Finite Group Databasebeta

← #97440

Subgroups — #97440

Order 512.

852 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 112 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×#2581C211yesmaximal, characteristic
2#10763C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7610C212yesmaximal
5#7609C212yesmaximal
6C22×C4×Q8C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C22×#101—21no—
9#2581—24no—
10#2581—24no—
11C2×#400—21no—
12C2×#395—21no—
13C2×#391C411yescharacteristic
14C2×#391C2211yescharacteristic
15C2×#389C411yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#387—21no—
18C2×#387—21no—
19C2×#382C411yescharacteristic
20#1817C2212yes—
21C2×#343—21no—
22#1728C2212yes—
23C2×#324—21no—
24C2×#324—21no—
25#1272—24no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C23×Q8D411yescharacteristic
29C22×M16—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—42no—
35C2×C4×Q8—42no—
36C2×C4×Q8—42no—
37C2×C4×Q8—42no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C42—21no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42C2311yescharacteristic
51C2×#101—24no—
52C2×#101—42no—
53C2×#101—42no—
54C2×C22⋊C8—22no—
55#400—28no—
56#395—28no—
57#391—22no—
58#391—22no—
59#391—22no—
60#391D414yes—
61#389—22no—
62#389—22no—
63#389—22no—
64#389D414yes—
65#387—28no—
66#387—28no—
67#382—22no—
68#382—22no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77#343—42no—
78#336—22no—
79#336—22no—
80#336—22no—
81#336—22no—
82#324—42no—
83#324—42no—
84C22×Q8—24no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—21no—
91C22×Q8—21no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—42no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8C22⋊C412yes—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C2×M16—24no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C22×C8—42no—
116C22×C8—42no—
117C4×Q8—48no—
118C4×Q8—48no—
119C4×Q8—48no—
120C4×Q8—48no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—24no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165#101—48no—
166C22⋊C8—42no—
167C22⋊C8—42no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—24no—
176C24C2×C22⋊C411yescharacteristic
177C2×Q8—24no—
178C2×Q8—24no—
179C2×Q8—24no—
180C2×Q8—22no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×Q8—44no—
185C2×Q8—42no—
186C2×Q8—48no—
187C2×Q8—48no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8#10114yescomplemented
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4#9912yes—
252C22×C4#9912yes—
253C22×C4#10412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4#12012yes—
257C22×C4#12012yes—
258C22×C4#13512yes—
259C22×C4AGL(1,ℤ/8)11yescharacteristic
260C22×C4#13711yescharacteristic
261M16—44no—
262M16—44no—
263M16—82no—
264M16—82no—
265C2×C8—44no—
266C2×C8—82no—
267C2×C8—82no—
268C2×C8—82no—
269C2×C8—82no—
270C2×C8—82no—
271C4⋊C4—24no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—48no—
279C4⋊C4—42no—
280C4⋊C4—48no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C42—24no—
285C42—24no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—48no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#37911yescharacteristic
296C23C2×#9911yescharacteristic
297C23C2×#9911yescharacteristic
298C23#41611yescharacteristic
299C23#41711yescharacteristic
300C23C2×#10411yescharacteristic
301C23#42111yescharacteristic
302Q8—44no—
303Q8—44no—
304Q8—48no—
305Q8—48no—
306Q8—48no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4C2×#10112yes—
373C2×C4#41312yes—
374C2×C4#44912yes—
375C2×C4#45212yes—
376C8—84no—
377C8—162no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22#150111yescharacteristic
391C22#150411yescharacteristic
392C22#151311yescharacteristic
393C22#151611yescharacteristic
394C22#178111yescharacteristic
395C22#178411yescharacteristic
396C22#178511yescharacteristic
397C22#178811yescharacteristic
398C22#178911yescharacteristic
399C22#179611yescharacteristic
400C22#179611yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—42no—
406C4—42no—
407C4—44no—
408C4—42no—
409C4—42no—
410C4—81no—
411C4—82no—
412C4—82no—
413C4—82no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2#833111yescharacteristic
419C2#836811yescharacteristic
420C2#837411yescharacteristic
421C2#839011yescharacteristic
422C2#839411yescharacteristic
423C2#852011yescharacteristic
424C2#855011yescharacteristic
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
195. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
#107632
#83032
#76102
#76092