Finite Group Databasebeta

← #129802

Subgroups — #129802

Order 512.

930 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 116 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×#2627C211yesmaximal, characteristic
2C2×#1354C212yesmaximal, complemented
3C2×#1353C212yesmaximal, complemented
4#10727C211yesmaximal, characteristic
5#8299C211yesmaximal, characteristic
6C22×C22⋊Q8C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407—21no—
9#2627—24no—
10#2627—24no—
11C2×#400C411yescharacteristic
12C2×#400C2211yescharacteristic
13C2×#398—21no—
14C2×#398—21no—
15C2×#395C411yescharacteristic
16C2×#395C2211yescharacteristic
17C2×#385—21no—
18C2×#381—21no—
19C2×#380C411yescharacteristic
20#1779—21no—
21C2×#343—21no—
22C2×#343C2212yes—
23C2×#324—21no—
24C2×#324C2212yes—
25#1354—24no—
26#1353—24no—
27C23×Q8C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—42no—
35C2×C22⋊Q8—42no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—42no—
43C22×C4⋊C4—42no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51#407—24no—
52#407—42no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—42no—
55C2×C22⋊C8—42no—
56#400—22no—
57#400—22no—
58#400—22no—
59#400D414yes—
60#398—28no—
61#398—28no—
62#395—22no—
63#395—22no—
64#395—22no—
65#395D414yes—
66#385—28no—
67#381—28no—
68#380—22no—
69#380—22no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95C2×C411yescharacteristic
77C2×#95C2×C411yescharacteristic
78#343—22no—
79#343—22no—
80#343—42no—
81#324—22no—
82#324—22no—
83#324—42no—
84C25C22⋊C411yescharacteristic
85C22×Q8—21no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8—82no—
89C22×Q8C22⋊C412yes—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—82no—
108C23×C4—82no—
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4SD1612yes—
113C23×C4Q1612yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C2×M16—24no—
118C2×M16—42no—
119C2×M16—42no—
120C2×M16—42no—
121C2×M16—42no—
122C2×M16—42no—
123C22×C8—42no—
124C22×C8—42no—
125C22⋊Q8—48no—
126C22⋊Q8—48no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—48no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C4⋊C4C2×D412yes—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C22⋊C8—48no—
163C22⋊C8—42no—
164C22⋊C8—42no—
165C22⋊C8—84no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—21no—
170#95—21no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95C22⋊C412yes—
175#95C22⋊C412yes—
176C24—21no—
177C24—21no—
178C24—21no—
179C24—21no—
180C24—42no—
181C24—42no—
182C24#9912yes—
183C24#10312yes—
184C24#10412yes—
185C24C2×C22⋊C411yescharacteristic
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—84no—
191C2×Q8—84no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
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—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4#9912yes—
257C22×C4#10312yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4#12012yes—
261C22×C4#12012yes—
262C22×C4C2×SD1611yescharacteristic
263C22×C4C2×Q1611yescharacteristic
264C22×C4#13712yes—
265M16—44no—
266M16—44no—
267M16—82no—
268M16—82no—
269C2×C8—44no—
270C2×C8—82no—
271C2×C8—82no—
272C2×C8—82no—
273C2×C8—84no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—48no—
282C4⋊C4—42no—
283C4⋊C4—84no—
284C4⋊C4—84no—
285C4⋊C4—84no—
286C22⋊C4—48no—
287C22⋊C4—48no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—44no—
301C23—44no—
302C23—42no—
303C23—42no—
304C23—44no—
305C23—44no—
306C23—42no—
307C23—42no—
308C23#32814yes—
309C23#37911yescharacteristic
310C23C2×#9911yescharacteristic
311C23C2×#9911yescharacteristic
312C23C2×#10311yescharacteristic
313C23#41911yescharacteristic
314C23C2×#10411yescharacteristic
315C23#42111yescharacteristic
316Q8—84no—
317Q8—84no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—81no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—88no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4#44812yes—
376C2×C4#45112yes—
377C8—84no—
378C8—162no—
379C22—22no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—24no—
388C22—24no—
389C22—24no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22C2×#32811yescharacteristic
399C22#150811yescharacteristic
400C22#151111yescharacteristic
401C22#152011yescharacteristic
402C22#178111yescharacteristic
403C22#178711yescharacteristic
404C22#178911yescharacteristic
405C22#212112yes—
406C22#213312yes—
407C4—22no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—82no—
416C4—81no—
417C4—82no—
418C4—82no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—44no—
424C2#832911yescharacteristic
425C2#1172212yes—
426C2#1181712yes—
427C2#1269011yescharacteristic
428C2#1270211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1354C2semidirect
3. C2×#1353C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#13542
C2×#13532
#107272
#82992