Finite Group Databasebeta

← #116182

Subgroups — #116182

Order 512.

749 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21932C211yesmaximal, characteristic
2C2×#1520C211yesmaximal, characteristic
3C2×#1508C211yesmaximal, characteristic
4#17397C211yesmaximal, characteristic
5#8107C211yesmaximal, characteristic
6#8057C211yesmaximal, characteristic
7#8019C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1851—22no—
16#1850—22no—
17#1797—22no—
18#1520—22no—
19#1520—22no—
20#1508—22no—
21#1508C412yescomplemented
22#1508C412yescomplemented
23C2×#328—22no—
24C2×#328—22no—
25#1472—22no—
26#1472—22no—
27#1463—22no—
28#1305—22no—
29#1298—22no—
30#1294—22no—
31C23×Q8Q811yescharacteristic
32C24×C4C2311yescharacteristic
33C22×M16—21no—
34C22×M16—21no—
35C22×M16—21no—
36C23×C8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C22⋊Q8D412yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4Q811yescharacteristic
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C42—21no—
56C2×C4⋊C8—24no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407—21no—
66#407—21no—
67#407C2×C412yes—
68#407D412yes—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8D412yes—
80C2×C22⋊C8D412yes—
81#380—21no—
82#380—21no—
83#380—21no—
84#380D412yes—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4C2×C411yescharacteristic
88C4×C22⋊C4C2×C411yescharacteristic
89C4×C22⋊C4D412yes—
90C2×#95—21no—
91C2×#95D412yes—
92#336—24no—
93#328—28no—
94#328—24no—
95#328—24no—
96#324—28no—
97C25#4111yescharacteristic
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—82no—
104C22×Q8C4⋊C412yes—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—41no—
109C23×C4—82no—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×Q811yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C23×C4#4111yescharacteristic
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—82no—
131C2×M16—82no—
132C2×M16—82no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—82no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C4⋊C412yes—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—82no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—82no—
188C4⋊C8—84no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205#95—22no—
206#95—22no—
207#95—41no—
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210C24—22no—
211C24—22no—
212C24—42no—
213C24—81no—
214C24C4×D411yescharacteristic
215C24#12111yescharacteristic
216C24C22⋊Q811yescharacteristic
217C24#12311yescharacteristic
218C24#12311yescharacteristic
219C24#12311yescharacteristic
220C24#12611yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—82no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C4⋊C411yescharacteristic
265C22×C4#11711yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4C22⋊Q811yescharacteristic
280C22×C4C22⋊Q811yescharacteristic
281C22×C4C22⋊Q811yescharacteristic
282C22×C4C22⋊Q811yescharacteristic
283C22×C4C22⋊Q811yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4#12411yescharacteristic
287C22×C4#12411yescharacteristic
288C22×C4#12411yescharacteristic
289C22×C4#12611yescharacteristic
290C22×C4#12711yescharacteristic
291M16—82no—
292M16—82no—
293M16—82no—
294M16—82no—
295M16—82no—
296M16—82no—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—82no—
304C2×C8—82no—
305C2×C8—82no—
306C2×C8—82no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C22⋊C4—24no—
318C22⋊C4—44no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C42—44no—
327C42—81no—
328C42—81no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—84no—
352C23—84no—
353C23#38511yescharacteristic
354C23#38611yescharacteristic
355C23#38811yescharacteristic
356C23#39411yescharacteristic
357C23#39611yescharacteristic
358C23#39711yescharacteristic
359C23#39911yescharacteristic
360C23#47112yes—
361C23#47312yes—
362Q8—82no—
363Q8—82no—
364Q8—82no—
365Q8—82no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—164no—
414C2×C4#38011yescharacteristic
415C2×C4#38511yescharacteristic
416C2×C4#38811yescharacteristic
417C2×C4#39011yescharacteristic
418C2×C4#39211yescharacteristic
419C2×C4#39311yescharacteristic
420C2×C4#39411yescharacteristic
421C2×C4#39611yescharacteristic
422C8—82no—
423C8—82no—
424C8—82no—
425C8—82no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—44no—
447C22—44no—
448C22—42no—
449C22—82no—
450C22—81no—
451C22—82no—
452C22—81no—
453C22#143911yescharacteristic
454C22#186112yes—
455C22#190812yes—
456C22#204611yescharacteristic
457C22#204711yescharacteristic
458C22#206412yes—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—44no—
467C4—82no—
468C4—82no—
469C4—82no—
470C4—162no—
471C4#203212yes—
472C2—22no—
473C2—21no—
474C2—22no—
475C2—22no—
476C2—21no—
477C2—41no—
478C2—41no—
479C2—41no—
480C2#1106211yescharacteristic
481C2#1108011yescharacteristic
482C2#1108911yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1508C4semidirect
22. #1508C4semidirect

Maximal subgroups

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

subgroupindex
#219322
C2×#15202
C2×#15082
#173972
#81072
#80572
#80192