Finite Group Databasebeta

← #99682

Subgroups — #99682

Order 512.

424 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774C211yesmaximal, characteristic
2#22798C211yesmaximal, characteristic
3#9874C211yesmaximal, characteristic, complemented
4#9677C211yesmaximal, characteristic, complemented
5#7560C211yesmaximal, characteristic
6#7031C211yesmaximal, characteristic, complemented
7#7027C211yesmaximal, characteristic, complemented
8C2×#433—21no—
9C4×C4⋊Q8C2211yescharacteristic
10#2293C2211yescharacteristic
11#2099C411yescharacteristic
12#2093C411yescharacteristic
13#2086—21no—
14#2067—21no—
15#2057C411yescharacteristic
16#2023C411yescharacteristic
17#1955—21no—
18#1924—21no—
19#1923—21no—
20#1842C2211yescharacteristic
21#1798—21no—
22C4×#107—21no—
23#1771—21no—
24#1767C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×#102—21no—
28#1744C2211yescharacteristic
29C2×Q8⋊C8—21no—
30C2×Q8⋊C8—21no—
31#1279—21no—
32#1279—21no—
33#1271—21no—
34#1270—21no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×D4D411yescharacteristic
38C2×#117—21no—
39#433—42no—
40#433—42no—
41C2×#107—21no—
42C2×#107—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×#102—21no—
52C2×#102—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×#97—21no—
58C2×#97—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62C2×C4×C8—21no—
63#391—21no—
64#390C2×C411yescharacteristic
65#390D411yescharacteristic
66#389—21no—
67#389—21no—
68#389C2×C411yescharacteristic
69#389D411yescharacteristic
70#388—21no—
71#388—21no—
72#384C2×C411yescharacteristic
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4D411yescharacteristic
76C4×C22⋊C4—21no—
77C43C2311yescharacteristic
78#340—21no—
79#340—21no—
80#340—21no—
81#340—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336C2×C411yescharacteristic
89#336C2×C411yescharacteristic
90Q8⋊C8—42no—
91Q8⋊C8—42no—
92C22×Q8—21no—
93C22×D4C22⋊C411yescharacteristic
94C23×C4—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—41no—
102C22×C8—41no—
103C4⋊Q8—42no—
104C4×Q8—41no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×D4—21no—
109C4×D4—21no—
110C4×D4—21no—
111C4×D4—21no—
112#117—41no—
113#117—41no—
114#117—41no—
115#117—41no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C42—21no—
133C2×C42—41no—
134C2×C42C22×C411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42#4111yescharacteristic
140C2×C42#4111yescharacteristic
141#107—42no—
142#107—42no—
143C4⋊C8—22no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—42no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8D812yes—
156#102—42no—
157#102—42no—
158C22⋊C8—42no—
159C22⋊C8—42no—
160C22⋊C8—81no—
161C22⋊C8—81no—
162#97—42no—
163#97—41no—
164#97—41no—
165C4×C8—22no—
166C4×C8—41no—
167C4×C8—42no—
168C4×C8—41no—
169C4×C8—41no—
170C4×C8—41no—
171C4×C8—41no—
172#95—21no—
173#95—21no—
174#95—21no—
175#95—41no—
176#95—41no—
177C24—21no—
178C2×Q8—22no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—41no—
187C2×D4—41no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4C4×D411yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12311yescharacteristic
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—81no—
227C2×C8—81no—
228C2×C8—81no—
229C2×C8—81no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—81no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—41no—
264C42—41no—
265C42—41no—
266C42—41no—
267C42—41no—
268C42—81no—
269C42—81no—
270C42C2×D811yescharacteristic
271C42#13511yescharacteristic
272C42#13511yescharacteristic
273C42#13511yescharacteristic
274C42AGL(1,ℤ/8)11yescharacteristic
275C42AGL(1,ℤ/8)11yescharacteristic
276C42#13711yescharacteristic
277C42#13711yescharacteristic
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#38611yescharacteristic
286Q8—42no—
287Q8—42no—
288Q8—81no—
289Q8—81no—
290D4—42no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4#41611yescharacteristic
325C2×C4#41911yescharacteristic
326C2×C4C4×D811yescharacteristic
327C2×C4#44211yescharacteristic
328C2×C4#44311yescharacteristic
329C2×C4#44411yescharacteristic
330C2×C4#44711yescharacteristic
331C2×C4#44911yescharacteristic
332C2×C4#45211yescharacteristic
333C2×C4#45211yescharacteristic
334C2×C4#45511yescharacteristic
335C2×C4#45611yescharacteristic
336C2×C4#45911yescharacteristic
337C2×C4#46311yescharacteristic
338C2×C4#48511yescharacteristic
339C2×C4#48511yescharacteristic
340C8—42no—
341C8—42no—
342C8—81no—
343C8—82no—
344C8—82no—
345C8—81no—
346C8—81no—
347C8—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#187311yescharacteristic
356C22#187611yescharacteristic
357C22#187911yescharacteristic
358C22#188111yescharacteristic
359C22#188211yescharacteristic
360C22#188411yescharacteristic
361C22#189111yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—41no—
373C4—81no—
374C4—81no—
375C4—82no—
376C4—81no—
377C4—161no—
378C4#164111yescharacteristic
379C4#164611yescharacteristic
380C4#164711yescharacteristic
381C4#165211yescharacteristic
382C2—161no—
383C2#898411yescharacteristic
384C2#899411yescharacteristic
385C2#899811yescharacteristic
386C2#903411yescharacteristic
387C2#904911yescharacteristic
388C2#905111yescharacteristic
389C2#905311yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC23#386
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #9874C2semidirect
4. #9677C2semidirect
6. #7031C2semidirect
7. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#237742
#227982
#98742
#96772
#75602
#70312
#70272