Finite Group Databasebeta

← #98536

Subgroups — #98536

Order 512.

960 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2#8560C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic
4#8340C212yesmaximal
5#7641C212yesmaximal
6C22×C4⋊Q8—21no—
7C22×C4⋊Q8C411yescharacteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2551—24no—
10#2551—24no—
11C2×#395—21no—
12C2×#395—21no—
13C2×#389—21no—
14C2×#389C411yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#384C411yescharacteristic
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20#1797C2212yes—
21C2×#332—24no—
22C2×#324—21no—
23C2×#324—21no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C23×Q8—21no—
28C23×Q8—21no—
29C2×C4⋊Q8—28no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—44no—
34C2×C4⋊Q8—44no—
35C2×C4⋊Q8—44no—
36C2×C4⋊Q8—42no—
37C2×C4⋊Q8—42no—
38C2×C4⋊Q8D414yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C42—21no—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42C2311yescharacteristic
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59#395—28no—
60#395—28no—
61#389—28no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389D414yes—
66#384—22no—
67#384—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95D411yescharacteristic
76C2×#95D411yescharacteristic
77#336—22no—
78#336—22no—
79#332—48no—
80#324—42no—
81#324—42no—
82C22×Q8—24no—
83C22×Q8—24no—
84C22×Q8—44no—
85C22×Q8—44no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—41no—
89C22×Q8—41no—
90C22×Q8—42no—
91C22×Q8—82no—
92C22×Q8—82no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C22×C8—42no—
105C22×C8—42no—
106C4⋊Q8—28no—
107C4⋊Q8—44no—
108C4⋊Q8—48no—
109C4⋊Q8—48no—
110C4⋊Q8—416no—
111C4⋊Q8—416no—
112C4⋊Q8—44no—
113C4⋊Q8—44no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4D812yes—
147C2×C4⋊C4SD1612yes—
148C2×C4⋊C4SD1612yes—
149C2×C4⋊C4Q1612yes—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—24no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C4⋊C8—44no—
164C4⋊C8—44no—
165C22⋊C8—42no—
166C22⋊C8—42no—
167C22⋊C8—42no—
168C22⋊C8—42no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—22no—
173#95—22no—
174C24C2×C22⋊C411yescharacteristic
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—48no—
178C2×Q8—44no—
179C2×Q8—44no—
180C2×Q8—84no—
181C2×Q8—84no—
182C2×Q8—84no—
183C2×Q8—84no—
184C2×Q8—88no—
185C2×Q8—88no—
186C2×Q8—88no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4#10112yes—
232C22×C4#10112yes—
233C22×C4#10212yes—
234C22×C4#10212yes—
235C22×C4#10312yes—
236C22×C4#10312yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4#12012yes—
240C22×C4#12012yes—
241C22×C4C2×D811yescharacteristic
242C22×C4C2×SD1611yescharacteristic
243C22×C4C2×SD1611yescharacteristic
244C22×C4C2×Q1611yescharacteristic
245C22×C4AGL(1,ℤ/8)12yes—
246C22×C4#13712yes—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—48no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—48no—
266C4⋊C4—48no—
267C4⋊C4—48no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—84no—
272C4⋊C4#9914yes—
273C4⋊C4#10114yes—
274C42—24no—
275C42—24no—
276C42—24no—
277C42—42no—
278C42—42no—
279C42—48no—
280C42—42no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23#37911yescharacteristic
286C23C2×#10111yescharacteristic
287C23C2×#10111yescharacteristic
288C23C2×#10211yescharacteristic
289C23C2×#10311yescharacteristic
290C23#41811yescharacteristic
291C23#41911yescharacteristic
292Q8—88no—
293Q8—88no—
294Q8—816no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#33214yescomplemented
353C2×C4C2×#9912yes—
354C2×C4C2×#10112yes—
355C2×C4#44712yes—
356C2×C4#44812yes—
357C2×C4#45012yes—
358C2×C4#45112yes—
359C8—162no—
360C8—162no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22C2×#33211yescharacteristic
374C22#154611yescharacteristic
375C22#154911yescharacteristic
376C22#155611yescharacteristic
377C22#178111yescharacteristic
378C22#178411yescharacteristic
379C22#178411yescharacteristic
380C22#178611yescharacteristic
381C22#178711yescharacteristic
382C22#179511yescharacteristic
383C22#179611yescharacteristic
384C4—22no—
385C4—22no—
386C4—24no—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—42no—
392C4—44no—
393C4—42no—
394C4—42no—
395C4—44no—
396C4—82no—
397C4—82no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#834011yescharacteristic
403C2#837511yescharacteristic
404C2#837911yescharacteristic
405C2#838811yescharacteristic
406C2#839611yescharacteristic
407C2#852511yescharacteristic
408C2#855111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
352. C2×C4#332semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#85602
#83402
#76412