Finite Group Databasebeta

← #117127

Subgroups — #117127

Order 512.

736 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic
2C2×#1848C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5#8043C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7#7849C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×#108—22no—
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16#1990—22no—
17#1990C414yes—
18#1848—22no—
19#1848—22no—
20C2×#340—22no—
21#1549—22no—
22#1549—22no—
23#1548—22no—
24#1548—22no—
25C2×#333—22no—
26C2×#332—22no—
27#1296—22no—
28#1296—22no—
29#1273—22no—
30C23×Q8—21no—
31C22×M16—21no—
32C22×M16—21no—
33C23×C8D412yes—
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—
45C2×C4⋊Q8Q812yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C42C2311yescharacteristic
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60C2×#108—28no—
61C2×#108—44no—
62C2×#108—44no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423D412yes—
70#423D412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76C2×C4⋊C8D412yes—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2×C412yes—
86C2×C4×C8D412yes—
87#384—21no—
88#384—21no—
89#384—21no—
90#384C2×C412yes—
91#340—28no—
92#333—28no—
93#332—28no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—82no—
101C22×Q8—82no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8C22⋊C414yes—
124C22×C8SD1614yes—
125C22×C8Q1614yes—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8C4⋊C414yes—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176#108—416no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—41no—
198C4×C8—41no—
199C4×C8—41no—
200C4×C8—41no—
201C24#12411yescharacteristic
202C2×Q8—24no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—84no—
207C2×Q8—84no—
208C2×Q8—84no—
209C2×Q8—84no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8—82no—
213C2×Q8—82no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12411yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12611yescharacteristic
243C22×C4#12611yescharacteristic
244C22×C4#12711yescharacteristic
245C22×C4C2×SD1612yes—
246C22×C4C2×Q1612yes—
247C22×C4AGL(1,ℤ/8)12yes—
248C22×C4#13712yes—
249M16—44no—
250M16—44no—
251M16—82no—
252M16—82no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—81no—
268C2×C8—81no—
269C2×C8—81no—
270C2×C8—81no—
271C2×C8#10318yes—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C42C2×C4⋊C411yescharacteristic
288C42#11711yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42#12011yescharacteristic
292C42#12011yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42C22⋊Q811yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42#12311yescharacteristic
302C42#12311yescharacteristic
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23#39011yescharacteristic
308C23#39311yescharacteristic
309C23#39711yescharacteristic
310C23#48712yes—
311C23#48912yes—
312Q8—84no—
313Q8—84no—
314Q8—84no—
315Q8—84no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—44no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—164no—
337C2×C4#38011yescharacteristic
338C2×C4#38511yescharacteristic
339C2×C4#38511yescharacteristic
340C2×C4#38611yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#39211yescharacteristic
344C2×C4#39411yescharacteristic
345C2×C4#39411yescharacteristic
346C2×C4#39611yescharacteristic
347C2×C4#39611yescharacteristic
348C2×C4#39911yescharacteristic
349C2×C4C2×#10312yes—
350C2×C4#41712yes—
351C2×C4#45012yes—
352C2×C4#45112yes—
353C2×C4#46012yes—
354C2×C4#46212yes—
355C2×C4#47112yes—
356C2×C4#47112yes—
357C2×C4#47212yes—
358C2×C4#47312yes—
359C2×C4#47512yes—
360C2×C4#47712yes—
361C2×C4#48312yes—
362C2×C4#48412yes—
363C2×C4#49211yescharacteristic
364C2×C4#49411yescharacteristic
365C2×C4#49611yescharacteristic
366C2×C4#49711yescharacteristic
367C8—28no—
368C8—82no—
369C8—82no—
370C8—82no—
371C8—82no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22#143911yescharacteristic
381C22#195611yescharacteristic
382C22#195911yescharacteristic
383C22#197711yescharacteristic
384C22#197711yescharacteristic
385C22#197811yescharacteristic
386C22#197911yescharacteristic
387C22#202412yes—
388C22#207212yes—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—84no—
394C4—162no—
395C4#180312yes—
396C4#187212yes—
397C4#201411yescharacteristic
398C4#201611yescharacteristic
399C4#205112yes—
400C4#205212yes—
401C4#206312yes—
402C4#206412yes—
403C4#209511yescharacteristic
404C4#209611yescharacteristic
405C2—22no—
406C2—22no—
407C2#976711yescharacteristic
408C2#977111yescharacteristic
409C2#977411yescharacteristic
410C2#977511yescharacteristic
411C2#1105011yescharacteristic
412C2#1112711yescharacteristic
413C2#1112811yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19902
C2×#18482
C2×#15492
C2×#15482
#80432
#78492