Finite Group Databasebeta

← #141197

Subgroups — #141197

Order 512.

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

normal subgroups only

This group has 598 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
1#15016C211yesmaximal, characteristic
2C2×#1352C212yesmaximal
3C2×#1344C212yesmaximal
4#10801C211yesmaximal, characteristic
5#8557C211yesmaximal, characteristic
6C2×#399C2211yescharacteristic
7C2×#397C2211yescharacteristic
8C2×#382C411yescharacteristic, complemented
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1445C411yescharacteristic, complemented
15#1445C411yescharacteristic, complemented
16#1442—21no—
17#1440—21no—
18#1436—21no—
19#1434C411yescharacteristic, complemented
20C22⋊#95—21no—
21C22⋊#95—21no—
22#1352—24no—
23#1344—24no—
24#1280—21no—
25C24×C4D411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—22no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4D411yescharacteristic
33C22×C22⋊C4C2311yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38#399—22no—
39#399—22no—
40#397—22no—
41#397—22no—
42#382—41no—
43#382—41no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—41no—
56C2×#95—41no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95D411yescharacteristic
61#342—22no—
62#342—42no—
63#342D414yes—
64#336—41no—
65#323—22no—
66#323—22no—
67C25C2×D411yescharacteristic
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—41no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—81no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—42no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—81no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—81no—
115C2×C22⋊C4C2×D412yes—
116C2×C22⋊C4#4112yes—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—42no—
122C2×C42—81no—
123C4⋊C8—82no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126#95—22no—
127#95—22no—
128#95—41no—
129#95—41no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—81no—
140#95—81no—
141#95—82no—
142#95—82no—
143C24—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—41no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24C2×C22⋊C411yescharacteristic
152C24#12112yes—
153C24#13512yes—
154C24AGL(1,ℤ/8)11yescharacteristic
155C24#13711yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
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—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—44no—
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—81no—
217C22×C4—81no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4#9911yescharacteristic
227C22×C4#9911yescharacteristic
228C22×C4C4×D412yes—
229C22×C4#12411yescharacteristic
230C22×C4#12711yescharacteristic
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—84no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C42—42no—
250C42—42no—
251C42—42no—
252C42—81no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—22no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23#39011yescharacteristic
284C23C2×#9911yescharacteristic
285C23#41011yescharacteristic
286C23#41611yescharacteristic
287C23#41711yescharacteristic
288C23#42111yescharacteristic
289C23#42111yescharacteristic
290C23#46312yes—
291C23#46412yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
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—84no—
325C2×C4—81no—
326C2×C4—84no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—84no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—162no—
360C2×C4—162no—
361C2×C4—162no—
362C2×C4—162no—
363C2×C4#35212yes—
364C8—162no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—82no—
390C22—81no—
391C22—81no—
392C22—82no—
393C22#151211yescharacteristic
394C22#151311yescharacteristic
395C22#151611yescharacteristic
396C22#152111yescharacteristic
397C22#196311yescharacteristic
398C22#196411yescharacteristic
399C22#198311yescharacteristic
400C22C2×#35211yescharacteristic
401C22#212311yescharacteristic
402C22#212611yescharacteristic
403C22#212711yescharacteristic
404C4—22no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—82no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—161no—
414C4—164no—
415C4—161no—
416C4—161no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—42no—
422C2—42no—
423C2#971611yescharacteristic
424C2#1182211yescharacteristic
425C2#1182711yescharacteristic
426C2#1184111yescharacteristic
427C2#1184611yescharacteristic
428C2#1322011yescharacteristic
429C2#1322511yescharacteristic
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 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#382C4semidirect
14. #1445C4semidirect
15. #1445C4semidirect
19. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#150162
C2×#13522
C2×#13442
#108012
#85572