Finite Group Databasebeta

← C2×#7189 #126913

Subgroups — C2×#7189 #126913

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#324C212yesmaximal
2C2×C4⋊#95C211yesmaximal, characteristic
3#7189C214yesmaximal, direct product factor
4C23×C4⋊C4C411yescharacteristic
5C23×C4⋊C4C2211yescharacteristic
6C23×C42C411yescharacteristic
7C22×C22⋊C8—22no—
8C22×#95—22no—
9C2×#324—22no—
10C2×#324—24no—
11C2×#324C418yescomplemented
12C2×#324C2214yes—
13C4⋊#95—22no—
14C4⋊#95C412yes—
15C4⋊#95C2212yes—
16C24×C4—21no—
17C24×C4—21no—
18C24×C4C2×C411yescharacteristic
19C24×C4D412yes—
20C23×C8—42no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4—41no—
24C22×C4⋊C4—42no—
25C22×C4⋊C4—41no—
26C22×C4⋊C4—41no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4—42no—
29C22×C4⋊C4C814yescomplemented
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C414yes—
33C22×C4⋊C4D412yes—
34C22×C4⋊C4D412yes—
35C22×C4⋊C4Q812yes—
36C22×C4⋊C4C2311yescharacteristic
37C22×C42—21no—
38C22×C42—22no—
39C22×C42—42no—
40C22×C42—42no—
41C22×C42C814yescomplemented
42C22×C42C2×C412yes—
43C2×C22⋊C8—28no—
44C2×C22⋊C8—24no—
45C2×C22⋊C8—42no—
46C2×C22⋊C8—44no—
47C2×#95—24no—
48C2×#95—24no—
49C2×#95—48no—
50#324—28no—
51#324—24no—
52#324C2×C418yescomplemented
53C25C22⋊C411yescharacteristic
54C23×C4—21no—
55C23×C4—22no—
56C23×C4—21no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—24no—
60C23×C4—24no—
61C23×C4—24no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—24no—
65C23×C4—22no—
66C23×C4—24no—
67C23×C4—22no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4—44no—
73C23×C4—44no—
74C23×C4—44no—
75C23×C4—44no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4—42no—
79C23×C4C4212yes—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C412yes—
82C23×C4C22⋊C414yes—
83C23×C4C4⋊C412yes—
84C23×C4C4⋊C414yes—
85C23×C4C2×C812yes—
86C23×C4C2×C812yes—
87C23×C4M1612yes—
88C23×C4M1612yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D412yes—
91C22×C8—48no—
92C22×C8—44no—
93C22×C8—84no—
94C22×C8—84no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C4212yes—
118C2×C4⋊C4C22⋊C414yes—
119C2×C4⋊C4C4⋊C414yes—
120C2×C4⋊C4C4⋊C412yes—
121C2×C4⋊C4C2×C814yescomplemented
122C2×C4⋊C4C22×C412yes—
123C2×C4⋊C4C2×D411yescharacteristic
124C2×C4⋊C4C2×Q811yescharacteristic
125C2×C42—24no—
126C2×C42—22no—
127C2×C42—48no—
128C2×C42—42no—
129C2×C42—42no—
130C2×C42—48no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42C2×C814yescomplemented
134C22⋊C8—28no—
135C22⋊C8—48no—
136C22⋊C8—44no—
137#95—48no—
138#95—48no—
139C24—22no—
140C24—22no—
141C24—22no—
142C24—22no—
143C24#9512yes—
144C24C22⋊C812yes—
145C24C22⋊C812yes—
146C24#9911yescharacteristic
147C24#9912yes—
148C24#9911yescharacteristic
149C24#10012yes—
150C24#10112yes—
151C24C2×C22⋊C411yescharacteristic
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—216no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4—48no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—416no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—88no—
219C22×C4—88no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4#9512yes—
225C22×C4#9514yes—
226C22×C4C4×C814yes—
227C22×C4#9714yes—
228C22×C4C22⋊C812yes—
229C22×C4C22⋊C812yes—
230C22×C4C4⋊C814yes—
231C22×C4C4⋊C814yes—
232C22×C4C2×C4211yescharacteristic
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C4⋊C411yescharacteristic
235C22×C4C2×C4⋊C412yes—
236C22×C4C22×C811yescharacteristic
237C22×C4C22×C811yescharacteristic
238C22×C4C2×M1611yescharacteristic
239C22×C4C2×M1611yescharacteristic
240C2×C8—48no—
241C2×C8—84no—
242C2×C8—84no—
243C2×C8—816no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—44no—
252C4⋊C4—48no—
253C4⋊C4—44no—
254C42—44no—
255C42—48no—
256C42—44no—
257C23—24no—
258C23—24no—
259C23—24no—
260C23—24no—
261C23—24no—
262C23—24no—
263C23—24no—
264C23—24no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—24no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23#32314yes—
278C23#32312yes—
279C23#32312yes—
280C23#32414yes—
281C23#32412yes—
282C23#32412yes—
283C23#33614yes—
284C23#34212yes—
285C23#34212yes—
286C23#34314yes—
287C23#35312yes—
288C23#35412yes—
289C23#35512yes—
290C23#35612yes—
291C23C2×#9511yescharacteristic
292C23C2×C22⋊C811yescharacteristic
293C23C2×C22⋊C811yescharacteristic
294C23C2×#9911yescharacteristic
295C23C2×#9911yescharacteristic
296C23C2×#10011yescharacteristic
297C23C2×#10111yescharacteristic
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—28no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—432no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—82no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—88no—
352C2×C4—88no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4#33614yes—
356C2×C4C2×C4×C811yescharacteristic
357C2×C4C2×#9711yescharacteristic
358C2×C4C2×C4⋊C811yescharacteristic
359C2×C4C2×C4⋊C811yescharacteristic
360C8—88no—
361C22—28no—
362C22—28no—
363C22—28no—
364C22—28no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—21no—
369C22—22no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—21no—
383C22#128014yes—
384C22#128014yes—
385C22#131814yes—
386C22#131914yes—
387C22#132414yes—
388C22#132514yes—
389C22#139314yes—
390C22#139414yes—
391C22C2×#32311yescharacteristic
392C22C2×#32311yescharacteristic
393C22C2×#32411yescharacteristic
394C22C2×#32411yescharacteristic
395C22C2×#33611yescharacteristic
396C22C2×#34211yescharacteristic
397C22C2×#34311yescharacteristic
398C22C2×#35311yescharacteristic
399C22C2×#35411yescharacteristic
400C22C2×#35511yescharacteristic
401C22C2×#35611yescharacteristic
402C4—24no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—48no—
408C4—42no—
409C4—44no—
410C4—42no—
411C4—84no—
412C2—21no—
413C2—24no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2#718918yesdirect product factor
418C2C2×#128011yescharacteristic
419C2C2×#131811yescharacteristic
420C2C2×#131911yescharacteristic
421C2C2×#132411yescharacteristic
422C2C2×#132511yescharacteristic
423C2C2×#139311yescharacteristic
424C2C2×#139411yescharacteristic
subgroupG/H
center, socleC24#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7189C2direct (N × H)
11. C2×#324C4semidirect
29. C22×C4⋊C4C8semidirect
41. C22×C42C8semidirect
52. #324C2×C4semidirect
121. C2×C4⋊C4C2×C8semidirect
133. C2×C42C2×C8semidirect
417. C2#7189direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3242
C2×C4⋊#952
#71892