Finite Group Databasebeta

← #100136

Subgroups — #100136

Order 512.

688 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 106 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#21912C211yesmaximal, characteristic, complemented
2#14001C211yesmaximal, characteristic
3#8701C211yesmaximal, characteristic, complemented
4#7592C212yesmaximal
5#7017C212yesmaximal, complemented
6C2×#390C411yescharacteristic
7C2×#390C411yescharacteristic
8C2×#390C2211yescharacteristic
9C2×#388—21no—
10C2×#388—21no—
11C2×#384C411yescharacteristic
12C2×C4×C4⋊C4C2211yescharacteristic
13#1916—24no—
14#1817C2212yes—
15#1817C2212yes—
16#1797—21no—
17C4×C22⋊C8—21no—
18#1439—21no—
19#1439—21no—
20C4⋊#95C411yescharacteristic
21C4×#95C2211yescharacteristic
22#1280—21no—
23#1280—21no—
24#1270—24no—
25C23×D4C2×C411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42C2311yescharacteristic
42C2×C4⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C4×C8—22no—
46#390—22no—
47#390—22no—
48#390—22no—
49#390—42no—
50#390C814yescomplemented
51#388—44no—
52#388—44no—
53#384—42no—
54C4×C4⋊C4—22no—
55C4×C4⋊C4—22no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95D411yescharacteristic
61C2×#95D411yescharacteristic
62#336—22no—
63#336—22no—
64#336—22no—
65#336—22no—
66#336—22no—
67#336D414yes—
68C25—21no—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4—22no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4C2×C812yes—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—24no—
88C22×C8—42no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4C2×C812yes—
104C2×C4⋊C4M1612yes—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—81no—
121C2×C22⋊C4—81no—
122C2×C22⋊C4—81no—
123C2×C22⋊C4—81no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—22no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42C2×C812yes—
139C2×C42C2×D412yes—
140C2×C42#4112yes—
141C4⋊C8—82no—
142C22⋊C8—42no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146C4×C8—82no—
147#95—24no—
148#95—22no—
149#95—22no—
150#95—42no—
151#95—42no—
152#95—42no—
153C24—22no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24C2×C22⋊C411yescharacteristic
169C2×D4—24no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—82no—
176C2×D4—82no—
177C2×D4—82no—
178C2×D4—82no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
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—41no—
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×C4C22⋊C812yes—
230C22×C4C22⋊C812yes—
231C22×C4#10012yes—
232C22×C4C4×D412yes—
233C22×C4#12011yescharacteristic
234C22×C4#12112yes—
235C22×C4#12311yescharacteristic
236C22×C4C22×C811yescharacteristic
237C22×C4C2×M1611yescharacteristic
238C22×C4#13112yes—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—44no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C22⋊C4—48no—
252C22⋊C4—48no—
253C22⋊C4—82no—
254C22⋊C4—84no—
255C22⋊C4—84no—
256C22⋊C4—84no—
257C22⋊C4—82no—
258C42—42no—
259C42—42no—
260C42—42no—
261C42—42no—
262C42—42no—
263C42—42no—
264C42—81no—
265C42—81no—
266C42—82no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23#38611yescharacteristic
307C23C2×C22⋊C811yescharacteristic
308C23#40811yescharacteristic
309C23#41011yescharacteristic
310C23#41011yescharacteristic
311C23C2×#10011yescharacteristic
312C23#41311yescharacteristic
313D4—84no—
314D4—84no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
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—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
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—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4C8×D412yes—
377C2×C4#43512yes—
378C2×C4#45312yes—
379C8—84no—
380C8—162no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22#145811yescharacteristic
409C22#146011yescharacteristic
410C22#146911yescharacteristic
411C22#147311yescharacteristic
412C22#186911yescharacteristic
413C22#188411yescharacteristic
414C22#188711yescharacteristic
415C22#189911yescharacteristic
416C22#190011yescharacteristic
417C22#191011yescharacteristic
418C22#191211yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—44no—
423C4—42no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—42no—
428C4—82no—
429C4—81no—
430C4—161no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—162no—
436C2#895911yescharacteristic
437C2#908011yescharacteristic
438C2#910211yescharacteristic
439C2#944611yescharacteristic
440C2#944811yescharacteristic
441C2#945711yescharacteristic
442C2#946411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×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
1. #21912C2semidirect
3. #8701C2semidirect
5. #7017C2semidirect
50. #390C8semidirect

Maximal subgroups

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

subgroupindex
#219122
#140012
#87012
#75922
#70172