Finite Group Databasebeta

← C2×#7718 #161681

Subgroups — C2×#7718 #161681

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic, complemented
3C2×#1817C211yesmaximal, characteristic, complemented
4#7718C218yesmaximal, direct product factor
5C2×#1271C212yesmaximal, complemented
6C2×#1270C212yesmaximal, complemented
7C22×#123—21no—
8C22×#121C411yescharacteristic
9C23×C42C2211yescharacteristic
10C22×C22⋊C8—21no—
11C22×C22⋊C8—21no—
12#2854—22no—
13#2854—22no—
14#2854C414yes—
15#2854C2214yes—
16C2×#390C411yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#389C411yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#388—21no—
21C2×#388—21no—
22C2×#386—21no—
23C2×#380C411yescharacteristic
24#1817—22no—
25#1817—22no—
26#1817C2214yescomplemented
27#1817C2214yescomplemented
28C2×#336—21no—
29C2×#336—21no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1271C2218yescomplemented
33#1270C2218yescomplemented
34C23×D4—21no—
35C23×D4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4D411yescharacteristic
39C23×C8—22no—
40C23×C8—22no—
41C2×#123—24no—
42C2×#123—24no—
43C2×#123—48no—
44C2×#121—22no—
45C2×#121—44no—
46C2×#121C2×C414yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C42—44no—
59C22×C42C2×C412yes—
60C22×C42C2×C411yescharacteristic
61C22×C42D412yes—
62C22×C42D412yes—
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—44no—
70C2×C22⋊C8—44no—
71#390—22no—
72#390—21no—
73#390—21no—
74#390C2×C412yes—
75#390C2×C414yes—
76#390C2312yes—
77#389—22no—
78#389—21no—
79#389—21no—
80#389C2×C412yes—
81#389C2×C414yes—
82#389C2312yes—
83#388—24no—
84#388—24no—
85#388—24no—
86#388—24no—
87#386—24no—
88#386—24no—
89#380—22no—
90#380C2×C414yes—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94#336—22no—
95#336—22no—
96#336—24no—
97#336—24no—
98#336C2314yes—
99#336C2314yes—
100C25—21no—
101C25C22⋊C411yescharacteristic
102C22×D4—24no—
103C22×D4—21no—
104C22×D4—24no—
105C22×D4—42no—
106C22×D4—42no—
107C22×D4—44no—
108C22×D4—44no—
109C22×D4—44no—
110C22×D4C22×C412yes—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C22×C8—24no—
136C22×C8—24no—
137C22×C8—44no—
138C22×C8—44no—
139C22×C8—44no—
140C22×C8—44no—
141C22×C8—44no—
142C22×C8—44no—
143#123—416no—
144#123—416no—
145#121—48no—
146#121—48no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C42—42no—
176C2×C42—44no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—44no—
180C2×C42—44no—
181C2×C42—42no—
182C2×C42—44no—
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22⋊C412yes—
186C2×C42C22×C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2411yescharacteristic
193C22⋊C8—48no—
194C22⋊C8—48no—
195#95—24no—
196#95—24no—
197#95—22no—
198#95—22no—
199C24—22no—
200C24—22no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24#9914yes—
214C24C2×C22⋊C412yes—
215C24C2×C22⋊C411yescharacteristic
216C2×D4—24no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—48no—
222C2×D4—48no—
223C2×D4—48no—
224C2×D4—42no—
225C2×D4—88no—
226C2×D4—816no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—416no—
264C22×C4—416no—
265C22×C4—416no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4#13114yes—
282C22×C4#13514yes—
283C22×C4AGL(1,ℤ/8)14yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—88no—
294C2×C8—88no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—48no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C22⋊C4—48no—
305C22⋊C4—48no—
306C22⋊C4—48no—
307C22⋊C4—48no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—41no—
316C42—42no—
317C42—41no—
318C42—41no—
319C42—42no—
320C42—42no—
321C42—41no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—28no—
329C23—28no—
330C23—28no—
331C23—44no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—84no—
343C23—84no—
344C23—84no—
345C23—84no—
346C23—84no—
347C23—84no—
348C23—84no—
349C23#40812yes—
350C23C2×#9914yes—
351C23C2×#9912yes—
352C23#41312yes—
353C23#41612yes—
354C23#41812yes—
355C23#42112yes—
356C23#42112yes—
357C23C22×C22⋊C411yescharacteristic
358D4—84no—
359D4—84no—
360D4—816no—
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—21no—
372C2×C4—21no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4#40812yes—
407C2×C4#41612yes—
408C2×C4#41812yes—
409C2×C4C2×#13112yes—
410C2×C4C2×#13512yes—
411C2×C4C2×AGL(1,ℤ/8)12yes—
412C8—84no—
413C8—84no—
414C22—24no—
415C22—24no—
416C22—24no—
417C22—416no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—168no—
430C22#146814yes—
431C22#148214yes—
432C22#149614yes—
433C22#151014yes—
434C22#151614yes—
435C22#153514yes—
436C22#155314yes—
437C22C2×#40811yescharacteristic
438C22C22×#9911yescharacteristic
439C22C2×#41311yescharacteristic
440C22C2×#41611yescharacteristic
441C22C2×#41811yescharacteristic
442C22C2×#42111yescharacteristic
443C22C2×#42111yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—82no—
454C4—82no—
455C4—82no—
456C2—44no—
457C2—162no—
458C2#771818yesdirect product factor
459C2C2×#146811yescharacteristic
460C2C2×#148211yescharacteristic
461C2C2×#149611yescharacteristic
462C2C2×#151011yescharacteristic
463C2C2×#151611yescharacteristic
464C2C2×#153511yescharacteristic
465C2C2×#155311yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1817C2semidirect
3. C2×#1817C2semidirect
4. #7718C2direct (N × H)
5. C2×#1271C2semidirect
6. C2×#1270C2semidirect
26. #1817C22semidirect
27. #1817C22semidirect
32. #1271C22semidirect
33. #1270C22semidirect
458. C2#7718direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#18172
#77182
C2×#12712
C2×#12702