Finite Group Databasebeta

← C2×#7732 #161705

Subgroups — C2×#7732 #161705

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2C2×#1831C211yesmaximal, characteristic, complemented
3C2×#1831C211yesmaximal, characteristic, complemented
4#7732C218yesmaximal, direct product factor
5C2×#1271C212yesmaximal, complemented
6C2×#1270C212yesmaximal
7C22×#126—21no—
8C22×C4⋊C8—21no—
9C22×C4⋊C8—21no—
10#2860—22no—
11#2860—22no—
12#2860C414yes—
13#2860C2214yes—
14C2×#390C411yescharacteristic
15C2×#390C411yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#389C411yescharacteristic
18C2×#389C411yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#388—21no—
21C2×#388—21no—
22C2×#388—21no—
23C2×C43C2211yescharacteristic
24#1831—22no—
25#1831—22no—
26#1831C2214yescomplemented
27#1831C2214yescomplemented
28C2×#336—21no—
29C2×#336—21no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1271C2218yescomplemented
33#1270C2218yes—
34C23×D4C2×C411yescharacteristic
35C23×C8—22no—
36C23×C8—22no—
37C2×#126—24no—
38C2×#126—24no—
39C2×#126—44no—
40C2×#126—44no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—24no—
61C2×C4⋊C8—24no—
62C2×C4⋊C8—44no—
63C2×C4⋊C8—44no—
64#390—22no—
65#390—22no—
66#390—21no—
67#390—21no—
68#390C2×C412yes—
69#390C2×C414yes—
70#390C2×C414yes—
71#390C2312yes—
72#389—22no—
73#389—22no—
74#389—21no—
75#389—21no—
76#389C2×C412yes—
77#389C2×C414yes—
78#389C2×C414yes—
79#389C2312yes—
80#388—24no—
81#388—24no—
82#388—24no—
83#388—24no—
84#388—24no—
85#388—24no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C43C2×C412yes—
90C43D412yes—
91C43D412yes—
92C43C2312yes—
93#336—22no—
94#336—22no—
95#336—24no—
96#336—24no—
97#336C2314yes—
98#336C2314yes—
99C25—21no—
100C22×D4—21no—
101C22×D4—24no—
102C22×D4—42no—
103C22×D4—42no—
104C22×D4C22×C412yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C22×C8—24no—
116C22×C8—24no—
117C22×C8—44no—
118C22×C8—44no—
119C22×C8—44no—
120C22×C8—44no—
121C22×C8—44no—
122C22×C8—44no—
123#126—416no—
124#126—48no—
125#126—48no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C22×C412yes—
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C22×C412yes—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C42—24no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42C22⋊C412yes—
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C412yes—
169C2×C42C22⋊C412yes—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42D814yes—
173C2×C42SD1614yes—
174C2×C42C22×C412yes—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2411yescharacteristic
184C4⋊C8—48no—
185C4⋊C8—48no—
186#95—24no—
187#95—24no—
188#95—22no—
189#95—22no—
190C24—22no—
191C24—42no—
192C24—42no—
193C24—42no—
194C24—42no—
195C24—42no—
196C24—42no—
197C24—42no—
198C24C2×C22⋊C411yescharacteristic
199C2×D4—24no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—88no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4#10214yes—
235C22×C4#10214yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4#13114yes—
244C22×C4C2×D812yes—
245C22×C4C2×SD1612yes—
246C22×C4AGL(1,ℤ/8)12yes—
247C22×C4#13712yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—88no—
258C2×C8—88no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—48no—
269C4⋊C4—48no—
270C22⋊C4—48no—
271C22⋊C4—48no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—44no—
277C22⋊C4—48no—
278C22⋊C4—48no—
279C22⋊C4—44no—
280C42—24no—
281C42—22no—
282C42—22no—
283C42—24no—
284C42—24no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42—41no—
289C42—42no—
290C42—41no—
291C42—41no—
292C42—42no—
293C42—42no—
294C42—41no—
295C42#10218yes—
296C42C2×D814yes—
297C42C2×SD1614yes—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—84no—
306C23—84no—
307C23—84no—
308C23—84no—
309C23—84no—
310C23—84no—
311C23—84no—
312C23#40812yes—
313C23#41012yes—
314C23#41312yes—
315C23C2×#10212yes—
316C23#41712yes—
317C23#42112yes—
318C23#42112yes—
319C23C22×C22⋊C411yescharacteristic
320D4—84no—
321D4—84no—
322C2×C4—28no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4#40812yes—
366C2×C4C2×#10212yes—
367C2×C4C2×#10214yes—
368C2×C4C2×#10214yes—
369C2×C4#41712yes—
370C2×C4C2×#13112yes—
371C2×C4C22×D811yescharacteristic
372C2×C4C22×SD1611yescharacteristic
373C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
374C2×C4C2×#13711yescharacteristic
375C8—84no—
376C8—84no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—168no—
385C22#147314yes—
386C22#148014yes—
387C22#149514yes—
388C22#151214yes—
389C22#151714yes—
390C22#153514yes—
391C22#154714yes—
392C22C2×#40811yescharacteristic
393C22C2×#41011yescharacteristic
394C22C2×#41311yescharacteristic
395C22C22×#10211yescharacteristic
396C22C2×#41711yescharacteristic
397C22C2×#42111yescharacteristic
398C22C2×#42111yescharacteristic
399C4—24no—
400C4—24no—
401C4—21no—
402C4—21no—
403C4—22no—
404C4—21no—
405C4—21no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—82no—
412C4—82no—
413C4—82no—
414C2—162no—
415C2#773218yesdirect product factor
416C2C2×#147311yescharacteristic
417C2C2×#148011yescharacteristic
418C2C2×#149511yescharacteristic
419C2C2×#151211yescharacteristic
420C2C2×#151711yescharacteristic
421C2C2×#153511yescharacteristic
422C2C2×#154711yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1831C2semidirect
3. C2×#1831C2semidirect
4. #7732C2direct (N × H)
5. C2×#1271C2semidirect
26. #1831C22semidirect
27. #1831C22semidirect
32. #1271C22semidirect
415. C2#7732direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28602
C2×#18312
#77322
C2×#12712
C2×#12702