Finite Group Databasebeta

← C2×#79917 #11659547

Subgroups — C2×#79917 #11659547

Order 768.

1982 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×#5578C211yesmaximal, characteristic
2C2×C3×C8×Q8C211yesmaximal, characteristic, complemented
3C2×#5024C211yesmaximal, characteristic, complemented
4#79917C2116yesmaximal, direct product factor
5C2×#4839C212yesmaximal
6C2×#4837C211yesmaximal, characteristic
7C2×#4828C212yesmaximal, complemented
8C2×#4831C211yesmaximal, characteristic, complemented
9C2×#4689C212yesmaximal
10C2×C4×D3×C8C211yesmaximal, characteristic
11C2×C8×Dic6C211yesmaximal, characteristic, complemented
12C2×C8×Dic6C212yesmaximal, complemented
13C2×#2960—31nomaximal, Sylow
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#946C411yescharacteristic
16C2×#944C411yescharacteristic
17C22×ΓL(1,𝔽25)C2212yes—
18C22×D3×C8C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20#5578C418yes—
21#5578C2218yes—
22C2×#897C411yescharacteristic
23C2×#896C412yes—
24C2×#889C412yes—
25C2×#888C411yescharacteristic
26C2×#872C2212yes—
27C2×C42×D3C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C2×C4×Dic6C2212yes—
30C3×C8×Q8C2218yescomplemented
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2212yes—
33C2×C3×C4×C8C2211yescharacteristic
34C2×C3×C4×C8C2212yes—
35C2×#815C2212yes—
36C2×Dic3⋊C8C2212yes—
37C2×Dic3⋊C8C2212yes—
38C2×Dic3⋊C8C2212yes—
39C2×C8×Dic3C2211yescharacteristic
40C2×C8×Dic3C2212yes—
41C2×C8×Dic3C2211yescharacteristic
42#5024C2218yescomplemented
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2212yes—
45C2×#803C2212yes—
46C2×C4×C3⋊C8C2211yescharacteristic
47#4839C22116yes—
48#4837C2218yes—
49#4831C2218yescomplemented
50#4828C22116yescomplemented
51#4689C22116yes—
52C4×D3×C8C2218yes—
53C8×Dic6C2218yescomplemented
54C8×Dic6C22116yescomplemented
55C2×#533—31no—
56#2960—316no—
57C2×#446—31no—
58C2×#446—32no—
59C2×#446—31no—
60C2×C8×Q8—32no—
61C2×C8×Q8—31no—
62C2×C8×Q8D311yescharacteristic, complemented
63C2×#424—32no—
64C2×#423—31no—
65C2×#405—32no—
66C22×C4×C8—31no—
67C22×C3×Q8C2×C411yescharacteristic
68C22×C4×D3C2×C411yescharacteristic
69C22×C4×D3C2×C411yescharacteristic
70C22×C4×D3C2311yescharacteristic
71C22×Dic6C2×C412yes—
72C22×Dic6C2×C411yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2312yes—
75C22×C3×C8C2311yescharacteristic
76C3×C4×Q8C2×C414yes—
77C3×C4×Q8C2314yes—
78C2×C3×C4⋊C4C2×C411yescharacteristic
79C2×C3×C4⋊C4C2×C412yes—
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2312yes—
82#946C2×C418yes—
83#944C2×C418yes—
84C2×(C2×D3)⋊C4C2×C412yes—
85C2×(C2×D3)⋊C4C2×C412yes—
86C2×C4⋊Dic3C2×C411yescharacteristic
87C2×C4⋊Dic3C2×C412yes—
88C2×Dic3⋊C4C2×C412yes—
89C2×Dic3⋊C4C2×C412yes—
90C2×Dic3⋊C4C2×C412yes—
91C2×Dic3⋊C4C2×C412yes—
92C2×Dic3⋊C4C2×C412yes—
93C2×C4×Dic3C2×C411yescharacteristic
94C2×C4×Dic3C2×C411yescharacteristic
95C2×C4×Dic3C2311yescharacteristic
96C2×C4×Dic3C2312yes—
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C2×ΓL(1,𝔽25)—28no—
101C2×ΓL(1,𝔽25)C2318yes—
102C2×D3×C8—24no—
103C2×D3×C8Q818yescomplemented
104C2×D3×C8C2314yes—
105C2×D3×C8C2314yes—
106#897C2×C418yes—
107#896C2×C4116yes—
108#889C2×C4116yes—
109#888C2×C418yes—
110#872C2×C418yes—
111#872C2318yes—
112C42×D3C2×C414yes—
113C42×D3C2314yes—
114C4×Dic6C2×C418yes—
115C4×Dic6C2×C414yes—
116C4×Dic6C2318yes—
117C4×Dic6C2314yes—
118C3×C4⋊C8C2318yes—
119C3×C4⋊C8C2314yes—
120C3×C4×C8C2314yes—
121C3×C4×C8C2318yes—
122#815C2318yes—
123Dic3⋊C8C2318yes—
124Dic3⋊C8C2318yes—
125Dic3⋊C8C2318yes—
126C8×Dic3C2314yes—
127C8×Dic3C2318yes—
128C8×Dic3C2314yes—
129C4⋊(C3⋊C8)C2314yes—
130C4⋊(C3⋊C8)C2318yes—
131#803C2318yes—
132C4×C3⋊C8C2314yes—
133C22×M16—32no—
134C23×C8—31no—
135C23×C8—31no—
136#533—38no—
137#533—38no—
138C2×C4⋊Q8—31no—
139C2×C4⋊Q8—31no—
140C2×#125—32no—
141C2×C22⋊Q8—31no—
142C2×C22⋊Q8—32no—
143C2×C22⋊Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×Q8—32no—
146C2×C4×Q8C2×D311yescharacteristic
147C2×#117—32no—
148C22×C42—31no—
149#446—38no—
150#446—316no—
151#446—38no—
152C8×Q8—316no—
153C8×Q8—38no—
154C8×Q8C2×D318yescomplemented
155#424—316no—
156#423—38no—
157C2×C4⋊C8—32no—
158C2×C4⋊C8—32no—
159C2×C4⋊C8—32no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—32no—
162C2×C4⋊C8C2×D311yescharacteristic
163C2×C4⋊C8C2×D312yes—
164#405—316no—
165C2×#97—32no—
166C2×#97—32no—
167C2×C4×C8—31no—
168C2×C4×C8—31no—
169C2×C4×C8—32no—
170C2×C4×C8—31no—
171C2×C4×C8—38no—
172C2×C4×C8C2×D311yescharacteristic
173C2×C4×C8C2×D312yes—
174C23×D3C22×C411yescharacteristic
175C2×C3×Q8—24no—
176C2×C3×Q8C22×C414yes—
177C22×C3×C4C22×C411yescharacteristic
178C22×C3×C4C22×C412yes—
179C22×C3×C4C22×C412yes—
180C22×C3×C4C22×C411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C22×Dic3C22×C411yescharacteristic
183C22×Dic3C22×C412yes—
184C22×Dic3C22×C412yes—
185C22×Dic3C22×C411yescharacteristic
186C22×Dic3C22×C411yescharacteristic
187C2×C4×D3—24no—
188C2×C4×D3—24no—
189C2×C4×D3C22×C414yes—
190C2×C4×D3C22×C414yes—
191C2×C4×D3C22×C412yes—
192C2×C4×D3C2×Q814yes—
193C2×C4×D3#4114yes—
194C2×C4×D3C2412yes—
195C2×Dic6—24no—
196C2×Dic6—28no—
197C2×Dic6C22×C418yes—
198C2×Dic6C22×C414yes—
199C2×C3×C8—22no—
200C2×C3×C8—24no—
201C2×C3×C8C2×Q814yes—
202C2×C3×C8C2412yes—
203C2×C3×C8C2414yes—
204C2×C3×C8C2412yes—
205C3×C4⋊C4C22×C414yes—
206C3×C4⋊C4C22×C418yes—
207C3×C42C22×C412yes—
208C3×C42C22×C414yes—
209C3×C42C2412yes—
210C3×C42C2414yes—
211(C2×D3)⋊C4C22×C418yes—
212(C2×D3)⋊C4C22×C418yes—
213C4⋊Dic3C22×C414yes—
214C4⋊Dic3C22×C418yes—
215Dic3⋊C4C22×C418yes—
216Dic3⋊C4C22×C418yes—
217Dic3⋊C4C22×C418yes—
218Dic3⋊C4C22×C418yes—
219Dic3⋊C4C22×C418yes—
220C4×Dic3C22×C412yes—
221C4×Dic3C22×C414yes—
222C4×Dic3C22×C414yes—
223C4×Dic3C22×C414yes—
224C4×Dic3C2412yes—
225C4×Dic3C2414yes—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—24no—
228C2×C3⋊C8C2×Q814yes—
229C2×C3⋊C8C2412yes—
230C2×C3⋊C8C2414yes—
231C2×C3⋊C8C2412yes—
232ΓL(1,𝔽25)—216no—
233D3×C8—28no—
234D3×C8C2×Q8116yescomplemented
235C22×Q8—32no—
236C22×Q8—31no—
237C22×Q8C4×D311yescharacteristic
238C23×C4—31no—
239C23×C4—31no—
240C23×C4—31no—
241C2×M16—38no—
242C2×M16—68no—
243C22×C8—31no—
244C22×C8—38no—
245C22×C8—34no—
246C22×C8—31no—
247C22×C8—32no—
248C22×C8—34no—
249C22×C8—64no—
250C22×C8C22×D311yescharacteristic
251C22×C8C22×D312yes—
252C22×C8C22×D311yescharacteristic
253C4⋊Q8—38no—
254C4⋊Q8—38no—
255#125—316no—
256C22⋊Q8—38no—
257C22⋊Q8—316no—
258C22⋊Q8—38no—
259C4×Q8—38no—
260C4×Q8—34no—
261C4×Q8—38no—
262C4×Q8—34no—
263C4×Q8C4×D314yes—
264C4×Q8C22×D314yes—
265#117—38no—
266#117—38no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4C4×D312yes—
275C2×C4⋊C4C4×D311yescharacteristic
276C2×C22⋊C4—32no—
277C2×C22⋊C4—32no—
278C2×C42—31no—
279C2×C42—31no—
280C2×C42—32no—
281C2×C42—31no—
282C2×C42—34no—
283C2×C42—34no—
284C2×C42C22×D312yes—
285C2×C42C22×D311yescharacteristic
286C4⋊C8—38no—
287C4⋊C8—38no—
288C4⋊C8—38no—
289C4⋊C8—34no—
290C4⋊C8—38no—
291C4⋊C8C22×D314yes—
292C4⋊C8C22×D318yes—
293#97—38no—
294#97—38no—
295C4×C8—34no—
296C4×C8—34no—
297C4×C8—38no—
298C4×C8—34no—
299C4×C8C22×D314yes—
300C4×C8C22×D318yes—
301C23×C3C23×C411yescharacteristic
302C22×D3—22no—
303C22×D3—22no—
304C22×D3C4×Q814yes—
305C22×D3C23×C412yes—
306C3×Q8—28no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4#13114yes—
311C2×C3×C4C23×C412yes—
312C2×C3×C4C23×C414yes—
313C2×C3×C4C23×C414yes—
314C2×C3×C4C23×C411yescharacteristic
315C2×C3×C4C23×C412yes—
316C2×C3×C4C22×Q812yes—
317C2×C3×C4C2×#4112yes—
318C2×C3×C4C2511yescharacteristic
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—22no—
322C2×Dic3C4×Q814yes—
323C2×Dic3#13114yes—
324C2×Dic3C23×C412yes—
325C2×Dic3C23×C414yes—
326C2×Dic3C23×C414yes—
327C2×Dic3C23×C412yes—
328C2×Dic3C23×C412yes—
329C4×D3—28no—
330C4×D3—28no—
331C4×D3C4×Q818yes—
332C4×D3C22×Q814yes—
333C4×D3C2×#4114yes—
334Dic6—28no—
335Dic6—216no—
336C3×C8—22no—
337C3×C8—24no—
338C3×C8C22×Q814yes—
339C3⋊C8—22no—
340C3⋊C8—24no—
341C3⋊C8C22×Q814yes—
342C24—31no—
343C2×Q8—24no—
344C2×Q8—38no—
345C2×Q8—34no—
346C2×Q8—68no—
347C2×Q8—64no—
348C2×Q8C2×C4×D314yes—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—32no—
352C22×C4—34no—
353C22×C4—34no—
354C22×C4—32no—
355C22×C4—31no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—34no—
359C22×C4—34no—
360C22×C4—64no—
361C22×C4—64no—
362C22×C4C2×C4×D311yescharacteristic
363C22×C4C2×C4×D312yes—
364C22×C4C2×C4×D312yes—
365C22×C4C2×C4×D311yescharacteristic
366C22×C4C23×D311yescharacteristic
367M16—616no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—34no—
371C2×C8—32no—
372C2×C8—316no—
373C2×C8—32no—
374C2×C8—34no—
375C2×C8—68no—
376C2×C8—62no—
377C2×C8—64no—
378C2×C8D3×Q814yes—
379C2×C8C23×D312yes—
380C2×C8C23×D314yes—
381C2×C8C23×D312yes—
382C4⋊C4—38no—
383C4⋊C4—38no—
384C4⋊C4—38no—
385C4⋊C4—38no—
386C4⋊C4—38no—
387C4⋊C4—38no—
388C4⋊C4—34no—
389C4⋊C4C2×C4×D318yes—
390C4⋊C4C2×C4×D314yes—
391C22⋊C4—38no—
392C22⋊C4—38no—
393C42—32no—
394C42—34no—
395C42—34no—
396C42—34no—
397C42—32no—
398C42—34no—
399C42C2×C4×D312yes—
400C42C2×C4×D314yes—
401C42C23×D312yes—
402C42C23×D314yes—
403C22×C3C2×C4×Q812yes—
404C22×C3C2×#13112yes—
405C22×C3C2×#13112yes—
406C22×C3C24×C411yescharacteristic
407C2×D3—28no—
408C2×D3—22no—
409C2×D3—22no—
410C2×D3C2×C4×Q814yes—
411C3×C4—24no—
412C3×C4—22no—
413C3×C4—24no—
414C3×C4C2×C4×Q812yes—
415C3×C4C2×#13114yes—
416C3×C4C23×Q811yescharacteristic
417C3×C4C22×#4111yescharacteristic
418Dic3—24no—
419Dic3—24no—
420Dic3—22no—
421Dic3C2×C4×Q814yes—
422Dic3C2×#13114yes—
423C23—32no—
424C23—34no—
425C23—62no—
426C23—62no—
427C23C22×C4×D311yescharacteristic
428Q8—28no—
429Q8—616no—
430Q8—68no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—34no—
437C2×C4—34no—
438C2×C4—34no—
439C2×C4—34no—
440C2×C4—38no—
441C2×C4—32no—
442C2×C4—34no—
443C2×C4—34no—
444C2×C4—68no—
445C2×C4—62no—
446C2×C4—64no—
447C2×C4—64no—
448C2×C4—68no—
449C2×C4#90114yes—
450C2×C4C22×C4×D312yes—
451C2×C4C22×C4×D314yes—
452C2×C4C22×C4×D314yes—
453C2×C4C22×C4×D312yes—
454C2×C4C22×C4×D311yescharacteristic
455C2×C4C2×D3×Q812yes—
456C2×C4D3×#4112yes—
457C2×C4C24×D311yescharacteristic
458C8—22no—
459C8—24no—
460C8—34no—
461C8—62no—
462C8—64no—
463C8C2×D3×Q814yes—
464C2×C3#296014yescomplemented
465C2×C3C22×C4×Q811yescharacteristic
466C2×C3C22×#13111yescharacteristic
467C2×C3C22×#13111yescharacteristic
468D3—24no—
469C22—34no—
470C22—68no—
471C22—62no—
472C22—62no—
473C22C4×D3×Q812yes—
474C22C2×#90112yes—
475C22D3×#13112yes—
476C22C23×C4×D311yescharacteristic
477C4—24no—
478C4—22no—
479C4—24no—
480C4—34no—
481C4—34no—
482C4—62no—
483C4—64no—
484C4—64no—
485C4C4×D3×Q812yes—
486C4C2×#90114yes—
487C4C22×D3×Q811yescharacteristic
488C4C2×D3×#4111yescharacteristic
489C3C2×#296011yescharacteristic, complemented, Sylow
490C2—64no—
491C2#7991714yesdirect product factor
492C2C2×C4×D3×Q811yescharacteristic
493C2C22×#90111yescharacteristic
494C2C2×D3×#13111yescharacteristic
subgroupG/H
centerC22×C8C22×D3
commutatorC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC2×C3×C8×Q8C2
socleC23×C3C23×C4
prime radicalC23×D3C22×C4
cosocleC2×C3×C4C25
hypercenterC2×C8×Q8D3
hypocenterC3C2×#2960
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×Q8C2semidirect
3. C2×#5024C2semidirect
4. #79917C2direct (N × H)
7. C2×#4828C2semidirect
8. C2×#4831C2semidirect
11. C2×C8×Dic6C2semidirect
12. C2×C8×Dic6C2semidirect
30. C3×C8×Q8C22semidirect
42. #5024C22semidirect
49. #4831C22semidirect
50. #4828C22semidirect
53. C8×Dic6C22semidirect
54. C8×Dic6C22semidirect
62. C2×C8×Q8D3semidirect
103. C2×D3×C8Q8semidirect
154. C8×Q8C2×D3semidirect
234. D3×C8C2×Q8semidirect
464. C2×C3#2960semidirect
489. C3C2×#2960semidirect
491. C2#79917direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55782
C2×C3×C8×Q82
C2×#50242
#799172
C2×#48392
C2×#48372
C2×#48282
C2×#48312
C2×#46892
C2×C4×D3×C82
C2×C8×Dic62
C2×#29603