Finite Group Databasebeta

← #119636

Subgroups — #119636

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20727C211yesmaximal, characteristic
2C4×#419C211yesmaximal, characteristic
3C2×#1487C211yesmaximal, characteristic
4#8097C211yesmaximal, characteristic
5#8084C211yesmaximal, characteristic
6#8063C211yesmaximal, characteristic
7#7870C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#419—21no—
14C2×C4×M16C2211yescharacteristic
15C4×#117C2211yescharacteristic
16#1821—21no—
17#1801C412yescomplemented
18#1801C412yescomplemented
19#1784—21no—
20#1763—22no—
21C4×#103—22no—
22C4×#101—22no—
23#1736—21no—
24#1487—22no—
25#1487—22no—
26C2×Q8⋊C8—22no—
27#1303—22no—
28#1302—22no—
29#1299—22no—
30#1276—22no—
31C23×Q8D411yescharacteristic
32C22×M16—21no—
33C22×M16—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×Q8—21no—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8D412yes—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C42—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431D412yes—
63#423—21no—
64#423—21no—
65#423—21no—
66#423C2×C412yes—
67#423C2×C411yescharacteristic
68#423C2×C411yescharacteristic
69#423D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73#419—28no—
74C2×#103—24no—
75C2×#101—22no—
76C2×#101—22no—
77#407—21no—
78#407—21no—
79C2×C22⋊C8—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C2×#97D412yes—
87C2×C4×C8—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#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×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—24no—
113C2×M16—21no—
114C2×M16—22no—
115C2×M16—21no—
116C2×M16—22no—
117C2×M16—42no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—82no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141#117—24no—
142#117—42no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C22⋊C4—41no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190#103—216no—
191#101—44no—
192#101—44no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#97—22no—
200#97—22no—
201#97—41no—
202#97—41no—
203C4×C8—22no—
204C4×C8—41no—
205C4×C8—41no—
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—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—81no—
240C22×C4—82no—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4#11711yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12411yescharacteristic
256C22×C4#12411yescharacteristic
257C22×C4#12611yescharacteristic
258C22×C4#12611yescharacteristic
259M16—44no—
260M16—44no—
261M16—42no—
262M16—42no—
263M16—42no—
264M16—42no—
265M16—82no—
266M16—82no—
267C2×C8—24no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—82no—
276C2×C8—82no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C22⋊C4—44no—
288C42—44no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—82no—
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12011yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42#12111yescharacteristic
305C42#12311yescharacteristic
306C42#12311yescharacteristic
307C42#12311yescharacteristic
308C42#12411yescharacteristic
309C42#12411yescharacteristic
310C42#12611yescharacteristic
311C42#12611yescharacteristic
312C23—22no—
313C23—22no—
314C23—41no—
315C23#38111yescharacteristic
316C23#39411yescharacteristic
317C23#39711yescharacteristic
318Q8—24no—
319Q8—24no—
320Q8—44no—
321Q8—44no—
322Q8—42no—
323Q8—42no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
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—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4#38611yescharacteristic
357C2×C4#38611yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#38811yescharacteristic
362C2×C4#39411yescharacteristic
363C2×C4#39411yescharacteristic
364C2×C4#39411yescharacteristic
365C2×C4#39711yescharacteristic
366C2×C4#39711yescharacteristic
367C2×C4#39711yescharacteristic
368C2×C4C2×#10412yes—
369C2×C4#42112yes—
370C2×C4#47112yes—
371C2×C4#47312yes—
372C8—44no—
373C8—82no—
374C8—82no—
375C8—81no—
376C8—81no—
377C8—82no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—44no—
386C22#144011yescharacteristic
387C22#182612yes—
388C22#205212yes—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—44no—
398C4—84no—
399C4—82no—
400C4#187112yes—
401C4#189611yescharacteristic
402C4#189711yescharacteristic
403C4#193012yes—
404C4#204611yescharacteristic
405C4#204711yescharacteristic
406C2—22no—
407C2—22no—
408C2—41no—
409C2#1115311yescharacteristic
410C2#1116411yescharacteristic
411C2#1120111yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1440
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
C4×#4192
C2×#14872
#80972
#80842
#80632
#78702