Finite Group Databasebeta

← #99740

Subgroups — #99740

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1831C211yesmaximal, characteristic
2#14703C211yesmaximal, characteristic
3#10795C211yesmaximal, characteristic
4C2×#1272C212yesmaximal
5#7025C212yesmaximal
6C22×C4⋊C8—21no—
7C2×#391C411yescharacteristic
8C2×#391C411yescharacteristic
9C2×#391C2211yescharacteristic
10C2×#389C411yescharacteristic
11C2×C43C2211yescharacteristic
12#1831—24no—
13C2×#336—21no—
14C2×#336—21no—
15C2×#336—21no—
16C2×#336C2212yes—
17C2×#336C2212yes—
18#1436—21no—
19#1436—21no—
20C4⋊#95C411yescharacteristic
21C4⋊#95C2211yescharacteristic
22#1434—21no—
23#1434—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C23×C8—22no—
27C23×C8—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42C2311yescharacteristic
39C2×C4⋊C8—24no—
40C2×C4⋊C8—42no—
41C2×C4⋊C8—42no—
42#391—22no—
43#391—22no—
44#391—22no—
45#391—42no—
46#391C814yescomplemented
47#389—42no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C43—22no—
60C43—22no—
61#336—24no—
62#336—24no—
63#336—22no—
64#336—22no—
65#336—22no—
66#336—42no—
67#336D414yes—
68C22×Q8—21no—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8C2×C812yes—
76C23×C4—21no—
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—24no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8—42no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8—42no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×C812yes—
114C2×C42—22no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—22no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—21no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—22no—
129C2×C42—21no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42C2×C812yes—
137C2×C42M1612yes—
138C2×C42SD1612yes—
139C2×C42Q1612yes—
140C2×C42C2×D412yes—
141C2×C42#4112yes—
142C4⋊C8—48no—
143#95—24no—
144#95—24no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—42no—
150#95—41no—
151#95—42no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157C24C2×C22⋊C411yescharacteristic
158C2×Q8—24no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—82no—
165C2×Q8—82no—
166C2×Q8—82no—
167C2×Q8—82no—
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—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
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×C4—41no—
230C22×C4C22⋊C812yes—
231C22×C4C22⋊C812yes—
232C22×C4#10012yes—
233C22×C4#10312yes—
234C22×C4#10312yes—
235C22×C4#10412yes—
236C22×C4C4×D412yes—
237C22×C4#12011yescharacteristic
238C22×C4#12112yes—
239C22×C4#12311yescharacteristic
240C22×C4C22×C811yescharacteristic
241C22×C4C2×M1611yescharacteristic
242C22×C4#13112yes—
243C22×C4C2×SD1611yescharacteristic
244C22×C4C2×Q1611yescharacteristic
245C22×C4AGL(1,ℤ/8)11yescharacteristic
246C22×C4AGL(1,ℤ/8)11yescharacteristic
247C22×C4#13711yescharacteristic
248C22×C4#13712yes—
249C22×C4#13711yescharacteristic
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—82no—
257C2×C8—82no—
258C2×C8—82no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—82no—
263C2×C8—82no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—84no—
269C42—24no—
270C42—24no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—42no—
276C42—42no—
277C42—44no—
278C42—44no—
279C42—42no—
280C42—42no—
281C42—42no—
282C42—42no—
283C23#38611yescharacteristic
284C23C2×C22⋊C811yescharacteristic
285C23#40811yescharacteristic
286C23#41011yescharacteristic
287C23#41011yescharacteristic
288C23C2×#10011yescharacteristic
289C23#41311yescharacteristic
290C23C2×#10311yescharacteristic
291C23#41711yescharacteristic
292C23#41711yescharacteristic
293C23#41911yescharacteristic
294C23C2×#10411yescharacteristic
295C23#42111yescharacteristic
296C23#42111yescharacteristic
297C23#42111yescharacteristic
298Q8—84no—
299Q8—84no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
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×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4Q8⋊C814yes—
384C2×C4C8×D412yes—
385C2×C4#43512yes—
386C2×C4#45011yescharacteristic
387C2×C4#45111yescharacteristic
388C2×C4#46812yes—
389C2×C4#47012yes—
390C2×C4#48311yescharacteristic
391C2×C4#48411yescharacteristic
392C8—84no—
393C8—84no—
394C22#145811yescharacteristic
395C22#146011yescharacteristic
396C22#146911yescharacteristic
397C22#147311yescharacteristic
398C22C2×Q8⋊C811yescharacteristic
399C22#147711yescharacteristic
400C22#147711yescharacteristic
401C22#147911yescharacteristic
402C22#148111yescharacteristic
403C22#148311yescharacteristic
404C22#149511yescharacteristic
405C22#149711yescharacteristic
406C22#150611yescharacteristic
407C22#150911yescharacteristic
408C22#151211yescharacteristic
409C22#151211yescharacteristic
410C22#151211yescharacteristic
411C22#151511yescharacteristic
412C22#151811yescharacteristic
413C22#151811yescharacteristic
414C22#152911yescharacteristic
415C22#153011yescharacteristic
416C22#153311yescharacteristic
417C22#153511yescharacteristic
418C22#154811yescharacteristic
419C22#155011yescharacteristic
420C22#155911yescharacteristic
421C22#156011yescharacteristic
422C22#186911yescharacteristic
423C22#187111yescharacteristic
424C22#187211yescharacteristic
425C22#189411yescharacteristic
426C22#189711yescharacteristic
427C22#189911yescharacteristic
428C22#190011yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—24no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—82no—
442C4—82no—
443C4—161no—
444C4—161no—
445C2#757711yescharacteristic
446C2#759011yescharacteristic
447C2#759611yescharacteristic
448C2#760111yescharacteristic
449C2#769611yescharacteristic
450C2#770911yescharacteristic
451C2#772211yescharacteristic
452C2#773311yescharacteristic
453C2#893711yescharacteristic
454C2#907311yescharacteristic
455C2#908011yescharacteristic
456C2#911611yescharacteristic
457C2#912411yescharacteristic
458C2#913511yescharacteristic
459C2#915211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
46. #391C8semidirect

Maximal subgroups

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

subgroupindex
C2×#18312
#147032
#107952
C2×#12722
#70252