Finite Group Databasebeta

← #120702

Subgroups — #120702

Order 512.

526 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1564C211yesmaximal, characteristic
2#10694C211yesmaximal, characteristic
3C2×#1281C211yesmaximal, characteristic
4C2×#1281C211yesmaximal, characteristic
5#7872C211yesmaximal, characteristic
6C2×#1274C211yesmaximal, characteristic
7C2×#1274C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×#97C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#376C2211yescharacteristic
15C2×#336C411yescharacteristic
16C2×#336C411yescharacteristic
17C2×#336C411yescharacteristic
18C2×#336C411yescharacteristic
19#1564—22no—
20#1564—22no—
21C2×#335—21no—
22C2×#334—21no—
23#1281—22no—
24#1281—22no—
25#1277—21no—
26#1277—21no—
27#1274—22no—
28#1274—22no—
29C23×C8C2×C411yescharacteristic
30C23×C8C2×C411yescharacteristic
31C23×C8C2×C411yescharacteristic
32C23×C8C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42C2311yescharacteristic
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431D412yes—
49#431D412yes—
50#431D412yes—
51#431Q812yes—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×#97—21no—
61C2×#97—21no—
62C2×#97—21no—
63C2×#97—21no—
64C2×#97—21no—
65C2×#97—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8Q811yescharacteristic
77C2×C4×C8Q811yescharacteristic
78#376—21no—
79#376—21no—
80#376—21no—
81#376—21no—
82C2×#95—21no—
83C2×#95—21no—
84#336—22no—
85#336—22no—
86#336—22no—
87#336C814yescomplemented
88#335—24no—
89#334—24no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4C22×C411yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8C4⋊C411yescharacteristic
122C22×C8C4⋊C411yescharacteristic
123C22×C8C4⋊C411yescharacteristic
124C22×C8C4⋊C411yescharacteristic
125C22×C8C2×C812yes—
126C22×C8C2×C812yes—
127C22×C8M1612yes—
128C22×C8M1612yes—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42C2×C812yes—
140C2×C42M1612yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×Q811yescharacteristic
146C2×C42C2×Q811yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4111yescharacteristic
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—41no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C22⋊C8—41no—
169#97—41no—
170#97—41no—
171#97—41no—
172#97—41no—
173C4×C8—21no—
174C4×C8—21no—
175C4×C8—22no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—22no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—41no—
186C4×C8—41no—
187C4×C8—41no—
188C4×C8—41no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193C24#11711yescharacteristic
194C22×C4—22no—
195C22×C4—22no—
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×C4C2×C4⋊C411yescharacteristic
213C22×C4#11711yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×Q811yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12411yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4#12411yescharacteristic
225C22×C4#12511yescharacteristic
226C22×C4#12511yescharacteristic
227C22×C4#12511yescharacteristic
228C22×C4#12611yescharacteristic
229C22×C4#12611yescharacteristic
230C22×C4C22×C811yescharacteristic
231C22×C4C2×M1611yescharacteristic
232C22×C4C2×M1611yescharacteristic
233C22×C4C2×M1611yescharacteristic
234C22×C4#13111yescharacteristic
235C22×C4#13111yescharacteristic
236C22×C4#13111yescharacteristic
237C22×C4#13111yescharacteristic
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—81no—
275C2×C8—81no—
276C2×C8—81no—
277C2×C8—81no—
278C2×C8—81no—
279C2×C8—81no—
280C2×C8—81no—
281C2×C8—81no—
282C2×C8C4⋊C812yes—
283C2×C8C4⋊C812yes—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—41no—
292C42#12011yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42#12311yescharacteristic
300C42#12311yescharacteristic
301C42#12311yescharacteristic
302C42#12411yescharacteristic
303C42#12511yescharacteristic
304C42#12511yescharacteristic
305C42#12611yescharacteristic
306C42#12611yescharacteristic
307C42C4⋊Q811yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23#38211yescharacteristic
313C23#38711yescharacteristic
314C23#38811yescharacteristic
315C23#43111yescharacteristic
316C23#43211yescharacteristic
317C23#43311yescharacteristic
318C23#43311yescharacteristic
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
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#38511yescharacteristic
350C2×C4#38711yescharacteristic
351C2×C4#38811yescharacteristic
352C2×C4#38911yescharacteristic
353C2×C4#39311yescharacteristic
354C2×C4#39411yescharacteristic
355C2×C4#39711yescharacteristic
356C2×C4#39811yescharacteristic
357C2×C4#39811yescharacteristic
358C2×C4#40011yescharacteristic
359C2×C4#40011yescharacteristic
360C2×C4#40011yescharacteristic
361C2×C4C2×C4⋊C811yescharacteristic
362C2×C4#42311yescharacteristic
363C2×C4#42411yescharacteristic
364C2×C4#42411yescharacteristic
365C2×C4#43111yescharacteristic
366C2×C4#43211yescharacteristic
367C2×C4#43311yescharacteristic
368C2×C4#43311yescharacteristic
369C2×C4C8×D411yescharacteristic
370C2×C4C8×D411yescharacteristic
371C2×C4C8×D411yescharacteristic
372C2×C4#43511yescharacteristic
373C2×C4#43511yescharacteristic
374C2×C4#43511yescharacteristic
375C2×C4#43511yescharacteristic
376C2×C4#43511yescharacteristic
377C2×C4#43511yescharacteristic
378C2×C4#43611yescharacteristic
379C2×C4#43611yescharacteristic
380C2×C4#43611yescharacteristic
381C2×C4C8×Q811yescharacteristic
382C2×C4#44611yescharacteristic
383C2×C4#44611yescharacteristic
384C2×C4#44611yescharacteristic
385C8—22no—
386C8—22no—
387C8—41no—
388C8—41no—
389C8—42no—
390C8—42no—
391C8—82no—
392C8—82no—
393C8—82no—
394C8—82no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22#144211yescharacteristic
408C22#183111yescharacteristic
409C22#183211yescharacteristic
410C22#183311yescharacteristic
411C22#183311yescharacteristic
412C22#183311yescharacteristic
413C22#183311yescharacteristic
414C22#191611yescharacteristic
415C22#191711yescharacteristic
416C22#192311yescharacteristic
417C22#192411yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—82no—
427C4—82no—
428C4#185011yescharacteristic
429C4#185111yescharacteristic
430C4#191611yescharacteristic
431C4#191711yescharacteristic
432C4#192311yescharacteristic
433C4#192411yescharacteristic
434C4#193911yescharacteristic
435C4#194011yescharacteristic
436C2—21no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2#878011yescharacteristic
441C2#878111yescharacteristic
442C2#878211yescharacteristic
443C2#878311yescharacteristic
444C2#1125511yescharacteristic
445C2#1125911yescharacteristic
446C2#1126011yescharacteristic
subgroupG/H
centerC22×C4#117
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#382
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#376 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
87. #336C8semidirect

Maximal subgroups

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

subgroupindex
C2×#15642
#106942
C2×#12812
#78722
C2×#12742