Finite Group Databasebeta

← C2×#7733 #161706

Subgroups — C2×#7733 #161706

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic
2C2×#1831C211yesmaximal, characteristic
3C2×#1831C211yesmaximal, characteristic
4#7733C218yesmaximal, direct product factor
5C2×#1272C212yesmaximal
6C2×#1271C212yesmaximal
7C22×C4⋊C8—21no—
8C22×C4⋊C8—21no—
9#2863—22no—
10#2863—22no—
11#2863C414yes—
12#2863C2214yes—
13C2×#391C411yescharacteristic
14C2×#391C411yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#389C411yescharacteristic
17C2×#389C411yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#387—21no—
20C2×#387—21no—
21C2×#387—21no—
22C2×#383—21no—
23C2×C43C2211yescharacteristic
24#1831—22no—
25#1831—22no—
26#1831C2214yes—
27#1831C2214yes—
28C2×#336—21no—
29C2×#336—21no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1272C2218yes—
33#1271C2218yes—
34C23×Q8C2×C411yescharacteristic
35C23×C8—22no—
36C23×C8—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42C2311yescharacteristic
50C2×C4⋊C8—24no—
51C2×C4⋊C8—24no—
52C2×C4⋊C8—44no—
53C2×C4⋊C8—44no—
54#391—22no—
55#391—22no—
56#391—21no—
57#391—21no—
58#391C2×C412yes—
59#391C2×C414yes—
60#391C2×C414yes—
61#391C2312yes—
62#389—22no—
63#389—22no—
64#389—21no—
65#389—21no—
66#389C2×C412yes—
67#389C2×C414yes—
68#389C2×C414yes—
69#389C2312yes—
70#387—24no—
71#387—24no—
72#387—24no—
73#387—24no—
74#387—24no—
75#387—24no—
76#383—24no—
77#383—24no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C43C2×C412yes—
88C43D412yes—
89C43D412yes—
90C43C2312yes—
91#336—22no—
92#336—22no—
93#336—24no—
94#336—24no—
95#336C2314yes—
96#336C2314yes—
97C22×Q8—21no—
98C22×Q8—24no—
99C22×Q8—42no—
100C22×Q8—42no—
101C22×Q8C22×C412yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C22×C8—24no—
114C22×C8—24no—
115C22×C8—44no—
116C22×C8—44no—
117C22×C8—44no—
118C22×C8—44no—
119C22×C8—44no—
120C22×C8—44no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C22×C412yes—
136C2×C4⋊C4C22×C412yes—
137C2×C4⋊C4C22×C412yes—
138C2×C42—24no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C412yes—
149C2×C42C22⋊C412yes—
150C2×C42C22⋊C412yes—
151C2×C42SD1614yes—
152C2×C42Q1614yes—
153C2×C42C22×C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2411yescharacteristic
163C4⋊C8—48no—
164C4⋊C8—48no—
165#95—22no—
166#95—24no—
167#95—24no—
168#95—22no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177C24C2×C22⋊C411yescharacteristic
178C2×Q8—24no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—88no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4#10314yes—
218C22×C4#10314yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4#13114yes—
227C22×C4C2×SD1612yes—
228C22×C4C2×Q1612yes—
229C22×C4AGL(1,ℤ/8)12yes—
230C22×C4#13712yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—88no—
241C2×C8—88no—
242C4⋊C4—24no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—48no—
248C4⋊C4—48no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C42—24no—
254C42—22no—
255C42—22no—
256C42—24no—
257C42—24no—
258C42—42no—
259C42—42no—
260C42—42no—
261C42—41no—
262C42—42no—
263C42—41no—
264C42—41no—
265C42—42no—
266C42—42no—
267C42—41no—
268C42#10318yes—
269C42C2×SD1614yes—
270C42C2×Q1614yes—
271C23#40812yes—
272C23#41012yes—
273C23#41312yes—
274C23C2×#10312yes—
275C23#41712yes—
276C23#42112yes—
277C23#42112yes—
278C23C22×C22⋊C411yescharacteristic
279Q8—84no—
280Q8—84no—
281C2×C4—28no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4#40812yes—
329C2×C4C2×#10312yes—
330C2×C4C2×#10314yes—
331C2×C4C2×#10314yes—
332C2×C4#41712yes—
333C2×C4C2×#13112yes—
334C2×C4C22×SD1611yescharacteristic
335C2×C4C22×Q1611yescharacteristic
336C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
337C2×C4C2×#13711yescharacteristic
338C8—84no—
339C8—84no—
340C22#147314yes—
341C22#148114yes—
342C22#149514yes—
343C22#151214yes—
344C22#151814yes—
345C22#153514yes—
346C22#155014yes—
347C22C2×#40811yescharacteristic
348C22C2×#41011yescharacteristic
349C22C2×#41311yescharacteristic
350C22C22×#10311yescharacteristic
351C22C2×#41711yescharacteristic
352C22C2×#42111yescharacteristic
353C22C2×#42111yescharacteristic
354C4—24no—
355C4—24no—
356C4—21no—
357C4—21no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—42no—
365C4—42no—
366C4—82no—
367C4—82no—
368C4—82no—
369C4—82no—
370C2#773318yesdirect product factor
371C2C2×#147311yescharacteristic
372C2C2×#148111yescharacteristic
373C2C2×#149511yescharacteristic
374C2C2×#151211yescharacteristic
375C2C2×#151811yescharacteristic
376C2C2×#153511yescharacteristic
377C2C2×#155011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
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
4. #7733C2direct (N × H)
370. C2#7733direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28632
C2×#18312
#77332
C2×#12722
C2×#12712