Finite Group Databasebeta

← #104102

Subgroups — #104102

Order 512.

672 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 100 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#29969C211yesmaximal, characteristic
2#25902C211yesmaximal, characteristic
3#15188C211yesmaximal, characteristic
4#9908C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7576C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#487—21no—
10#2857C2211yescharacteristic
11C2×#391—21no—
12C2×#391—21no—
13C2×#389C2211yescharacteristic
14#2100—21no—
15#2087—22no—
16#2057—21no—
17#2024—22no—
18#1990—21no—
19#1872—22no—
20#1852—21no—
21#1824—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24C4×C22⋊C8C2211yescharacteristic
25C2×#332—22no—
26#1442—22no—
27#1441—22no—
28C4×#95C2211yescharacteristic
29#1294—22no—
30#1271—22no—
31C23×Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—44no—
37C2×C4⋊Q8D411yescharacteristic
38C2×C22⋊Q8—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D412yes—
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#487—44no—
47C2×#107—22no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D412yes—
50C2×#103—24no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C4×C8—21no—
55#391—21no—
56#391—42no—
57#391—44no—
58#391—42no—
59#389—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#384D411yescharacteristic
66#380—21no—
67#380—21no—
68C2×#95—24no—
69C2×#95—24no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95D412yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—24no—
77#336—21no—
78#336—21no—
79#336D412yes—
80#332—44no—
81C22×Q8—24no—
82C22×Q8—21no—
83C22×Q8—42no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8—41no—
89C22×Q8—41no—
90C22×Q8—41no—
91C22×Q8—41no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—41no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—44no—
108C4⋊Q8—44no—
109C22⋊Q8—44no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4SD1612yes—
134C2×C22⋊C4—41no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144#107—82no—
145C4⋊C8—42no—
146C4⋊C8—42no—
147#103—48no—
148C22⋊C8—42no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—42no—
152C4×C8—42no—
153#95—22no—
154#95—42no—
155#95—48no—
156#95—48no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95Q1614yes—
162C24#12011yescharacteristic
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—42no—
172C2×Q8—82no—
173C2×Q8—82no—
174C2×Q8—82no—
175C2×Q8—82no—
176C2×Q8—82no—
177C2×Q8—81no—
178C2×Q8—81no—
179C2×Q8—82no—
180C2×Q8—81no—
181C2×Q8—81no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4#12011yescharacteristic
215C22×C4#12011yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12711yescharacteristic
220C22×C4C2×SD1611yescharacteristic
221C22×C4C2×Q1612yes—
222C22×C4#13511yescharacteristic
223C22×C4AGL(1,ℤ/8)11yescharacteristic
224C22×C4#13711yescharacteristic
225C22×C4#13712yes—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—81no—
233C2×C8—81no—
234C2×C8—81no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C22⋊C4—82no—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—41no—
260C42—82no—
261C42—82no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23#39211yescharacteristic
267C23#45011yescharacteristic
268C23#45112yes—
269C23#45211yescharacteristic
270C23#45412yes—
271Q8—84no—
272Q8—84no—
273Q8—84no—
274Q8—84no—
275Q8—84no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4—84no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4#44811yescharacteristic
324C2×C4#44911yescharacteristic
325C2×C4#45412yes—
326C2×C4#45412yes—
327C2×C4B(4,𝔽2)12yes—
328C2×C4#46212yes—
329C2×C4#46511yescharacteristic
330C2×C4#46811yescharacteristic
331C2×C4#49411yescharacteristic
332C2×C4#49711yescharacteristic
333C8—82no—
334C8—82no—
335C8—161no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22#163612yes—
345C22#163712yes—
346C22#200011yescharacteristic
347C22#200311yescharacteristic
348C22#200311yescharacteristic
349C22#200311yescharacteristic
350C22#200911yescharacteristic
351C22#201611yescharacteristic
352C22#202111yescharacteristic
353C4—22no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—22no—
358C4—21no—
359C4—41no—
360C4—84no—
361C4—82no—
362C4—84no—
363C4—161no—
364C4—161no—
365C4—161no—
366C4#159712yes—
367C4#159912yes—
368C2—22no—
369C2—22no—
370C2#992311yescharacteristic
371C2#995711yescharacteristic
372C2#1005911yescharacteristic
373C2#1009511yescharacteristic
374C2#1024711yescharacteristic
375C2#1029511yescharacteristic
376C2#1032711yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C22×C42 ▷ 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
#299692
#259022
#151882
#99082
#83402
#75762
#70282