Finite Group Databasebeta

← #98541

Subgroups — #98541

Order 512.

802 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2446C211yesmaximal, characteristic
2#8560C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic
4#8342C212yesmaximal
5#7641C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C22×C4×Q8C411yescharacteristic
8#2446—24no—
9#2446—24no—
10C2×#400—21no—
11C2×#398—21no—
12C2×#391C411yescharacteristic
13C2×#391C411yescharacteristic
14C2×#391C411yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#387—21no—
17C2×#387—21no—
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20#1797C2212yes—
21C2×#333—24no—
22C2×#324—21no—
23C2×#324—21no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C23×Q8C2×C411yescharacteristic
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—44no—
31C2×C4⋊Q8D414yes—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—44no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C42—21no—
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42C2311yescharacteristic
47C2×C4⋊C8—22no—
48C2×C4⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51#400—28no—
52#398—28no—
53#391—22no—
54#391—22no—
55#391—22no—
56#391—22no—
57#391—22no—
58#391—22no—
59#389—22no—
60#389D414yes—
61#387—28no—
62#387—28no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76#333—48no—
77#324—42no—
78#324—42no—
79C22×Q8—24no—
80C22×Q8—21no—
81C22×Q8—21no—
82C22×Q8—42no—
83C22×Q8—42no—
84C22×Q8—42no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8—41no—
89C22×Q8C22⋊C412yes—
90C22×Q8C22⋊C412yes—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C22×C8—42no—
103C22×C8—42no—
104C4⋊Q8—28no—
105C4⋊Q8—44no—
106C4⋊Q8—48no—
107C4×Q8—48no—
108C4×Q8—44no—
109C4×Q8—44no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4SD1612yes—
133C2×C4⋊C4Q1612yes—
134C2×C42—24no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—42no—
140C2×C42—41no—
141C2×C42—41no—
142C2×C42—44no—
143C2×C42—42no—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C4⋊C8—44no—
147C4⋊C8—44no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—22no—
160#95—22no—
161C24C2×C22⋊C411yescharacteristic
162C2×Q8—24no—
163C2×Q8—22no—
164C2×Q8—24no—
165C2×Q8—48no—
166C2×Q8—48no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—84no—
177C2×Q8—84no—
178C2×Q8#9914yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4#10312yes—
228C22×C4#10312yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4#12012yes—
232C22×C4#12012yes—
233C22×C4C2×SD1611yescharacteristic
234C22×C4C2×Q1611yescharacteristic
235C22×C4#13512yes—
236C22×C4AGL(1,ℤ/8)11yescharacteristic
237C22×C4#13712yes—
238C22×C4#13711yescharacteristic
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—48no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C42—24no—
259C42—42no—
260C42—42no—
261C42—42no—
262C42—42no—
263C42—48no—
264C42—42no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23#37911yescharacteristic
270C23#41311yescharacteristic
271C23#41311yescharacteristic
272C23C2×#10311yescharacteristic
273C23#41611yescharacteristic
274C23#41711yescharacteristic
275C23#41911yescharacteristic
276Q8—44no—
277Q8—44no—
278Q8—48no—
279Q8—84no—
280Q8—84no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4C2×#9912yes—
340C2×C4#41312yes—
341C2×C4#44812yes—
342C2×C4#44912yes—
343C2×C4#45112yes—
344C2×C4#45212yes—
345C8—162no—
346C8—162no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22#154211yescharacteristic
360C22#155011yescharacteristic
361C22#155411yescharacteristic
362C22#155911yescharacteristic
363C22#178111yescharacteristic
364C22#178511yescharacteristic
365C22#178511yescharacteristic
366C22#178711yescharacteristic
367C22#178811yescharacteristic
368C22#179611yescharacteristic
369C22#179611yescharacteristic
370C4—22no—
371C4—22no—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—44no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—82no—
381C4—81no—
382C4—82no—
383C4—82no—
384C2—21no—
385C2—21no—
386C2—21no—
387C2—21no—
388C2#834311yescharacteristic
389C2#837711yescharacteristic
390C2#837911yescharacteristic
391C2#839211yescharacteristic
392C2#839411yescharacteristic
393C2#852711yescharacteristic
394C2#854811yescharacteristic
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

Maximal subgroups

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

subgroupindex
C2×#24462
#85602
#83422
#76412