Finite Group Databasebeta

← #97139

Subgroups — #97139

Order 512.

760 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 138 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
1#35167C211yesmaximal, characteristic
2#11202C211yesmaximal, characteristic
3#9127C212yesmaximal
4#8558C211yesmaximal, characteristic
5#7667C212yesmaximal
6C2×#544C412yes—
7C22×C4⋊C8C2211yescharacteristic
8#2857—21no—
9#2857C2212yes—
10#2659—21no—
11#2659C412yes—
12#2552—22no—
13#2551—22no—
14#2550—24no—
15#2316—21no—
16#2285—21no—
17C2×C4×C4⋊C4C2211yescharacteristic
18#1797C2212yes—
19C2×#336—21no—
20C2×#336—21no—
21C2×#336C2211yescharacteristic
22C2×#333—22no—
23C2×#324—21no—
24#1296—22no—
25#1280—21no—
26#1272—24no—
27C23×C8—21no—
28C23×C8—21no—
29#544—44no—
30#544—44no—
31C2×C4⋊Q8—24no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8D412yes—
34C2×#125—24no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8C2×C414yes—
37C2×#117—22no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42C22×C42C2311yescharacteristic
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—42no—
46C2×C22⋊C8—22no—
47#398—24no—
48#398—24no—
49#395—24no—
50#395—24no—
51#391—22no—
52#391D414yes—
53#389—24no—
54#389—24no—
55#389—22no—
56#389—22no—
57#389—22no—
58#389—22no—
59#384—21no—
60#384—21no—
61#384—21no—
62#384D412yes—
63#382—22no—
64#380—22no—
65#380—22no—
66#380—22no—
67#380—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C2×#95C2×C411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76#336—42no—
77#336—42no—
78#333—44no—
79#324—42no—
80C22×Q8—22no—
81C22×Q8C22⋊C414yes—
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—42no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—416no—
107C4⋊Q8—81no—
108C4⋊Q8—81no—
109#125—416no—
110C22⋊Q8—44no—
111C22⋊Q8—44no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114#117—44no—
115#117—44no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4C22⋊C414yes—
135C2×C4⋊C4SD1612yes—
136C2×C4⋊C4Q1612yes—
137C2×C22⋊C4—24no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—24no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42SD1612yes—
150C2×C42Q1612yes—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C4⋊C8—24no—
154C4⋊C8—42no—
155C4⋊C8—42no—
156C4⋊C8—42no—
157C22⋊C8—42no—
158C22⋊C8—42no—
159#95—24no—
160#95—24no—
161#95—22no—
162#95—22no—
163#95—24no—
164#95—24no—
165C24C2×C22⋊C411yescharacteristic
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—44no—
170C2×Q8—82no—
171C2×Q8—82no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4#10012yes—
201C22×C4#10012yes—
202C22×C4#10312yes—
203C22×C4#10312yes—
204C22×C4#10312yes—
205C22×C4#10312yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4#12011yescharacteristic
208C22×C4#12012yes—
209C22×C4#12011yescharacteristic
210C22×C4C2×SD1611yescharacteristic
211C22×C4C2×SD1611yescharacteristic
212C22×C4C2×Q1611yescharacteristic
213C22×C4C2×Q1611yescharacteristic
214C22×C4#13711yescharacteristic
215C22×C4#13711yescharacteristic
216C22×C4#13712yes—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C2×C8—81no—
227C2×C8—81no—
228C2×C8—81no—
229C2×C8—81no—
230C2×C8—81no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C4⋊C4—28no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—48no—
242C4⋊C4—48no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—42no—
262C42—48no—
263C42—41no—
264C42—41no—
265C42—81no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23#37911yescharacteristic
271C23C2×#10011yescharacteristic
272C23C2×#10011yescharacteristic
273C23C2×#10311yescharacteristic
274C23C2×#10311yescharacteristic
275C23#41911yescharacteristic
276C23#41911yescharacteristic
277Q8—164no—
278Q8—164no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4#33314yescomplemented
327C2×C4#44811yescharacteristic
328C2×C4#44812yes—
329C2×C4#44811yescharacteristic
330C2×C4#45111yescharacteristic
331C2×C4#45112yes—
332C2×C4#45111yescharacteristic
333C2×C4#45411yescharacteristic
334C2×C4#45611yescharacteristic
335C2×C4B(4,𝔽2)11yescharacteristic
336C2×C4#45811yescharacteristic
337C8—82no—
338C8—82no—
339C8—162no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22C2×#33311yescharacteristic
353C22#154811yescharacteristic
354C22#154811yescharacteristic
355C22#155811yescharacteristic
356C22#178311yescharacteristic
357C22#178711yescharacteristic
358C22#178711yescharacteristic
359C22#179011yescharacteristic
360C22#179111yescharacteristic
361C22#179311yescharacteristic
362C22#179411yescharacteristic
363C4—24no—
364C4—22no—
365C4—22no—
366C4—21no—
367C4—21no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—84no—
373C4—84no—
374C4#159911yescharacteristic
375C4#160511yescharacteristic
376C4#161111yescharacteristic
377C4#161711yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2#834211yescharacteristic
383C2#835111yescharacteristic
384C2#835611yescharacteristic
385C2#840611yescharacteristic
386C2#841011yescharacteristic
387C2#845911yescharacteristic
388C2#853611yescharacteristic
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
326. C2×C4#333semidirect

Maximal subgroups

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

subgroupindex
#351672
#112022
#91272
#85582
#76672