Finite Group Databasebeta

← #97137

Subgroups — #97137

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#9129C212yesmaximal
4#8558C211yesmaximal, characteristic
5#7665C212yesmaximal
6C2×#544—21no—
7C22×C4⋊C8C2211yescharacteristic
8#2857—21no—
9#2857C2212yes—
10#2659—21no—
11#2659C412yes—
12#2552—22no—
13#2551—22no—
14#2550—24no—
15#2316C412yes—
16#2285—21no—
17C2×C4×C4⋊C4C2211yescharacteristic
18#1797C2212yes—
19C2×#336—21no—
20C2×#336—21no—
21C2×#336C2211yescharacteristic
22C2×#332—22no—
23C2×#324—21no—
24#1296—22no—
25#1280—21no—
26#1271—24no—
27C23×C8—21no—
28C23×C8—21no—
29#544—44no—
30#544—42no—
31#544—42no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D412yes—
35C2×#125—24no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×#117—22no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C42D411yescharacteristic
42C22×C42D411yescharacteristic
43C22×C42C2311yescharacteristic
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—42no—
47C2×C22⋊C8—22no—
48#398—24no—
49#398—24no—
50#395—24no—
51#395—24no—
52#391—22no—
53#391—22no—
54#389—22no—
55#389—22no—
56#389—24no—
57#389—24no—
58#389—22no—
59#389D414yes—
60#384—21no—
61#384—21no—
62#384—21no—
63#384D412yes—
64#382—22no—
65#380—22no—
66#380—22no—
67#380—22no—
68#380C2×C414yes—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C2×#95C2×C411yescharacteristic
74C2×#95D411yescharacteristic
75#336—22no—
76#336—22no—
77#336—42no—
78#336—42no—
79#332—44no—
80#324—42no—
81C22×Q8—22no—
82C22×Q8—22no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C22×C8—22no—
91C22×C8—22no—
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—41no—
104C22×C8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—48no—
108C4⋊Q8—48no—
109C4⋊Q8—81no—
110C4⋊Q8—81no—
111#125—48no—
112#125—48no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C22⋊Q8—42no—
116C22⋊Q8—42no—
117C22⋊Q8—42no—
118C22⋊Q8—42no—
119#117—44no—
120#117—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4C22⋊C414yes—
139C2×C4⋊C4C22⋊C414yes—
140C2×C4⋊C4D812yes—
141C2×C4⋊C4SD1612yes—
142C2×C22⋊C4—24no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—24no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42SD1612yes—
155C2×C42Q1612yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C4⋊C8—24no—
159C4⋊C8—42no—
160C4⋊C8—42no—
161C4⋊C8—42no—
162C22⋊C8—42no—
163C22⋊C8—42no—
164#95—24no—
165#95—24no—
166#95—22no—
167#95—24no—
168#95—22no—
169#95—24no—
170C24C2×C22⋊C411yescharacteristic
171C2×Q8—44no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—82no—
175C2×Q8—82no—
176C2×Q8—82no—
177C2×Q8—82no—
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—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4#10112yes—
207C22×C4#10112yes—
208C22×C4#10212yes—
209C22×C4#10212yes—
210C22×C4#10312yes—
211C22×C4#10312yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4#12012yes—
214C22×C4#12011yescharacteristic
215C22×C4#12011yescharacteristic
216C22×C4C2×D811yescharacteristic
217C22×C4C2×SD1611yescharacteristic
218C22×C4C2×SD1611yescharacteristic
219C22×C4C2×Q1611yescharacteristic
220C22×C4AGL(1,ℤ/8)12yes—
221C22×C4#13711yescharacteristic
222C22×C4#13711yescharacteristic
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—81no—
230C2×C8—81no—
231C2×C8—81no—
232C2×C8—81no—
233C2×C8—81no—
234C2×C8—81no—
235C2×C8—81no—
236C2×C8—81no—
237C2×C8—82no—
238C2×C8—82no—
239C2×C8—82no—
240C4⋊C4—28no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—48no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—42no—
270C42—48no—
271C42—41no—
272C42—41no—
273C42—81no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#37911yescharacteristic
279C23C2×#10111yescharacteristic
280C23C2×#10111yescharacteristic
281C23C2×#10211yescharacteristic
282C23C2×#10311yescharacteristic
283C23#41811yescharacteristic
284C23#41911yescharacteristic
285Q8—162no—
286Q8—162no—
287Q8—164no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4#33214yescomplemented
336C2×C4#44712yes—
337C2×C4#44811yescharacteristic
338C2×C4#44811yescharacteristic
339C2×C4#45012yes—
340C2×C4#45111yescharacteristic
341C2×C4#45111yescharacteristic
342C2×C4#45411yescharacteristic
343C2×C4#45611yescharacteristic
344C2×C4B(4,𝔽2)11yescharacteristic
345C2×C4#45811yescharacteristic
346C8—82no—
347C8—82no—
348C8—162no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22C2×#33211yescharacteristic
362C22#154611yescharacteristic
363C22#154911yescharacteristic
364C22#155611yescharacteristic
365C22#178411yescharacteristic
366C22#178611yescharacteristic
367C22#178711yescharacteristic
368C22#179011yescharacteristic
369C22#179111yescharacteristic
370C22#179311yescharacteristic
371C22#179411yescharacteristic
372C4—24no—
373C4—22no—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—84no—
382C4—84no—
383C4#159911yescharacteristic
384C4#160511yescharacteristic
385C4#161111yescharacteristic
386C4#161711yescharacteristic
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2#834011yescharacteristic
392C2#835011yescharacteristic
393C2#835411yescharacteristic
394C2#840611yescharacteristic
395C2#841011yescharacteristic
396C2#846011yescharacteristic
397C2#853711yescharacteristic
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
335. C2×C4#332semidirect

Maximal subgroups

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

subgroupindex
#351672
#112022
#91292
#85582
#76652