Finite Group Databasebeta

← #118575

Subgroups — #118575

Order 512.

587 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17820C211yesmaximal, characteristic
2#17449C211yesmaximal, characteristic
3C2×#1487C211yesmaximal, characteristic
4#17365C211yesmaximal, characteristic
5#8061C211yesmaximal, characteristic
6#8007C211yesmaximal, characteristic
7#7886C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#1574—22no—
16#1571—21no—
17#1570—21no—
18#1569—22no—
19#1491—22no—
20#1487—22no—
21#1487—21no—
22#1487—21no—
23#1487—22no—
24#1479—22no—
25C2×Q8⋊C8—22no—
26#1467C414yes—
27#1463—22no—
28#1461—22no—
29#1299—22no—
30#1293—22no—
31#1278—22no—
32C23×Q8Q811yescharacteristic
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—
46C22×C4⋊C4—21no—
47C22×C4⋊C4Q811yescharacteristic
48C22×C42—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#432C2×C412yes—
58#431—21no—
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#431—21no—
68#431D412yes—
69#431D412yes—
70#423—21no—
71#423—21no—
72#423—21no—
73#423—21no—
74#423—21no—
75#423—21no—
76#423D412yes—
77#423D411yescharacteristic
78#423D411yescharacteristic
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82#407—21no—
83#407—21no—
84#407—21no—
85#407D412yes—
86C2×C22⋊C8—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16—21no—
91C2×C4×C8D412yes—
92#335—24no—
93#334—24no—
94Q8⋊C8—28no—
95Q8⋊C8—24no—
96Q8⋊C8—24no—
97#324—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—44no—
104C22×Q8C4⋊C412yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×Q811yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—22no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—22no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—82no—
118C22×C8—22no—
119C22×C8—41no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4C4⋊C412yes—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—22no—
173C4⋊C8—21no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8C22⋊C414yes—
192#97—22no—
193#97—22no—
194#97—41no—
195#97—41no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—41no—
201C4×C8—41no—
202C24C22⋊Q811yescharacteristic
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—44no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—41no—
215C2×Q8—41no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—81no—
227C22×C4—82no—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C4⋊C411yescharacteristic
230C22×C4#11711yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4#12611yescharacteristic
245C22×C4#12611yescharacteristic
246M16—42no—
247M16—42no—
248M16—82no—
249M16—82no—
250M16—82no—
251M16—82no—
252M16—82no—
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—82no—
261C2×C8—82no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—42no—
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42#12011yescharacteristic
279C42#12011yescharacteristic
280C42#12011yescharacteristic
281C42#12311yescharacteristic
282C42#12311yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12311yescharacteristic
286C42#12411yescharacteristic
287C42#12411yescharacteristic
288C42#12411yescharacteristic
289C42#12711yescharacteristic
290C23—22no—
291C23—41no—
292C23—41no—
293C23#38011yescharacteristic
294C23#39611yescharacteristic
295C23#39611yescharacteristic
296Q8—24no—
297Q8—24no—
298Q8—42no—
299Q8—42no—
300Q8—44no—
301Q8—44no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—24no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38611yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#39011yescharacteristic
336C2×C4#39211yescharacteristic
337C2×C4#39311yescharacteristic
338C2×C4#39411yescharacteristic
339C2×C4#39411yescharacteristic
340C2×C4#39711yescharacteristic
341C2×C4#39911yescharacteristic
342C2×C4#47112yes—
343C2×C4#47312yes—
344C8—44no—
345C8—82no—
346C8—82no—
347C8—82no—
348C8—82no—
349C8—81no—
350C8—81no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—44no—
360C22#143911yescharacteristic
361C22#206412yes—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—44no—
372C4—82no—
373C4#193012yes—
374C4#201911yescharacteristic
375C4#202011yescharacteristic
376C4#209112yes—
377C2—22no—
378C2—41no—
379C2—41no—
380C2#1109411yescharacteristic
381C2#1110111yescharacteristic
382C2#1114511yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
#178202
#174492
C2×#14872
#173652
#80612
#80072
#78862