Finite Group Databasebeta

← #117121

Subgroups — #117121

Order 512.

798 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21195C211yesmaximal, characteristic
2C2×#1846C211yesmaximal, characteristic, complemented
3C2×#1517C212yesmaximal
4#8026C212yesmaximal, complemented
5#7849C211yesmaximal, characteristic, complemented
6C2×#532C2211yescharacteristic
7C2×#431C2212yes—
8C22×#107—21no—
9C22×#106—21no—
10C2×#416—21no—
11C2×#416—21no—
12C22×C4×C8C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2212yes—
15#1958—22no—
16#1957C412yescomplemented
17#1956C412yescomplemented
18#1846—24no—
19#1517—24no—
20#1517—24no—
21C2×#327—24no—
22#1294—24no—
23#1273—22no—
24C22×#41—21no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27#532—22no—
28#532—22no—
29#532—21no—
30#532—21no—
31#532D412yes—
32#532Q812yes—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×#127C2×C411yescharacteristic
35C2×#124—21no—
36C2×#121—22no—
37C2×C4×D4—22no—
38C2×#117—21no—
39C22×C4⋊C4—22no—
40C22×C42C2311yescharacteristic
41#431—22no—
42#431—22no—
43#431—22no—
44#431—22no—
45C2×#107—24no—
46C2×#107—42no—
47C2×#107—42no—
48C2×#106—24no—
49C2×#106—42no—
50C2×#106—42no—
51C2×C4⋊C8—22no—
52#416—42no—
53#416—42no—
54#416—42no—
55#416—42no—
56C2×#103—22no—
57C2×#103—22no—
58C2×#102—22no—
59C2×#102—22no—
60C2×C22⋊C8D414yes—
61C2×C4×C8—22no—
62C2×C4×C8—22no—
63C2×C4×C8—22no—
64C2×C4×C8C2×C412yes—
65C2×C4×C8D412yes—
66#384C2×C412yes—
67#382—22no—
68#382—22no—
69#382—22no—
70#382—21no—
71#382D414yes—
72#380—22no—
73C4×C4⋊C4—22no—
74C2×#95—22no—
75C2×#95—22no—
76#327—216no—
77C2×#41—22no—
78C2×#41—22no—
79C2×#41—41no—
80C2×#41—41no—
81C2×#41—41no—
82C2×#41—41no—
83C2×#41—41no—
84C2×#41—41no—
85C2×#41—41no—
86C2×#41—41no—
87C2×#41—41no—
88C2×#41—41no—
89C22×Q8—21no—
90C22×D4—21no—
91C22×D4—22no—
92C22×D4—41no—
93C23×C4—41no—
94C23×C4—42no—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
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—24no—
106C22×C8C22⋊C412yes—
107C22×C8C22⋊C412yes—
108C22×C8D812yes—
109C22×C8D812yes—
110C22×C8SD1612yes—
111C22×C8SD1612yes—
112C4⋊Q8—21no—
113C4⋊Q8—22no—
114C4⋊Q8C4⋊C412yes—
115#127—21no—
116#127—22no—
117#127C4⋊C412yes—
118#124—22no—
119#124—22no—
120#124—42no—
121#121—24no—
122#121—24no—
123#121—42no—
124#121—42no—
125C4×D4—24no—
126C4×D4—24no—
127C4×D4—42no—
128C4×D4—42no—
129#117—41no—
130#117—41no—
131#117—41no—
132#117—41no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—41no—
144C2×C42—41no—
145C2×C42C22×C411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D412yes—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×Q811yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4112yes—
153C2×C42#4112yes—
154#107—48no—
155#106—48no—
156C4⋊C8—42no—
157C4⋊C8—42no—
158#103—44no—
159#103—44no—
160#102—44no—
161#102—44no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C4×C8—24no—
165C4×C8—24no—
166C4×C8—42no—
167C4×C8—42no—
168#95—24no—
169#95—24no—
170#95—42no—
171#95—42no—
172#95—42no—
173C24—41no—
174C24#12711yescharacteristic
175#41—42no—
176#41—42no—
177#41—42no—
178#41—42no—
179#41—42no—
180#41—42no—
181#41—42no—
182#41—42no—
183#41—81no—
184#41—81no—
185#41—81no—
186#41—81no—
187#41—81no—
188#41—81no—
189#41—81no—
190#41—81no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—22no—
198C2×D4—24no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—84no—
208C2×D4—81no—
209C2×D4—81no—
210C2×D4—81no—
211C2×D4—81no—
212C22×C4—44no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—88no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C4×D412yes—
227C22×C4#12012yes—
228C22×C4#12011yescharacteristic
229C22×C4#12112yes—
230C22×C4#12112yes—
231C22×C4#12111yescharacteristic
232C22×C4C22⋊Q812yes—
233C22×C4C22⋊Q811yescharacteristic
234C22×C4#12312yes—
235C22×C4#12311yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×SD1611yescharacteristic
240C22×C4C2×SD1611yescharacteristic
241C22×C4AGL(1,ℤ/8)11yescharacteristic
242C22×C4AGL(1,ℤ/8)11yescharacteristic
243C22×C4#13711yescharacteristic
244C22×C4#13711yescharacteristic
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—48no—
258C2×C8—44no—
259C2×C8#10214yes—
260C2×C8#10214yes—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—84no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C42—81no—
275C42—81no—
276C42C2×C4⋊C411yescharacteristic
277C42#11711yescharacteristic
278C42C4×D412yes—
279C42#12111yescharacteristic
280C42#12112yes—
281C42C22⋊Q811yescharacteristic
282C42C22⋊Q812yes—
283C42#12312yes—
284C42#12412yes—
285C42#12612yes—
286C23—24no—
287C23—42no—
288C23—81no—
289C23—81no—
290C23—84no—
291C23#39011yescharacteristic
292C23#39211yescharacteristic
293C23#39911yescharacteristic
294C23#49211yescharacteristic
295C23#49311yescharacteristic
296C23#49611yescharacteristic
297C23#49711yescharacteristic
298Q8—42no—
299Q8—42no—
300Q8—81no—
301Q8—81no—
302D4—42no—
303D4—42no—
304D4—84no—
305D4—84no—
306D4—82no—
307D4—82no—
308D4—82no—
309D4—82no—
310D4—81no—
311D4—81no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—88no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—164no—
330C2×C4—164no—
331C2×C4#38011yescharacteristic
332C2×C4#38512yes—
333C2×C4#38611yescharacteristic
334C2×C4#38812yes—
335C2×C4#39311yescharacteristic
336C2×C4#39412yes—
337C2×C4#39612yes—
338C2×C4#39711yescharacteristic
339C2×C4C2×#10211yescharacteristic
340C2×C4C2×#10211yescharacteristic
341C2×C4#41711yescharacteristic
342C2×C4#41711yescharacteristic
343C2×C4#44711yescharacteristic
344C2×C4#44711yescharacteristic
345C2×C4#44811yescharacteristic
346C2×C4#44811yescharacteristic
347C2×C4#45911yescharacteristic
348C2×C4#45911yescharacteristic
349C2×C4#46111yescharacteristic
350C2×C4#46111yescharacteristic
351C2×C4#47411yescharacteristic
352C2×C4#47411yescharacteristic
353C2×C4#47611yescharacteristic
354C2×C4#47611yescharacteristic
355C2×C4#48011yescharacteristic
356C2×C4#48011yescharacteristic
357C2×C4#48111yescharacteristic
358C2×C4#48111yescharacteristic
359C2×C4#48611yescharacteristic
360C2×C4#48611yescharacteristic
361C2×C4#48911yescharacteristic
362C2×C4#48911yescharacteristic
363C8—24no—
364C8—24no—
365C8—84no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—82no—
373C22—84no—
374C22—162no—
375C22—162no—
376C22#143911yescharacteristic
377C22#195711yescharacteristic
378C22#195911yescharacteristic
379C22#197614yes—
380C22#201111yescharacteristic
381C22#201311yescharacteristic
382C22#209311yescharacteristic
383C22#209411yescharacteristic
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—82no—
389C4—164no—
390C4—162no—
391C4#180211yescharacteristic
392C4#180211yescharacteristic
393C4#187411yescharacteristic
394C4#187411yescharacteristic
395C4#202311yescharacteristic
396C4#202311yescharacteristic
397C4#207111yescharacteristic
398C4#207111yescharacteristic
399C2—21no—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—162no—
404C2#975612yes—
405C2#975812yes—
406C2#1104911yescharacteristic
407C2#1112612yes—
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1846C2semidirect
4. #8026C2semidirect
5. #7849C2semidirect
16. #1957C4semidirect
17. #1956C4semidirect

Maximal subgroups

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

subgroupindex
#211952
C2×#18462
C2×#15172
#80262
#78492