Finite Group Databasebeta

← #117329

Subgroups — #117329

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17806C211yesmaximal, characteristic, complemented
2#17717C211yesmaximal, characteristic
3C2×#1502C211yesmaximal, characteristic
4#17384C211yesmaximal, characteristic
5#7950C211yesmaximal, characteristic, complemented
6#7892C211yesmaximal, characteristic, complemented
7#7884C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1574—22no—
16#1573—22no—
17#1564—22no—
18#1553—22no—
19#1549C411yescharacteristic, complemented
20#1549C411yescharacteristic, complemented
21#1545C411yescharacteristic, complemented
22#1545C411yescharacteristic, complemented
23#1540—22no—
24#1502—22no—
25#1502—22no—
26C2×#327—22no—
27#1471—22no—
28#1469—22no—
29#1460—22no—
30#1285—22no—
31#1279—22no—
32#1278—22no—
33C22×#41—21no—
34#532—21no—
35#532—21no—
36#532—21no—
37#532—21no—
38#532—21no—
39#532—21no—
40#532—21no—
41#532—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#127C2×C411yescharacteristic
44C2×#124—21no—
45C2×#121—21no—
46C2×#121D412yes—
47C2×C4×D4—21no—
48C2×C4×D4Q812yes—
49C22×C4⋊C4—21no—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432D412yes—
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#431—21no—
68#431C2×C411yescharacteristic
69#431C2×C411yescharacteristic
70#431D412yes—
71#431D412yes—
72#431D412yes—
73#423—21no—
74#423C2×C411yescharacteristic
75#423C2×C411yescharacteristic
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8D412yes—
79#407—21no—
80#407—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C4×M16—21no—
85C4×M16—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D412yes—
93C2×#95—21no—
94#335—24no—
95#334—24no—
96#332—22no—
97#332—22no—
98#331—22no—
99#331—22no—
100#327—28no—
101#323—28no—
102C2×#41—41no—
103C2×#41—41no—
104C2×#41—41no—
105C2×#41—41no—
106C2×#41—82no—
107C2×#41—82no—
108C22×Q8—21no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4C4⋊C412yes—
113C23×C4—22no—
114C23×C4—41no—
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C2×M16—21no—
119C2×M16—21no—
120C2×M16—41no—
121C22×C8—22no—
122C22×C8—41no—
123C22×C8—41no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128#127—21no—
129#127—21no—
130#127—21no—
131#127—21no—
132#124—41no—
133#124—41no—
134#124—41no—
135#124—41no—
136#121—22no—
137#121—22no—
138#121—22no—
139#121—22no—
140#121—41no—
141#121—41no—
142#121—41no—
143#121—41no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4C4⋊C412yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42—82no—
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×Q811yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8C22⋊C412yes—
187C4⋊C8C22⋊C412yes—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#97—22no—
200#97—41no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210#95—22no—
211#95—41no—
212C24—22no—
213C24#12311yescharacteristic
214#41—82no—
215#41—82no—
216#41—84no—
217#41—84no—
218#41—82no—
219#41—82no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—82no—
236C2×D4—82no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—84no—
247C22×C4—81no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12411yescharacteristic
266C22×C4#12611yescharacteristic
267C22×C4#12611yescharacteristic
268M16—42no—
269M16—42no—
270M16—82no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—82no—
281C2×C8—82no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C42—82no—
294C42—82no—
295C42C2×C22⋊C411yescharacteristic
296C42#11711yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42#12011yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42#12311yescharacteristic
309C42#12411yescharacteristic
310C42#12711yescharacteristic
311C23—22no—
312C23—21no—
313C23—21no—
314C23—41no—
315C23—42no—
316C23—44no—
317C23—82no—
318C23—82no—
319C23#38511yescharacteristic
320C23#39411yescharacteristic
321C23#39711yescharacteristic
322Q8—82no—
323Q8—82no—
324D4—82no—
325D4—82no—
326D4—82no—
327D4—84no—
328D4—82no—
329D4—82no—
330D4—82no—
331D4—82no—
332D4—81no—
333D4—81no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—88no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4#38011yescharacteristic
362C2×C4#38511yescharacteristic
363C2×C4#38611yescharacteristic
364C2×C4#38811yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#39011yescharacteristic
367C2×C4#39211yescharacteristic
368C2×C4#39311yescharacteristic
369C2×C4#39411yescharacteristic
370C2×C4#39611yescharacteristic
371C2×C4#39611yescharacteristic
372C2×C4#39911yescharacteristic
373C2×C4#44312yes—
374C2×C4#44412yes—
375C2×C4#47112yes—
376C2×C4#47212yes—
377C8—42no—
378C8—42no—
379C8—82no—
380C8—82no—
381C8—82no—
382C8—81no—
383C8—81no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—164no—
397C22#143911yescharacteristic
398C22#186612yes—
399C22#204511yescharacteristic
400C22#204611yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—84no—
410C4—82no—
411C4#190712yes—
412C4#192912yes—
413C4#197411yescharacteristic
414C4#197511yescharacteristic
415C4#206312yes—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—41no—
420C2—162no—
421C2#1106111yescharacteristic
422C2#1112511yescharacteristic
423C2#1113811yescharacteristic
subgroupG/H
centerC2×C4#396
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
1. #17806C2semidirect
5. #7950C2semidirect
6. #7892C2semidirect
7. #7884C2semidirect
19. #1549C4semidirect
20. #1549C4semidirect
21. #1545C4semidirect
22. #1545C4semidirect

Maximal subgroups

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

subgroupindex
#178062
#177172
C2×#15022
#173842
#79502
#78922
#78842