Finite Group Databasebeta

← #97612

Subgroups — #97612

Order 512.

1000 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 132 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×#2429C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3#8311C212yesmaximal, complemented
4C4×#323C211yesmaximal, characteristic
5#7575C212yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic
7C22×C22⋊Q8C411yescharacteristic
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2429—24no—
11#2429—24no—
12C2×#398—21no—
13C2×#396—21no—
14C2×#391C2211yescharacteristic
15C2×#389—21no—
16C2×#389C411yescharacteristic
17C2×#388—21no—
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×Q8⋊C8—24no—
22C2×#323—21no—
23C2×#323—21no—
24#1280—21no—
25#1280—21no—
26#1272—24no—
27C23×Q8C2×C411yescharacteristic
28C23×Q8C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—44no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—44no—
38C2×C22⋊Q8—44no—
39C2×C4×Q8—22no—
40C2×C4×Q8—44no—
41C2×C4×Q8D414yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4D411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C4×C8—22no—
61#398—28no—
62#396—28no—
63#391—22no—
64#391D414yes—
65#389—28no—
66#389—22no—
67#389—22no—
68#388—28no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95D411yescharacteristic
75C2×#95D411yescharacteristic
76#336—22no—
77#336—22no—
78Q8⋊C8—48no—
79#323—24no—
80#323—42no—
81C25C22⋊C411yescharacteristic
82C22×Q8—24no—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8—42no—
90C22×Q8—42no—
91C22×Q8—42no—
92C22×Q8C22⋊C412yes—
93C22×Q8C22⋊C412yes—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C22×C8—42no—
111C22×C8—42no—
112C4⋊Q8—48no—
113C4⋊Q8—44no—
114C4⋊Q8—44no—
115C22⋊Q8—48no—
116C22⋊Q8—44no—
117C22⋊Q8—48no—
118C22⋊Q8—44no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121C4×Q8—28no—
122C4×Q8—44no—
123C4×Q8—48no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4SD1612yes—
153C2×C22⋊C4Q1612yes—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—24no—
157C2×C42—42no—
158C2×C42—44no—
159C2×C42—42no—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C4⋊C8—44no—
163C22⋊C8—24no—
164C22⋊C8—42no—
165C22⋊C8—42no—
166C22⋊C8—42no—
167C4×C8—44no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—22no—
172#95—22no—
173C24—21no—
174C24—21no—
175C24—21no—
176C24—21no—
177C24—21no—
178C24—21no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24#10312yes—
184C24C2×C22⋊C411yescharacteristic
185C2×Q8—24no—
186C2×Q8—22no—
187C2×Q8—24no—
188C2×Q8—24no—
189C2×Q8—24no—
190C2×Q8—48no—
191C2×Q8—48no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—84no—
200C2×Q8—84no—
201C2×Q8#10014yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
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—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—84no—
250C22×C4—84no—
251C22×C4#10312yes—
252C22×C4#10412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4#12012yes—
256C22×C4#12012yes—
257C22×C4#13112yes—
258C22×C4#13112yes—
259C22×C4C2×SD1611yescharacteristic
260C22×C4C2×Q1611yescharacteristic
261C22×C4#13712yes—
262C2×C8—44no—
263C2×C8—82no—
264C2×C8—82no—
265C2×C8—82no—
266C2×C8—82no—
267C2×C8—82no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—48no—
271C4⋊C4—42no—
272C4⋊C4—48no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—48no—
277C4⋊C4—42no—
278C4⋊C4#9914yes—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—44no—
282C22⋊C4—48no—
283C22⋊C4—48no—
284C42—24no—
285C42—42no—
286C42—48no—
287C42—42no—
288C42—42no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23#37911yescharacteristic
317C23#40811yescharacteristic
318C23#40811yescharacteristic
319C23C2×#10311yescharacteristic
320C23#41911yescharacteristic
321C23C2×#10411yescharacteristic
322C23#42111yescharacteristic
323Q8—28no—
324Q8—48no—
325Q8—44no—
326Q8—84no—
327Q8—84no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—84no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4#40812yes—
380C2×C4#40812yes—
381C2×C4C2×#9912yes—
382C2×C4C2×#10012yes—
383C2×C4#44812yes—
384C2×C4#45112yes—
385C8—84no—
386C8—162no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#148111yescharacteristic
411C22#148511yescharacteristic
412C22#149311yescharacteristic
413C22#149711yescharacteristic
414C22#178011yescharacteristic
415C22#178111yescharacteristic
416C22#178311yescharacteristic
417C22#178711yescharacteristic
418C22#178911yescharacteristic
419C22#179611yescharacteristic
420C22#179611yescharacteristic
421C4—22no—
422C4—22no—
423C4—22no—
424C4—44no—
425C4—42no—
426C4—44no—
427C4—42no—
428C4—42no—
429C4—81no—
430C4—82no—
431C4—82no—
432C4#147114yes—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—82no—
438C2#830711yescharacteristic
439C2#831411yescharacteristic
440C2#832711yescharacteristic
441C2#836411yescharacteristic
442C2#836911yescharacteristic
443C2#837911yescharacteristic
444C2#839111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ 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
3. #8311C2semidirect
5. #7575C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
#85572
#83112
C4×#3232
#75752