Finite Group Databasebeta

← #117237

Subgroups — #117237

Order 512.

619 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17819C211yesmaximal, characteristic, complemented
2C2×#1553C211yesmaximal, characteristic
3#17544C211yesmaximal, characteristic
4#17384C211yesmaximal, characteristic
5#8071C211yesmaximal, characteristic, complemented
6#7866C211yesmaximal, characteristic, complemented
7#7863C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1574—22no—
16#1574—22no—
17#1568—22no—
18#1553—22no—
19#1553—22no—
20C2×#332C411yescharacteristic, complemented
21C2×#332C411yescharacteristic, complemented
22C2×#331C411yescharacteristic, complemented
23C2×#331C411yescharacteristic, complemented
24#1507—22no—
25#1505—22no—
26#1502—22no—
27#1471—22no—
28#1469—22no—
29#1460—22no—
30#1300—22no—
31#1276—22no—
32#1275—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—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432D412yes—
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#431D412yes—
69#431D412yes—
70#431D412yes—
71#423—21no—
72#423—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2×C411yescharacteristic
75C2×C4⋊C8C2×C411yescharacteristic
76C2×C4⋊C8C2×C411yescharacteristic
77C2×C4⋊C8C2×C411yescharacteristic
78#407—21no—
79#407—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C4×M16—21no—
84C4×M16—21no—
85C2×#97—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92#376D412yes—
93C43—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×C4C2×D411yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C2×M16—41no—
118C2×M16—41no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
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—
152#117—24no—
153#117—42no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4C4⋊C412yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—41no—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42—41no—
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—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8C22⋊C412yes—
188C4⋊C8C22⋊C412yes—
189C22⋊C8—22no—
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—
201#97—41no—
202#97—41no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—41no—
208C4×C8—41no—
209#95—22no—
210#95—22no—
211C24—22no—
212C24#12311yescharacteristic
213#41—82no—
214#41—82no—
215#41—82no—
216#41—82no—
217#41—84no—
218#41—84no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—81no—
222C2×Q8—81no—
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—81no—
234C2×D4—81no—
235C2×D4—82no—
236C2×D4—84no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—81no—
246C22×C4—84no—
247C22×C4C2×C4⋊C411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12411yescharacteristic
262C22×C4#12411yescharacteristic
263C22×C4#12611yescharacteristic
264C22×C4#12611yescharacteristic
265M16—82no—
266M16—82no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—81no—
277C2×C8—81no—
278C2×C8—82no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—44no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—24no—
290C42—42no—
291C42—81no—
292C42—81no—
293C42—82no—
294C42C2×C22⋊C411yescharacteristic
295C42#11711yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42#12011yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42C22⋊Q811yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42#12311yescharacteristic
308C42#12411yescharacteristic
309C42#12711yescharacteristic
310C23—22no—
311C23—41no—
312C23—41no—
313C23—42no—
314C23—44no—
315C23—82no—
316C23—82no—
317C23#38511yescharacteristic
318C23#39411yescharacteristic
319C23#39711yescharacteristic
320Q8—82no—
321Q8—81no—
322Q8—81no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—84no—
327D4—84no—
328D4—81no—
329D4—81no—
330D4—82no—
331D4—81no—
332D4—81no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4#38011yescharacteristic
363C2×C4#38511yescharacteristic
364C2×C4#38611yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#39011yescharacteristic
368C2×C4#39211yescharacteristic
369C2×C4#39311yescharacteristic
370C2×C4#39411yescharacteristic
371C2×C4#39611yescharacteristic
372C2×C4#39611yescharacteristic
373C2×C4#39911yescharacteristic
374C2×C4#44312yes—
375C2×C4#44412yes—
376C8—42no—
377C8—42no—
378C8—82no—
379C8—82no—
380C8—82no—
381C8—81no—
382C8—81no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—164no—
396C22#143911yescharacteristic
397C22#186612yes—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—84no—
408C4—82no—
409C4#190311yescharacteristic
410C4#190311yescharacteristic
411C4#192912yes—
412C4#197411yescharacteristic
413C4#197511yescharacteristic
414C2—22no—
415C2—41no—
416C2—41no—
417C2—162no—
418C2#1105811yescharacteristic
419C2#1112511yescharacteristic
420C2#1113811yescharacteristic
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. #17819C2semidirect
5. #8071C2semidirect
6. #7866C2semidirect
7. #7863C2semidirect
20. C2×#332C4semidirect
21. C2×#332C4semidirect
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
#178192
C2×#15532
#175442
#173842
#80712
#78662
#78632