Finite Group Databasebeta

← #110918

Subgroups — #110918

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20546C211yesmaximal, characteristic
2C2×#1550C211yesmaximal, characteristic
3#17672C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#8097C211yesmaximal, characteristic
6#8082C211yesmaximal, characteristic
7#7990C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1771—22no—
16#1767—21no—
17C4×C4⋊C8—21no—
18#1752—21no—
19C8×C22⋊C4—21no—
20#1550—22no—
21#1550—21no—
22#1550—21no—
23C2×#333C412yescomplemented
24C2×#332C412yescomplemented
25#1537C414yescomplemented
26#1531—22no—
27#1528—22no—
28#1508—21no—
29#1508—21no—
30#1506C414yes—
31#1503—22no—
32#1303—22no—
33#1301—22no—
34#1290—22no—
35C22×M16—21no—
36C23×C8—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533C2×C412yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432C2×C412yes—
61#432C2×C412yes—
62#432D412yes—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2×C412yes—
79C2×C4⋊C8C2×C412yes—
80C2×C4⋊C8D412yes—
81C2×C4⋊C8D411yescharacteristic
82C2×C4⋊C8D411yescharacteristic
83#407—21no—
84#407—21no—
85C2×C22⋊C8—21no—
86C4×M16—21no—
87C4×M16—21no—
88C2×#97—22no—
89C2×#97D412yes—
90C2×C4×C8—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D412yes—
93C4×C22⋊C4—21no—
94C4×C22⋊C4D411yescharacteristic
95C4×C22⋊C4D411yescharacteristic
96#376C2×C412yes—
97C43—21no—
98#336—22no—
99#336—22no—
100#333—22no—
101#333—22no—
102#332—22no—
103#332—22no—
104#330—28no—
105#328—24no—
106#328—24no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—82no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—82no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8C4212yes—
131C4⋊Q8C4212yes—
132#125—22no—
133#125—22no—
134#125—41no—
135#125—41no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152#117—22no—
153#117—22no—
154#117—44no—
155#117—41no—
156#117—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—41no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—41no—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8C22⋊C412yes—
199C4⋊C8C22⋊C412yes—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—22no—
207#97—22no—
208#97—22no—
209#97—82no—
210#97C22⋊C414yes—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—82no—
214#95—22no—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217C24#12311yescharacteristic
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—81no—
223C2×Q8—81no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4#11711yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12411yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12611yescharacteristic
251C22×C4#12611yescharacteristic
252M16—82no—
253M16—81no—
254M16—81no—
255M16—82no—
256M16—82no—
257C2×C8—24no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—82no—
268C2×C8—82no—
269C2×C8—82no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—44no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C22⋊C4—24no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C42—41no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—44no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42C2×C4211yescharacteristic
297C42C2×C22⋊C411yescharacteristic
298C42C2×C22⋊C411yescharacteristic
299C42#11711yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42C4×D411yescharacteristic
303C42C4×D411yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42#12111yescharacteristic
307C42#12111yescharacteristic
308C42#12311yescharacteristic
309C42#12311yescharacteristic
310C42#12411yescharacteristic
311C42#12711yescharacteristic
312C23—22no—
313C23—22no—
314C23—41no—
315C23#38611yescharacteristic
316C23#38811yescharacteristic
317C23#38811yescharacteristic
318Q8—82no—
319Q8—82no—
320Q8—82no—
321Q8—81no—
322Q8—81no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4C4×C22⋊C411yescharacteristic
356C2×C4C4×C22⋊C411yescharacteristic
357C2×C4C4×C22⋊C411yescharacteristic
358C2×C4#38611yescharacteristic
359C2×C4#38611yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#38811yescharacteristic
362C2×C4#38811yescharacteristic
363C2×C4#38811yescharacteristic
364C2×C4#39011yescharacteristic
365C2×C4#39011yescharacteristic
366C2×C4#39011yescharacteristic
367C2×C4#47112yes—
368C2×C4#47312yes—
369C8—44no—
370C8—42no—
371C8—42no—
372C8—82no—
373C8—82no—
374C8—82no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—44no—
383C22#143711yescharacteristic
384C22#190812yes—
385C22#193212yes—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—44no—
396C4—82no—
397C4—82no—
398C4—82no—
399C4#190411yescharacteristic
400C4#190511yescharacteristic
401C4#193012yes—
402C4#197711yescharacteristic
403C4#197911yescharacteristic
404C2—22no—
405C2—22no—
406C2—41no—
407C2#1094211yescharacteristic
408C2#1094611yescharacteristic
409C2#1095411yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ 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
23. C2×#333C4semidirect
24. C2×#332C4semidirect
25. #1537C4semidirect

Maximal subgroups

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

subgroupindex
#205462
C2×#15502
#176722
#175452
#80972
#80822
#79902