Finite Group Databasebeta

← C2×#9192 #197420

Subgroups — C2×#9192 #197420

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2921C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic
3C2×#1548C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9192C218yesmaximal, direct product factor
6C2×#1286C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C22×#117C2211yescharacteristic
9#2921—24no—
10#2921—22no—
11#2921C2214yes—
12C2×#433—24no—
13C2×#431C2212yes—
14C2×#423C2212yes—
15C2×#407—21no—
16C2×C4×M16—21no—
17#2628—22no—
18#2628—22no—
19#2628C414yes—
20#2628C2214yes—
21C2×#400—21no—
22C2×#399—21no—
23C2×#395C412yes—
24C2×#393C412yes—
25C2×#389—21no—
26C2×#382C2211yescharacteristic
27C2×#380—21no—
28#1548—24no—
29#1548C2218yescomplemented
30C2×#333—24no—
31#1458—22no—
32#1458C2214yes—
33C2×#324—21no—
34C2×#323—21no—
35#1286C2218yescomplemented
36C23×Q8C2×C412yes—
37C24×C4D411yescharacteristic
38C22×M16—22no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2312yes—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117C2×C412yes—
52C2×#117D412yes—
53C2×#117D412yes—
54C2×#117C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66#433—216no—
67#433—216no—
68#431—22no—
69#431—22no—
70#431—22no—
71#431C2314yes—
72#423—22no—
73#423—22no—
74#423D414yes—
75#423C2314yes—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8D414yes—
79#407—24no—
80#407—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—22no—
83C4×M16—24no—
84C4×M16—24no—
85C2×#97—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88#400—24no—
89#400—24no—
90#399—24no—
91#399—24no—
92#395—24no—
93#395C2×C418yes—
94#393—24no—
95#393C2×C418yes—
96#389—24no—
97#389—24no—
98#382—21no—
99#382—21no—
100#382C2×C412yes—
101#382C2312yes—
102#380—24no—
103#380—24no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95C2×C412yes—
108#333—216no—
109#324—24no—
110#323—24no—
111C25C22⋊C411yescharacteristic
112C22×Q8—22no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—84no—
116C22×Q8C22×C414yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—44no—
136C2×M16—44no—
137C22×C8—24no—
138C22×C8—42no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147#117—24no—
148#117—24no—
149#117—24no—
150#117—24no—
151#117—24no—
152#117—24no—
153#117—24no—
154#117—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—84no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—84no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22×C412yes—
187C2×C22⋊C4C2×D412yes—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42C22⋊C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42C2411yescharacteristic
202C4⋊C8—24no—
203C4⋊C8—24no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—42no—
207C4⋊C8C2×D418yes—
208C22⋊C8—216no—
209C22⋊C8—24no—
210C22⋊C8—42no—
211#97—28no—
212C4×C8—28no—
213C4×C8—24no—
214C4×C8—42no—
215#95—28no—
216#95—22no—
217#95—28no—
218#95—22no—
219#95—22no—
220#95C22×C414yes—
221C24—21no—
222C24—22no—
223C24—21no—
224C24—22no—
225C24—42no—
226C24—42no—
227C24C2×C22⋊C411yescharacteristic
228C24C2×C22⋊C412yes—
229C2×Q8—28no—
230C2×Q8—44no—
231C2×Q8—88no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×Q8—88no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C4×D412yes—
275C22×C4C4×D412yes—
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12112yes—
279C22×C4#12112yes—
280C22×C4#12311yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4#13514yes—
283C22×C4#13714yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—44no—
288M16—44no—
289M16—48no—
290C2×C8—28no—
291C2×C8—48no—
292C2×C8—44no—
293C2×C8—84no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—88no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—88no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42C4×D412yes—
317C42C4×D412yes—
318C42#12012yes—
319C42#12112yes—
320C42#12112yes—
321C42#12312yes—
322C42C22×D412yes—
323C42C2×#4112yes—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—48no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—81no—
338C23—84no—
339C23#38611yescharacteristic
340C23#38611yescharacteristic
341C23#41612yes—
342C23#41912yes—
343C23C22×C22⋊C411yescharacteristic
344Q8—88no—
345Q8—88no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—88no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4#38612yes—
381C2×C4#38612yes—
382C2×C4#38611yescharacteristic
383C2×C4#38611yescharacteristic
384C2×C4#41612yes—
385C2×C4#41912yes—
386C2×C4#45214yes—
387C2×C4#45214yes—
388C2×C4#48514yes—
389C2×C4#48514yes—
390C2×C4C2×C4×D412yes—
391C2×C4C2×#12011yescharacteristic
392C2×C4C2×#12112yes—
393C2×C4C2×#12311yescharacteristic
394C2×C4C2×#13512yes—
395C2×C4C2×#13712yes—
396C8—48no—
397C8—84no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—24no—
403C22—24no—
404C22—44no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—82no—
409C22—84no—
410C22—84no—
411C22#187312yes—
412C22#187312yes—
413C22#190912yes—
414C22#191412yes—
415C22C2×#38611yescharacteristic
416C22C2×#41611yescharacteristic
417C22C2×#41911yescharacteristic
418C4—22no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—84no—
426C4—84no—
427C4#187312yes—
428C4#187312yes—
429C4C2×#45212yes—
430C4C2×#48512yes—
431C2—21no—
432C2—21no—
433C2—22no—
434C2—82no—
435C2#919214yesdirect product factor
436C2C2×#187311yescharacteristic
437C2C2×#190911yescharacteristic
438C2C2×#191411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1548C2semidirect
5. #9192C2direct (N × H)
6. C2×#1286C2semidirect
29. #1548C22semidirect
35. #1286C22semidirect
435. C2#9192direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29212
C2×#26282
C2×#15482
C2×#14582
#91922
C2×#12862