Finite Group Databasebeta

← #134483

Subgroups — #134483

Order 512.

664 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 96 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#14230C211yesmaximal, characteristic
2C2×#1352C212yesmaximal
3#10801C211yesmaximal, characteristic
4#8557C211yesmaximal, characteristic
5#7250C212yesmaximal
6C2×#399C2211yescharacteristic
7C2×#389C411yescharacteristic, complemented
8C2×#387—21no—
9C2×#385—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1797—21no—
12C2×#342—21no—
13C2×#342C2212yes—
14C2×#323C2212yes—
15#1445C411yescharacteristic, complemented
16#1444C411yescharacteristic, complemented
17#1439C2211yescharacteristic
18#1437—21no—
19C4⋊#95—21no—
20C22⋊#95—21no—
21C4×#95C411yescharacteristic, complemented
22#1352—24no—
23#1280—21no—
24C24×C4D411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—22no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4D411yescharacteristic
34C22×C22⋊C4C2311yescharacteristic
35C22×C42—21no—
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C2×C4⋊C8—41no—
39C2×C22⋊C8—22no—
40#399—21no—
41#399—21no—
42#399D412yes—
43#399Q812yes—
44#389—41no—
45#389—41no—
46#387—44no—
47#385—44no—
48C4×C22⋊C4—41no—
49C4×C22⋊C4—41no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—41no—
57C2×#95—41no—
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95D411yescharacteristic
62#342—24no—
63#342—22no—
64#342—22no—
65#336—41no—
66#323—22no—
67#323—22no—
68C25C2×D411yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—41no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—81no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—42no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4C4⋊C412yes—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—81no—
122C2×C22⋊C4C2×D411yescharacteristic
123C2×C22⋊C4C2×Q811yescharacteristic
124C2×C22⋊C4#4111yescharacteristic
125C2×C22⋊C4#4111yescharacteristic
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—81no—
134C2×C42—81no—
135C4⋊C8—82no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#95—24no—
139#95—21no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—82no—
149#95—82no—
150#95C4⋊C412yes—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24C2×C22⋊C411yescharacteristic
160C24#12111yescharacteristic
161C24C22⋊Q811yescharacteristic
162C24#13512yes—
163C24#13712yes—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
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—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
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—42no—
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—81no—
224C22×C4—81no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4#9911yescharacteristic
234C22×C4#9911yescharacteristic
235C22×C4#10412yes—
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—82no—
248C4⋊C4—84no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—84no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C42—42no—
264C42—81no—
265C42—81no—
266C42—81no—
267C42—84no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—22no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23#38011yescharacteristic
299C23C2×#9911yescharacteristic
300C23#41011yescharacteristic
301C23#41611yescharacteristic
302C23#41911yescharacteristic
303C23C2×#10411yescharacteristic
304C23#42111yescharacteristic
305C23#46412yes—
306C23#47912yes—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—24no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—84no—
354C2×C4—81no—
355C2×C4—84no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—84no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—162no—
377C2×C4—162no—
378C2×C4—162no—
379C2×C4—162no—
380C2×C4#35312yes—
381C8—162no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—81no—
407C22—81no—
408C22—82no—
409C22—82no—
410C22#150011yescharacteristic
411C22#150811yescharacteristic
412C22#151511yescharacteristic
413C22#151611yescharacteristic
414C22#179911yescharacteristic
415C22#180511yescharacteristic
416C22#180611yescharacteristic
417C22#212111yescharacteristic
418C22#212311yescharacteristic
419C22C2×#35311yescharacteristic
420C22#212611yescharacteristic
421C4—22no—
422C4—44no—
423C4—42no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4—81no—
431C4—81no—
432C4—82no—
433C4—164no—
434C4—161no—
435C2—21no—
436C2—21no—
437C2—21no—
438C2—21no—
439C2—42no—
440C2—42no—
441C2#857411yescharacteristic
442C2#1178811yescharacteristic
443C2#1178911yescharacteristic
444C2#1184311yescharacteristic
445C2#1184611yescharacteristic
446C2#1279611yescharacteristic
447C2#1279711yescharacteristic
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
7. C2×#389C4semidirect
15. #1445C4semidirect
16. #1444C4semidirect
21. C4×#95C4semidirect

Maximal subgroups

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

subgroupindex
#142302
C2×#13522
#108012
#85572
#72502