Finite Group Databasebeta

← #101179

Subgroups — #101179

Order 512.

658 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 116 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#21930C211yesmaximal, characteristic
2#15161C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7643C212yesmaximal, complemented
5#7039C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#399C411yescharacteristic
8C2×#393C411yescharacteristic
9C2×#391C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1850—24no—
12#1797—21no—
13#1797C2212yes—
14#1797C2212yes—
15C2×#323—21no—
16#1446C411yescharacteristic
17#1445C411yescharacteristic
18#1441—21no—
19#1439—21no—
20#1436—21no—
21C4⋊#95C2211yescharacteristic
22C22⋊#95—21no—
23#1280—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C23×C8—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42C2311yescharacteristic
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—42no—
44C2×C22⋊C8—42no—
45#399—42no—
46#393—42no—
47#391—22no—
48#391—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
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×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#336—22no—
64#336—22no—
65#336D414yes—
66#323—42no—
67C25C22⋊C411yescharacteristic
68C22×Q8—22no—
69C22×Q8—42no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—81no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4SD1612yes—
87C23×C4Q1612yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—24no—
92C22×C8—44no—
93C22×C8—42no—
94C22×C8—82no—
95C22×C8—82no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4—81no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—81no—
115C2×C22⋊C4—81no—
116C2×C22⋊C4SD1612yes—
117C2×C22⋊C4Q1612yes—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—81no—
121C2×C42—81no—
122C2×C42C2×D412yes—
123C2×C42#4112yes—
124C4⋊C8—82no—
125C4⋊C8—82no—
126C4⋊C8—82no—
127C22⋊C8—44no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132#95—22no—
133#95—22no—
134#95—41no—
135#95—41no—
136#95—42no—
137#95—42no—
138#95—41no—
139#95—42no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—81no—
144#95—81no—
145#95—81no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24#10012yes—
155C24#10312yes—
156C24#10312yes—
157C24C2×C22⋊C411yescharacteristic
158C2×Q8—24no—
159C2×Q8—42no—
160C2×Q8—82no—
161C2×Q8—82no—
162C2×Q8—84no—
163C2×Q8—84no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
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—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4#10312yes—
226C22×C4#10312yes—
227C22×C4C4×D412yes—
228C22×C4#12011yescharacteristic
229C22×C4#12112yes—
230C22×C4#12311yescharacteristic
231C22×C4C2×SD1611yescharacteristic
232C22×C4C2×SD1611yescharacteristic
233C22×C4C2×Q1611yescharacteristic
234C22×C4C2×Q1611yescharacteristic
235C22×C4#13512yes—
236C22×C4AGL(1,ℤ/8)11yescharacteristic
237C22×C4#13711yescharacteristic
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—88no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—82no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—44no—
251C22⋊C4—84no—
252C22⋊C4—84no—
253C42—42no—
254C42—42no—
255C42—42no—
256C42—81no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23#33314yes—
284C23#38611yescharacteristic
285C23C2×#10011yescharacteristic
286C23#41311yescharacteristic
287C23C2×#10311yescharacteristic
288C23C2×#10311yescharacteristic
289C23#41611yescharacteristic
290C23#41711yescharacteristic
291Q8—84no—
292Q8—84no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
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—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
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—81no—
347C2×C4—81no—
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#45011yescharacteristic
359C2×C4#45111yescharacteristic
360C2×C4#46512yes—
361C2×C4#46712yes—
362C2×C4#48311yescharacteristic
363C2×C4#48411yescharacteristic
364C8—84no—
365C8—162no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22C2×#33311yescharacteristic
390C22#154211yescharacteristic
391C22#155011yescharacteristic
392C22#155111yescharacteristic
393C22#187211yescharacteristic
394C22#188411yescharacteristic
395C22#189111yescharacteristic
396C22#189211yescharacteristic
397C22#190011yescharacteristic
398C22#191111yescharacteristic
399C22#191311yescharacteristic
400C4—22no—
401C4—22no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—82no—
410C4—82no—
411C4—81no—
412C4—161no—
413C4—161no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—44no—
419C2#903311yescharacteristic
420C2#912711yescharacteristic
421C2#919911yescharacteristic
422C2#938311yescharacteristic
423C2#944711yescharacteristic
424C2#948311yescharacteristic
425C2#951511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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
4. #7643C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219302
#151612
#85572
#76432
#70392