Finite Group Databasebeta

← #117191

Subgroups — #117191

Order 512.

688 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21815C211yesmaximal, characteristic
2#21390C211yesmaximal, characteristic
3#17819C211yesmaximal, characteristic, complemented
4#17478C211yesmaximal, characteristic
5#8048C211yesmaximal, characteristic, complemented
6#7924C211yesmaximal, characteristic, complemented
7#7869C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C4×#117C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1987C411yescharacteristic, complemented
16#1987C411yescharacteristic, complemented
17#1982C411yescharacteristic, complemented
18#1982C411yescharacteristic, complemented
19#1980—22no—
20#1962—22no—
21#1875—22no—
22#1789—22no—
23C4×#102—22no—
24#1574—22no—
25#1574—22no—
26#1568—22no—
27#1496—22no—
28#1492—22no—
29#1478—22no—
30#1297—22no—
31#1282—22no—
32#1276—22no—
33C22×#41—21no—
34#532—21no—
35#532—21no—
36#532—21no—
37#532—21no—
38#532—21no—
39#532—21no—
40#532D412yes—
41#532Q812yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#127C2×C411yescharacteristic
44C2×#124—21no—
45C2×#121—21no—
46C2×#121—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×#117—21no—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431D412yes—
68#423D412yes—
69#423D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2×C411yescharacteristic
72C2×C4⋊C8C2×C411yescharacteristic
73C2×#104—28no—
74C2×#102—22no—
75C2×#102—22no—
76#407—21no—
77#407—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8D412yes—
80C4×M16—21no—
81C4×M16—21no—
82C2×C4×C8—21no—
83C2×C4×C8D412yes—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C411yescharacteristic
88C4×C4⋊C4C2×C411yescharacteristic
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4D412yes—
93#376—21no—
94#376—21no—
95C43—21no—
96C43—21no—
97#340—22no—
98#340—22no—
99#335—24no—
100#334—24no—
101D4⋊C8—28no—
102C2×#41—22no—
103C2×#41—22no—
104C2×#41—41no—
105C2×#41—41no—
106C2×#41—84no—
107C22×Q8—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—41no—
112C23×C4—41no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4#4111yescharacteristic
116C2×M16—22no—
117C2×M16—22no—
118C22×C8—41no—
119C22×C8C22⋊C412yes—
120C22×C8C22⋊C412yes—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8C4⋊C412yes—
125#127—21no—
126#127—21no—
127#127—21no—
128#127C4⋊C412yes—
129#124—22no—
130#124—22no—
131#124—41no—
132#124—41no—
133#121—22no—
134#121—22no—
135#121—22no—
136#121—22no—
137#121—41no—
138#121—41no—
139#121—41no—
140#121—41no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149#117—24no—
150#117—42no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×Q811yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—41no—
191#104—88no—
192#102—44no—
193#102—44no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202#97—41no—
203#97—41no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210#95—21no—
211#95—21no—
212#95—22no—
213#95—41no—
214#95—41no—
215C24—41no—
216C24#12111yescharacteristic
217#41—48no—
218#41—84no—
219#41—84no—
220#41—84no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—82no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—82no—
237C2×D4—84no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—84no—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12411yescharacteristic
266C22×C4#12611yescharacteristic
267C22×C4#12711yescharacteristic
268M16—44no—
269M16—82no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—81no—
278C2×C8—81no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—44no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C42—44no—
296C42—44no—
297C42—44no—
298C42—42no—
299C42—42no—
300C42—81no—
301C42—81no—
302C42—82no—
303C42C2×C4⋊C411yescharacteristic
304C42#11711yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42#12011yescharacteristic
308C42#12011yescharacteristic
309C42#12111yescharacteristic
310C42#12111yescharacteristic
311C42C22⋊Q811yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42#12311yescharacteristic
315C42#12311yescharacteristic
316C42#12311yescharacteristic
317C42#12411yescharacteristic
318C42#12611yescharacteristic
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23—82no—
325C23—82no—
326C23—81no—
327C23—81no—
328C23#39011yescharacteristic
329C23#39411yescharacteristic
330C23#39611yescharacteristic
331Q8—44no—
332Q8—82no—
333D4—44no—
334D4—44no—
335D4—44no—
336D4—82no—
337D4—84no—
338D4—84no—
339D4—81no—
340D4—81no—
341D4—82no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—84no—
376C2×C4#38011yescharacteristic
377C2×C4#38511yescharacteristic
378C2×C4#38511yescharacteristic
379C2×C4#38611yescharacteristic
380C2×C4#38811yescharacteristic
381C2×C4#38811yescharacteristic
382C2×C4#39211yescharacteristic
383C2×C4#39311yescharacteristic
384C2×C4#39411yescharacteristic
385C2×C4#39611yescharacteristic
386C2×C4#39711yescharacteristic
387C2×C4#39911yescharacteristic
388C2×C4#44312yes—
389C2×C4#44412yes—
390C8—42no—
391C8—42no—
392C8—82no—
393C8—82no—
394C8—82no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—162no—
409C22—162no—
410C22#143911yescharacteristic
411C22#197411yescharacteristic
412C22#197511yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—82no—
422C4—82no—
423C4—84no—
424C4—82no—
425C4#186612yes—
426C4#188411yescharacteristic
427C4#188411yescharacteristic
428C4#192912yes—
429C4#200812yes—
430C2—21no—
431C2—21no—
432C2—41no—
433C2—41no—
434C2—162no—
435C2#1105611yescharacteristic
436C2#1112511yescharacteristic
437C2#1112911yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17819C2semidirect
5. #8048C2semidirect
6. #7924C2semidirect
7. #7869C2semidirect
15. #1987C4semidirect
16. #1987C4semidirect
17. #1982C4semidirect
18. #1982C4semidirect

Maximal subgroups

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

subgroupindex
#218152
#213902
#178192
#174782
#80482
#79242
#78692