Finite Group Databasebeta

← #117293

Subgroups — #117293

Order 512.

643 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21969C211yesmaximal, characteristic, complemented
2C2×#1551C211yesmaximal, characteristic
3#17544C211yesmaximal, characteristic
4#17463C211yesmaximal, characteristic
5#7949C211yesmaximal, characteristic, complemented
6#7921C211yesmaximal, characteristic, complemented
7#7891C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1950—22no—
16#1950—22no—
17#1810—22no—
18#1551—22no—
19#1551—22no—
20C2×#333C411yescharacteristic, complemented
21C2×#333C411yescharacteristic, complemented
22C2×#331C411yescharacteristic, complemented
23C2×#331C411yescharacteristic, complemented
24#1507—22no—
25#1505—22no—
26#1502—22no—
27#1484—22no—
28#1480—22no—
29#1478—22no—
30#1285—22no—
31#1281—22no—
32#1279—22no—
33C22×#41—21no—
34C23×C8—21no—
35#532—21no—
36#532—21no—
37#532—21no—
38#532—21no—
39#532—21no—
40#532—21no—
41#532D412yes—
42#532Q812yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#124—21no—
46C2×#121—21no—
47C2×#121—21no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431D412yes—
68C2×#108—24no—
69#423—21no—
70#423—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2×C411yescharacteristic
76C2×C4⋊C8C2×C411yescharacteristic
77C2×C4⋊C8C2×C411yescharacteristic
78C2×C4⋊C8C2×C411yescharacteristic
79C2×C4⋊C8D412yes—
80C2×C4⋊C8D412yes—
81#407—21no—
82#407D412yes—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C4×M16—21no—
86C4×M16D412yes—
87C2×#97—21no—
88C2×C4×C8—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4D412yes—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93#376—21no—
94C43—21no—
95#341—24no—
96#333—22no—
97#333—22no—
98#331—22no—
99#331—22no—
100#327—28no—
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×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C2×M16—22no—
117C2×M16—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—41no—
125C22×C8—82no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8C4⋊C412yes—
130#127—21no—
131#127—21no—
132#127—21no—
133#127C4⋊C412yes—
134#124—22no—
135#124—22no—
136#124—41no—
137#124—41no—
138#121—22no—
139#121—22no—
140#121—22no—
141#121—22no—
142#121—41no—
143#121—41no—
144#121—41no—
145#121—41no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154#117—24no—
155#117—42no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42—22no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×Q811yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177#108—84no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8C22⋊C412yes—
193C4⋊C8C22⋊C412yes—
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—22no—
203#97—41no—
204#97—41no—
205#97—41no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—41no—
209C4×C8—41no—
210#95—22no—
211#95—41no—
212C24—41no—
213C24#12411yescharacteristic
214#41—84no—
215#41—84no—
216#41—84no—
217#41—84no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—82no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—81no—
232C2×D4—81no—
233C2×D4—82no—
234C2×D4—84no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—84no—
245C22×C4#11711yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4#12411yescharacteristic
262C22×C4#12611yescharacteristic
263M16—42no—
264M16—42no—
265M16—42no—
266M16—42no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—44no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—82no—
278C2×C8—81no—
279C2×C8—81no—
280C2×C8—82no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C42—44no—
293C42—82no—
294C42—81no—
295C42—81no—
296C42C2×C22⋊C411yescharacteristic
297C42C2×C4⋊C411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12011yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42#12111yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42#12311yescharacteristic
309C42#12311yescharacteristic
310C42#12611yescharacteristic
311C42#12711yescharacteristic
312C23—22no—
313C23—22no—
314C23—41no—
315C23—42no—
316C23—82no—
317C23—82no—
318C23—82no—
319C23#38811yescharacteristic
320C23#39311yescharacteristic
321C23#39411yescharacteristic
322Q8—82no—
323Q8—82no—
324D4—44no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—84no—
329D4—84no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4#38011yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38611yescharacteristic
367C2×C4#38811yescharacteristic
368C2×C4#39011yescharacteristic
369C2×C4#39211yescharacteristic
370C2×C4#39411yescharacteristic
371C2×C4#39611yescharacteristic
372C2×C4#39611yescharacteristic
373C2×C4#39711yescharacteristic
374C2×C4#39911yescharacteristic
375C2×C4#44312yes—
376C2×C4#44412yes—
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—44no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—164no—
396C22#143911yescharacteristic
397C22#192912yes—
398C22#204812yes—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—84no—
409C4—82no—
410C4#186612yes—
411C4#190411yescharacteristic
412C4#190511yescharacteristic
413C4#197411yescharacteristic
414C4#197511yescharacteristic
415C2—22no—
416C2—22no—
417C2—41no—
418C2—162no—
419C2#1105711yescharacteristic
420C2#1112511yescharacteristic
421C2#1114211yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21969C2semidirect
5. #7949C2semidirect
6. #7921C2semidirect
7. #7891C2semidirect
20. C2×#333C4semidirect
21. C2×#333C4semidirect
22. C2×#331C4semidirect
23. C2×#331C4semidirect

Maximal subgroups

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

subgroupindex
#219692
C2×#15512
#175442
#174632
#79492
#79212
#78912