Finite Group Databasebeta

← C2×#7799 #165705

Subgroups — C2×#7799 #165705

Order 512.

1039 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2727C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#2083C211yesmaximal, characteristic, complemented
4C2×#2071C211yesmaximal, characteristic
5#7799C218yesmaximal, direct product factor
6C2×#1271C211yesmaximal, characteristic, complemented
7C2×#1271C211yesmaximal, characteristic, complemented
8C2×#1270C211yesmaximal, characteristic
9C22×#126—21no—
10C22×#107—21no—
11C22×#102—21no—
12#2727—22no—
13#2727—22no—
14#2727—22no—
15#2727C2214yes—
16C2×#399—21no—
17C2×#396—21no—
18C2×#394—21no—
19C2×#390C2211yescharacteristic
20C2×#389—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388—21no—
24C2×#384C2211yescharacteristic
25#2099—22no—
26#2099C2214yescomplemented
27#2083—22no—
28#2083C2214yescomplemented
29#2071—22no—
30#2071C2214yes—
31C2×#340—21no—
32C2×#340—21no—
33C2×#340—21no—
34C2×#340—21no—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1271C2214yescomplemented
39#1271C2214yescomplemented
40#1270C2214yes—
41C23×D4D411yescharacteristic
42C23×C8—21no—
43C23×C8—21no—
44C23×C8—21no—
45C2×#126—24no—
46C2×#126—24no—
47C2×#126—44no—
48C2×#126—44no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—44no—
71C2×#102—24no—
72C2×#102—44no—
73#399—24no—
74#399—24no—
75#396—24no—
76#396—24no—
77#394—24no—
78#394—24no—
79#390—21no—
80#390—21no—
81#390D412yes—
82#390C2312yes—
83#389—24no—
84#389—24no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389D412yes—
90#389D412yes—
91#389C2312yes—
92#389C2312yes—
93#388—24no—
94#388—24no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384C2312yes—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336C2312yes—
110#336C2312yes—
111#336C2312yes—
112C25—21no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—42no—
117C22×D4—42no—
118C22×D4C2×D412yes—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—41no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C22×C8—24no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—42no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140#126—48no—
141#126—416no—
142#126—48no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4D814yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C42—24no—
200C2×C42—42no—
201C2×C42—44no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2411yescharacteristic
206#107—48no—
207#102—48no—
208#95—24no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213C24—22no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24#12011yescharacteristic
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—48no—
227C2×D4—42no—
228C2×D4—81no—
229C2×D4—81no—
230C2×D4—84no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4C2×D812yes—
265C22×C4#13512yes—
266C22×C4#13512yes—
267C22×C4AGL(1,ℤ/8)12yes—
268C22×C4AGL(1,ℤ/8)12yes—
269C22×C4#13712yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—24no—
274C2×C8—48no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—84no—
284C2×C8—84no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—48no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C4⋊C4—84no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4C2×D814yes—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—48no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—84no—
336C23—84no—
337C23—84no—
338C23—84no—
339C23—84no—
340C23—84no—
341C23—84no—
342C23#44712yes—
343C23#44912yes—
344C23#45212yes—
345C23#45412yes—
346C23#45512yes—
347C23#45612yes—
348C23#45812yes—
349C23C2×#12011yescharacteristic
350D4—44no—
351D4—82no—
352D4—84no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—48no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—164no—
395C2×C4#44712yes—
396C2×C4#44912yes—
397C2×C4#45212yes—
398C2×C4C22×D811yescharacteristic
399C2×C4C2×#13511yescharacteristic
400C2×C4C2×#13511yescharacteristic
401C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
402C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C8—44no—
405C8—82no—
406C8—82no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—168no—
415C22#161014yes—
416C22#161214yes—
417C22#161514yes—
418C22#162514yes—
419C22#164114yes—
420C22#165014yes—
421C22#166414yes—
422C22C2×#44711yescharacteristic
423C22C2×#44911yescharacteristic
424C22C2×#45211yescharacteristic
425C22C2×#45511yescharacteristic
426C22C2×#45411yescharacteristic
427C22C2×#45611yescharacteristic
428C22C2×#45811yescharacteristic
429C4—21no—
430C4—21no—
431C4—22no—
432C4—21no—
433C4—21no—
434C4—22no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—44no—
439C4—82no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—162no—
444C2—162no—
445C2#779918yesdirect product factor
446C2C2×#161511yescharacteristic
447C2C2×#161011yescharacteristic
448C2C2×#161211yescharacteristic
449C2C2×#162511yescharacteristic
450C2C2×#165011yescharacteristic
451C2C2×#164111yescharacteristic
452C2C2×#166411yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2099C2semidirect
3. C2×#2083C2semidirect
5. #7799C2direct (N × H)
6. C2×#1271C2semidirect
7. C2×#1271C2semidirect
26. #2099C22semidirect
28. #2083C22semidirect
38. #1271C22semidirect
39. #1271C22semidirect
445. C2#7799direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#27272
C2×#20992
C2×#20832
C2×#20712
#77992
C2×#12712
C2×#12702