Finite Group Databasebeta

← #138401

Subgroups — #138401

Order 512.

708 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 90 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×#1820C211yesmaximal, characteristic
2#14907C211yesmaximal, characteristic
3C2×#1344C212yesmaximal
4#8300C211yesmaximal, characteristic
5#7250C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#398C411yescharacteristic, complemented
8C2×#397C2211yescharacteristic
9C2×#389C411yescharacteristic, complemented
10C2×#385—21no—
11C2×#381—21no—
12#1820—24no—
13#1817—21no—
14C2×#342—21no—
15C2×#342C2212yes—
16C2×#324—21no—
17C2×#323C2212yes—
18#1441—21no—
19#1441C411yescharacteristic, complemented
20#1440—21no—
21#1439C2211yescharacteristic
22#1436—21no—
23#1434C411yescharacteristic, complemented
24#1344—24no—
25C24×C4D411yescharacteristic
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—42no—
29C2×#117—42no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C22⋊C4—22no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C22⋊C4D411yescharacteristic
40C22×C22⋊C4C2311yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42D411yescharacteristic
43C2×C22⋊C8—22no—
44#398—42no—
45#397—22no—
46#397—22no—
47#389—42no—
48#385—44no—
49#381—24no—
50#381—42no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—42no—
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62#342—24no—
63#342—22no—
64#342D414yes—
65#336—42no—
66#324—42no—
67#323—22no—
68#323—22no—
69C25C2×D411yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4—42no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C22×C8—42no—
90#117—48no—
91#117—48no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4C2×D412yes—
129C2×C22⋊C4#4112yes—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—41no—
135C2×C42—41no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#95—22no—
139#95—41no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—48no—
148#95—42no—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—41no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24C2×C22⋊C411yescharacteristic
158C24#12011yescharacteristic
159C24#12311yescharacteristic
160C24#13512yes—
161C24AGL(1,ℤ/8)12yes—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—44no—
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—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4#9912yes—
238C22×C4#9912yes—
239C22×C4C4×D412yes—
240C22×C4#12112yes—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—48no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—82no—
262C42—44no—
263C42—44no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23#38611yescharacteristic
295C23C2×#9911yescharacteristic
296C23C2×#9911yescharacteristic
297C23#41611yescharacteristic
298C23#41811yescharacteristic
299C23#42111yescharacteristic
300C23#42111yescharacteristic
301C23#44912yes—
302C23#48212yes—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—48no—
313C2×C4—48no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—84no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—162no—
371C2×C4—162no—
372C2×C4—162no—
373C2×C4—162no—
374C8—162no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—22no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—42no—
399C22—82no—
400C22—81no—
401C22—81no—
402C22—82no—
403C22#151011yescharacteristic
404C22#151011yescharacteristic
405C22#151611yescharacteristic
406C22#151611yescharacteristic
407C22#187511yescharacteristic
408C22#189711yescharacteristic
409C22#189811yescharacteristic
410C22#212011yescharacteristic
411C22#212311yescharacteristic
412C22#212611yescharacteristic
413C22#212911yescharacteristic
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—82no—
419C4—82no—
420C4—81no—
421C4—81no—
422C4—161no—
423C4—164no—
424C4—161no—
425C2—21no—
426C2—21no—
427C2—21no—
428C2—21no—
429C2—42no—
430C2—42no—
431C2#915811yescharacteristic
432C2#1179811yescharacteristic
433C2#1180111yescharacteristic
434C2#1184211yescharacteristic
435C2#1184611yescharacteristic
436C2#1308511yescharacteristic
437C2#1309111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#398C4semidirect
9. C2×#389C4semidirect
19. #1441C4semidirect
23. #1434C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18202
#149072
C2×#13442
#83002
#72502