Finite Group Databasebeta

← C2×#71147 #10770939

Subgroups — C2×#71147 #10770939

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic
2C2×C3×#476C211yesmaximal, characteristic
3C2×#5091C211yesmaximal, characteristic, complemented
4C2×#5079C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic
6#71147C218yesmaximal, direct product factor
7C2×#4482C211yesmaximal, characteristic, complemented
8C2×#4484C211yesmaximal, characteristic, complemented
9C2×#1638—31nomaximal, Sylow
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#936—21no—
13C2×#926—21no—
14C2×#924—21no—
15#5542—22no—
16#5542—22no—
17#5542—22no—
18#5542C2214yes—
19C2×#887—21no—
20C2×#879—22no—
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#476—22no—
24C3×#476C2214yes—
25C2×C3×#106—21no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102—21no—
28C2×#835—22no—
29#5091—22no—
30#5091C2214yescomplemented
31#5079—22no—
32#5079C2214yescomplemented
33C2×#810—21no—
34C2×#807—21no—
35C2×C4⋊(C3⋊C8)C2211yescharacteristic
36C2×C4×C3⋊C8C2211yescharacteristic
37D4⋊(C3⋊C8)C2214yes—
38#4484C2214yescomplemented
39#4482C2214yescomplemented
40C2×#556—31no—
41C2×#487—31no—
42C2×#477—31no—
43C2×#476D311yescharacteristic
44#1638—38no—
45C2×#332—31no—
46C2×Q8⋊C8—31no—
47C2×D4⋊C8—31no—
48C22×C3×Q8—21no—
49C22×C3×D4D411yescharacteristic
50C23×C3×C4—21no—
51C22×Dic6—21no—
52C22×Dic6D411yescharacteristic
53C22×C3×C8—21no—
54C3×C4⋊Q8—21no—
55C3×C4⋊Q8—21no—
56C3×C4⋊Q8—21no—
57C3×C4⋊Q8C2312yes—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4D412yes—
61C3×C4×D4C2312yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×C3×C42C2311yescharacteristic
67C2×#217—22no—
68C2×#217—22no—
69#936—24no—
70#936—24no—
71#926—24no—
72#926—24no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3D411yescharacteristic
75C2×C4⋊Dic3D412yes—
76#924—24no—
77#924—24no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—21no—
80C2×C4×Dic3—22no—
81C2×C4×Dic3—21no—
82C22×C3⋊C8—21no—
83C22×C3⋊C8—21no—
84#887—24no—
85#887—24no—
86#879—28no—
87#879—28no—
88#869—21no—
89#869—21no—
90#869D412yes—
91#869C2312yes—
92C4×Dic6—21no—
93C4×Dic6—21no—
94C4×Dic6D412yes—
95C4×Dic6C2312yes—
96C3×#106—24no—
97C3×C4⋊C8—21no—
98C3×C4⋊C8C2312yes—
99C3×#102—24no—
100#835—28no—
101#810—24no—
102#807—24no—
103C4⋊(C3⋊C8)—21no—
104C4⋊(C3⋊C8)C2312yes—
105C4×C3⋊C8—21no—
106C4×C3⋊C8C2312yes—
107#556—34no—
108#556—62no—
109#556—62no—
110#556—62no—
111C2×C4⋊Q8—31no—
112C2×C4⋊Q8C2×D311yescharacteristic
113C2×#126—62no—
114C2×#124—61no—
115C2×C22⋊Q8—61no—
116C2×C4×Q8—31no—
117C2×C4×Q8—61no—
118C2×C4×D4C2×D311yescharacteristic
119C2×#117—61no—
120#487—34no—
121#487—62no—
122#477—34no—
123#477—62no—
124#476—22no—
125#476C2×D314yes—
126C2×#107—61no—
127C2×#106—21no—
128C2×C4⋊C8—31no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#103—61no—
131C2×#103—62no—
132C2×#102—21no—
133C2×C4×C8—31no—
134#332—34no—
135Q8⋊C8—34no—
136D4⋊C8—34no—
137C24×C3—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—44no—
140C2×C3×Q8—41no—
141C2×C3×D4—21no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4C2×D412yes—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—41no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—22no—
157C2×Dic6—22no—
158C2×Dic6—22no—
159C2×Dic6—21no—
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×Dic6—42no—
163C2×Dic6—42no—
164C2×Dic6C2×D412yes—
165C2×C3×C8—22no—
166C2×C3×C8—42no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4C2×D412yes—
173C3×C22⋊C4—22no—
174C3×C22⋊C4—22no—
175C3×C42C2×D411yescharacteristic
176C3×C42C2×D411yescharacteristic
177C3×C42C2×D411yescharacteristic
178C3×C42C2411yescharacteristic
179#217—28no—
180#217—28no—
181C4⋊Dic3—21no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3C2×D412yes—
186C4⋊Dic3C2×D414yes—
187Dic3⋊C4—28no—
188Dic3⋊C4—22no—
189Dic3⋊C4—22no—
190C4×Dic3—28no—
191C4×Dic3—22no—
192C4×Dic3—22no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—42no—
196C2×C3⋊C8—42no—
197C22×Q8—21no—
198C22×Q8—31no—
199C22×Q8—61no—
200C22×D4#6711yescharacteristic
201C23×C4—21no—
202C22×C8—21no—
203C22×C8—61no—
204C22×C8—61no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—32no—
209C4⋊Q8—32no—
210C4⋊Q8—61no—
211C4⋊Q8—61no—
212C4⋊Q8C22×D312yes—
213#126—68no—
214#126—68no—
215#124—64no—
216#124—64no—
217C22⋊Q8—64no—
218C22⋊Q8—64no—
219C4×Q8—32no—
220C4×Q8—32no—
221C4×Q8—64no—
222C4×Q8—64no—
223C4×Q8—61no—
224C4×Q8—61no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4#6712yes—
228C4×D4C22×D312yes—
229#117—64no—
230#117—64no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4#6711yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C42—62no—
243C2×C42—61no—
244C2×C42C22×D311yescharacteristic
245#107—64no—
246#106—24no—
247C4⋊C8—21no—
248C4⋊C8—32no—
249C4⋊C8—61no—
250C4⋊C8C22×D312yes—
251#103—64no—
252#103—68no—
253#102—24no—
254C4×C8—32no—
255C4×C8—61no—
256C23×C3—22no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3#12011yescharacteristic
261C3×Q8—44no—
262C3×Q8—82no—
263C3×D4—22no—
264C3×D4—22no—
265C3×D4—42no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—44no—
272C2×C3×C4—42no—
273C2×C3×C4—82no—
274C2×C3×C4#12011yescharacteristic
275C2×C3×C4#12011yescharacteristic
276C2×C3×C4#12011yescharacteristic
277C2×C3×C4#13512yes—
278C2×C3×C4#13512yes—
279C2×C3×C4#13712yes—
280C2×C3×C4#13712yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C22×D411yescharacteristic
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—24no—
287C2×Dic3—42no—
288C2×Dic3—44no—
289C2×Dic3—42no—
290Dic6—22no—
291Dic6—22no—
292Dic6—42no—
293Dic6—42no—
294Dic6—42no—
295Dic6—44no—
296C3×C8—42no—
297C3⋊C8—42no—
298C3⋊C8—42no—
299C24—21no—
300C2×Q8—22no—
301C2×Q8—32no—
302C2×Q8—44no—
303C2×Q8—41no—
304C2×Q8—61no—
305C2×Q8—62no—
306C2×Q8—62no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—122no—
310C2×Q8—122no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4C2×#6712yes—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—61no—
322C22×C4—62no—
323C22×C4—61no—
324C22×C4D3×D411yescharacteristic
325C22×C4D3×D411yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C2×C8—22no—
328C2×C8—42no—
329C2×C8—62no—
330C2×C8—62no—
331C2×C8—122no—
332C2×C8—122no—
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—34no—
337C4⋊C4—32no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—68no—
341C4⋊C4—61no—
342C4⋊C4—62no—
343C4⋊C4—62no—
344C4⋊C4—62no—
345C4⋊C4—62no—
346C4⋊C4—62no—
347C4⋊C4C2×#6712yes—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—68no—
351C22⋊C4—68no—
352C42—68no—
353C42—62no—
354C42—62no—
355C42D3×D411yescharacteristic
356C42D3×D411yescharacteristic
357C42C2×#6711yescharacteristic
358C42C23×D311yescharacteristic
359C22×C3—42no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—84no—
363C22×C3#45212yes—
364C22×C3#45212yes—
365C22×C3#45412yes—
366C22×C3C2×#12011yescharacteristic
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—42no—
370C3×C4—82no—
371C3×C4#45212yes—
372C3×C4#45212yes—
373C3×C4C2×#13511yescharacteristic
374C3×C4C2×#13511yescharacteristic
375C3×C4C2×#13711yescharacteristic
376C3×C4C2×#13711yescharacteristic
377Dic3—42no—
378Dic3—44no—
379Dic3—42no—
380C23—22no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23#93711yescharacteristic
385Q8—44no—
386Q8—62no—
387Q8—62no—
388Q8—82no—
389Q8—122no—
390Q8—122no—
391Q8—124no—
392Q8—122no—
393D4—22no—
394D4—22no—
395D4—42no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—62no—
404C2×C4—64no—
405C2×C4—62no—
406C2×C4—82no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—124no—
410C2×C4#91512yes—
411C2×C4#91612yes—
412C2×C4#93711yescharacteristic
413C2×C4#93711yescharacteristic
414C2×C4#93711yescharacteristic
415C2×C4#94212yes—
416C2×C4#95012yes—
417C2×C4C2×D3×D411yescharacteristic
418C2×C4C2×D3×D411yescharacteristic
419C2×C4C22×#6711yescharacteristic
420C8—42no—
421C8—122no—
422C8—122no—
423C2×C3—82no—
424C2×C3#163814yescomplemented
425C2×C3C2×#45211yescharacteristic
426C2×C3C2×#45211yescharacteristic
427C2×C3C2×#45411yescharacteristic
428C22—42no—
429C22—42no—
430C22—42no—
431C22—84no—
432C22#504712yes—
433C22#517312yes—
434C22#520112yes—
435C22C2×#93711yescharacteristic
436C4—21no—
437C4—21no—
438C4—42no—
439C4—82no—
440C4—124no—
441C4—122no—
442C4—122no—
443C4#504712yes—
444C4#517312yes—
445C4C2×#91511yescharacteristic
446C4C2×#91611yescharacteristic
447C4C2×#94211yescharacteristic
448C4C2×#95011yescharacteristic
449C3C2×#163811yescharacteristic, complemented, Sylow
450C2—82no—
451C2#7114714yesdirect product factor
452C2C2×#504711yescharacteristic
453C2C2×#517311yescharacteristic
454C2C2×#520111yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC2×C3×#476C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenterC2×#476D3
hypocenterC3C2×#1638
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
3. C2×#5091C2semidirect
4. C2×#5079C2semidirect
6. #71147C2direct (N × H)
7. C2×#4482C2semidirect
8. C2×#4484C2semidirect
30. #5091C22semidirect
32. #5079C22semidirect
38. #4484C22semidirect
39. #4482C22semidirect
424. C2×C3#1638semidirect
449. C3C2×#1638semidirect
451. C2#71147direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55422
C2×C3×#4762
C2×#50912
C2×#50792
C2×D4⋊(C3⋊C8)2
#711472
C2×#44822
C2×#44842
C2×#16383