Finite Group Databasebeta

← #119618

Subgroups — #119618

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20727C211yesmaximal, characteristic
2C2×#1763C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4#8098C211yesmaximal, characteristic
5#8088C211yesmaximal, characteristic
6#8088C211yesmaximal, characteristic
7#7950C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C22×#103—21no—
14C22×#97C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1821—21no—
17#1801C412yescomplemented
18#1801C412yescomplemented
19#1784—21no—
20#1763—22no—
21#1763—22no—
22C4×#101—22no—
23C2×#340—21no—
24#1491—22no—
25#1487—22no—
26#1479—22no—
27#1303—22no—
28#1302—22no—
29#1302—22no—
30#1285—22no—
31C23×Q8D411yescharacteristic
32C23×C8—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8D412yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4D411yescharacteristic
48C22×C42—21no—
49C22×C42—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#431D412yes—
67#423—21no—
68#423—21no—
69#423C2×C412yes—
70#423C2×C411yescharacteristic
71#423C2×C411yescharacteristic
72C2×#103—28no—
73C2×#103—24no—
74C2×#101—22no—
75C2×#101—22no—
76#407—21no—
77#407—21no—
78#407—21no—
79#407—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C2×#97—21no—
85C2×#97—21no—
86C2×#97—21no—
87C2×#97—21no—
88C2×#97—21no—
89C2×#97D412yes—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4D412yes—
94C2×#95—21no—
95#343—22no—
96#343—22no—
97#340—24no—
98Q8⋊C8—28no—
99C22×Q8—21no—
100C22×Q8—24no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—42no—
106C22×Q8C22⋊C412yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—41no—
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—21no—
114C2×M16—22no—
115C2×M16—21no—
116C2×M16—41no—
117C2×M16—41no—
118C22×C8—24no—
119C22×C8—22no—
120C22×C8—42no—
121C22×C8—41no—
122C22×C8—41no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—82no—
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186#103—216no—
187#101—44no—
188#101—44no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—41no—
201#97—41no—
202#97—41no—
203#97—41no—
204#97—41no—
205C4×C8—22no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209#95—22no—
210#95—41no—
211C24#12311yescharacteristic
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8#10414yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4#11711yescharacteristic
247C22×C4#11711yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4#12611yescharacteristic
262C22×C4#12611yescharacteristic
263M16—42no—
264M16—42no—
265M16—42no—
266M16—42no—
267M16—81no—
268M16—81no—
269M16—82no—
270C2×C8—24no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—42no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—82no—
296C42—82no—
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42#12011yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42#12111yescharacteristic
305C42#12111yescharacteristic
306C42#12311yescharacteristic
307C42#12311yescharacteristic
308C42#12311yescharacteristic
309C42#12411yescharacteristic
310C42#12411yescharacteristic
311C42#12611yescharacteristic
312C42#12611yescharacteristic
313C23—21no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23#38111yescharacteristic
318C23#39411yescharacteristic
319C23#39711yescharacteristic
320Q8—24no—
321Q8—24no—
322Q8—44no—
323Q8—44no—
324Q8—42no—
325Q8—42no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—88no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4#38611yescharacteristic
362C2×C4#38611yescharacteristic
363C2×C4#38811yescharacteristic
364C2×C4#38811yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#39411yescharacteristic
368C2×C4#39411yescharacteristic
369C2×C4#39411yescharacteristic
370C2×C4#39711yescharacteristic
371C2×C4#39711yescharacteristic
372C2×C4#39711yescharacteristic
373C2×C4C2×#10412yes—
374C2×C4#42112yes—
375C8—44no—
376C8—82no—
377C8—82no—
378C8—82no—
379C8—81no—
380C8—81no—
381C22—24no—
382C22—24no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22#144011yescharacteristic
390C22#182612yes—
391C22#204811yescharacteristic
392C22#204911yescharacteristic
393C22#208212yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—44no—
405C4—84no—
406C4—82no—
407C4#187112yes—
408C4#189611yescharacteristic
409C4#189711yescharacteristic
410C2—21no—
411C2—21no—
412C2—22no—
413C2—22no—
414C2#1115211yescharacteristic
415C2#1115811yescharacteristic
416C2#1120111yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
17. #1801C4semidirect
18. #1801C4semidirect

Maximal subgroups

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

subgroupindex
#207272
C2×#17632
#174492
#80982
#80882
#79502