Finite Group Databasebeta

← #116128

Subgroups — #116128

Order 512.

721 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21934C211yesmaximal, characteristic
2#17578C211yesmaximal, characteristic
3C2×#1508C211yesmaximal, characteristic
4#17397C211yesmaximal, characteristic
5#8115C211yesmaximal, characteristic
6#8051C211yesmaximal, characteristic
7#8009C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1851—22no—
16#1851—22no—
17#1797—22no—
18#1520—22no—
19#1511—22no—
20#1508—22no—
21#1508—22no—
22#1508—22no—
23C2×#328C412yescomplemented
24C2×#328C412yescomplemented
25#1472—22no—
26#1472—22no—
27#1463—22no—
28#1306—22no—
29#1297—22no—
30#1293—22no—
31C23×Q8Q811yescharacteristic
32C24×C4C2311yescharacteristic
33C22×M16—21no—
34C22×M16—21no—
35C22×M16—21no—
36C22×M16—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—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4Q811yescharacteristic
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C42—21no—
55C2×C4⋊C8—24no—
56#407—21no—
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#407—21no—
68#407—21no—
69#407—21no—
70#407—21no—
71#407D412yes—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8C2×C412yes—
78C2×C22⋊C8D412yes—
79C2×C22⋊C8D412yes—
80#381D412yes—
81#380—21no—
82#380—21no—
83#379—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4D412yes—
89C4×C22⋊C4D412yes—
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C411yescharacteristic
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—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
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—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—82no—
133C2×M16—82no—
134C2×M16—82no—
135C2×M16—82no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—41no—
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—
156#117—82no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4C4⋊C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—41no—
185C2×C42—82no—
186C4⋊C8—84no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203#95—21no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95C22⋊C412yes—
208#95C22⋊C412yes—
209C24—42no—
210C24—41no—
211C24—41no—
212C24—81no—
213C24C4×D411yescharacteristic
214C24#12111yescharacteristic
215C24C22⋊Q811yescharacteristic
216C24#12311yescharacteristic
217C24#12311yescharacteristic
218C24#12311yescharacteristic
219C24#12611yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C22×C4—22no—
231C22×C4—24no—
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—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—82no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—82no—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4#11711yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4C22⋊Q811yescharacteristic
279C22×C4C22⋊Q811yescharacteristic
280C22×C4C22⋊Q811yescharacteristic
281C22×C4C22⋊Q811yescharacteristic
282C22×C4#12311yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4#12411yescharacteristic
285C22×C4#12411yescharacteristic
286C22×C4#12411yescharacteristic
287C22×C4#12611yescharacteristic
288C22×C4#12711yescharacteristic
289M16—82no—
290M16—82no—
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—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—82no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—82no—
321C22⋊C4—81no—
322C22⋊C4—81no—
323C22⋊C4—82no—
324C42—44no—
325C42—82no—
326C42—82no—
327C23—22no—
328C23—22no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
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
360Q8—82no—
361Q8—82no—
362Q8—82no—
363Q8—82no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—41no—
373C2×C4—41no—
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—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—84no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—84no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4#38011yescharacteristic
412C2×C4#38511yescharacteristic
413C2×C4#38811yescharacteristic
414C2×C4#39011yescharacteristic
415C2×C4#39211yescharacteristic
416C2×C4#39311yescharacteristic
417C2×C4#39411yescharacteristic
418C2×C4#39611yescharacteristic
419C8—82no—
420C8—82no—
421C8—82no—
422C8—82no—
423C22—21no—
424C22—21no—
425C22—21no—
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—42no—
443C22—42no—
444C22—44no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22#143911yescharacteristic
453C22#186112yes—
454C22#190211yescharacteristic
455C22#190211yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—44no—
464C4—82no—
465C4—82no—
466C4—82no—
467C4—162no—
468C4#203212yes—
469C2—21no—
470C2—21no—
471C2—22no—
472C2—41no—
473C2—41no—
474C2—41no—
475C2—41no—
476C2—41no—
477C2#1105911yescharacteristic
478C2#1108011yescharacteristic
479C2#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
23. C2×#328C4semidirect
24. C2×#328C4semidirect

Maximal subgroups

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

subgroupindex
#219342
#175782
C2×#15082
#173972
#81152
#80512
#80092