2#7868012yes—
483C2#7939111yescharacteristic
484C2#7939811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1750C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1750D3
hypocenterC3#25612
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
1. C2×#5518C2semidirect
3. C2×#4715C2semidirect
4. C2×#4697C2semidirect
5. C3×#1750C2semidirect
6. #69964C2semidirect
8. #69827C2semidirect
9. #69771C2semidirect
10. #68686C2semidirect
11. #68685C2semidirect
54. #1750D3semidirect
98. #870D4semidirect
100. #869D4semidirect
477. C3#25612semidirect

Maximal subgroups

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

subgroupindex
C2×#55182
#822872
C2×#47152
C2×#46972
C3×#17502
#699642
#699622
#698272
#697712
#686862
#686852
#256123