Finite Group Databasebeta

← #72319384

Subgroups — #72319384

Order 1024.

536 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#2179C211yesmaximal, characteristic
2#151979C211yesmaximal, characteristic, complemented
3C2×#7389C212yesmaximal
4#135715C211yesmaximal, characteristic, complemented
5#94057C212yesmaximal, complemented
6C2×#2179C411yescharacteristic
7C2×#2179C2211yescharacteristic
8C4×#508C411yescharacteristic
9C4×#443—21no—
10#13627—21no—
11C2×#1398C2212yes—
12C4×#368C411yescharacteristic
13#12671C411yescharacteristic
14#12671C2211yescharacteristic
15#12642—21no—
16#12642—21no—
17C4×#359—21no—
18C4×#346C2211yescharacteristic
19C2×#1368—21no—
20C2×#1368C2212yes—
21#8805—21no—
22#7466—21no—
23#7389—24no—
24C2×#508C2×C411yescharacteristic
25C2×#443—21no—
26C2×#443—21no—
27C4×#135—21no—
28C4×#131—41no—
29C2×M64—22no—
30#2179—22no—
31#2179—22no—
32#2179—21no—
33#2179—21no—
34#2179D412yes—
35#2179Q812yes—
36C4×Q32C812yescomplemented
37C4×SD32—22no—
38C4×D16C812yescomplemented
39C2×#368C2×C411yescharacteristic
40C2×#368C2×C411yescharacteristic
41#2158C2×C411yescharacteristic
42C2×#359—21no—
43C2×#359—21no—
44#2138—21no—
45C2×#346C2×C411yescharacteristic
46C2×#346D411yescharacteristic
47C2×#346D411yescharacteristic
48C2×#346C2311yescharacteristic
49C2×C4×C16C2×C411yescharacteristic
50C2×C4×C16D411yescharacteristic
51#1842—41no—
52C4×#108—21no—
53#1772—21no—
54#1772—21no—
55#1764—41no—
56#1764—41no—
57C4×#104—41no—
58C42×C8D411yescharacteristic
59C2×#334—22no—
60C2×#334—41no—
61C4⋊C32—24no—
62#1398—22no—
63#1398—22no—
64#1382—21no—
65#1382—21no—
66#1381—21no—
67#1368—22no—
68#1368—22no—
69#1368—42no—
70C2×#135—21no—
71C2×#131—41no—
72C2×#131—41no—
73C4×#41—41no—
74#508—21no—
75#508—21no—
76#508C4⋊C412yes—
77#508M1612yes—
78C2×Q32C2×C811yescharacteristic
79C2×SD32—21no—
80C2×D16C2×C811yescharacteristic
81C22×C16C22⋊C411yescharacteristic
82C22×C16C22⋊C411yescharacteristic
83C22×C16C22×C411yescharacteristic
84C22×C16C2×D411yescharacteristic
85#443—24no—
86#443—22no—
87#443—22no—
88#443—42no—
89#443—41no—
90#443—41no—
91C4×Q16—22no—
92C4×SD16—42no—
93C4×D8—22no—
94C2×#108—21no—
95C2×#108—21no—
96#427—21no—
97C2×C4⋊C8—42no—
98C2×C4⋊C8—82no—
99C2×#104—41no—
100C2×#104—41no—
101#416—41no—
102#405—41no—
103#405—41no—
104#405—81no—
105#405—81no—
106C4×M16—41no—
107C4×M16—81no—
108C2×C4×C8—81no—
109C2×C4×C8C22⋊C411yescharacteristic
110C2×C4×C8C22⋊C411yescharacteristic
111C2×C4×C8D811yescharacteristic
112C2×C4×C8SD1611yescharacteristic
113C2×C4×C8SD1611yescharacteristic
114C2×C4×C8C2×D411yescharacteristic
115C43D811yescharacteristic
116M64—42no—
117M64—42no—
118C2×C32—24no—
119#368—21no—
120#368—21no—
121#368C4⋊C412yes—
122#368M1612yes—
123#367—21no—
124#366C2×C812yes—
125#359—24no—
126#359—22no—
127#359—22no—
128#358—22no—
129#357—22no—
130#346—21no—
131#346—21no—
132#346—21no—
133#346—21no—
134#346—41no—
135#346C2×D411yescharacteristic
136#346C2×Q811yescharacteristic
137#346#4111yescharacteristic
138#346#4111yescharacteristic
139C4×C16—22no—
140C4×C16C2×C812yes—
141C4×C16M1612yes—
142#341—41no—
143#339—41no—
144#339—41no—
145#334—42no—
146#334—42no—
147#334—82no—
148C2×#41—41no—
149#135—22no—
150#135—22no—
151#135—41no—
152#135—41no—
153C2×Q16—21no—
154C2×SD16—41no—
155C2×D8—21no—
156#131—44no—
157#131—42no—
158#131—42no—
159#131—81no—
160#131—81no—
161#131—82no—
162C2×M16—41no—
163C2×M16—41no—
164C2×M16—81no—
165C2×M16—81no—
166C22×C8—81no—
167C22×C8—81no—
168C22×C8—82no—
169C22×C8#10211yescharacteristic
170C22×C8#10211yescharacteristic
171C22×C8C2×C22⋊C411yescharacteristic
172C22×C8C2×SD1611yescharacteristic
173C4×Q8—42no—
174C4×D4—42no—
175C4×D4—82no—
176#117—41no—
177#117—81no—
178C2×C42—22no—
179C2×C42—81no—
180C2×C42#10211yescharacteristic
181C2×C42#10211yescharacteristic
182C2×C42C2×D811yescharacteristic
183Q32—21no—
184Q32C4⋊C812yescomplemented
185SD32—22no—
186SD32—22no—
187D16—21no—
188D16C4⋊C812yescomplemented
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16C22⋊C812yes—
197C2×C16C2×C4⋊C411yescharacteristic
198C2×C16#11711yescharacteristic
199C2×C16#12111yescharacteristic
200C2×C16C22⋊Q811yescharacteristic
201C2×C16C22×C811yescharacteristic
202C2×C16C2×M1611yescharacteristic
203C2×C16C2×M1611yescharacteristic
204C2×C16C2×M1611yescharacteristic
205#108—22no—
206#108—22no—
207#108—41no—
208#108—41no—
209#107—22no—
210#106—41no—
211C4⋊C8—82no—
212C4⋊C8—82no—
213C4⋊C8—164no—
214#104—44no—
215#104—42no—
216#104—42no—
217#103—42no—
218#102—42no—
219#97—81no—
220#97—81no—
221#97—81no—
222C4×C8—22no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—41no—
228C4×C8—81no—
229C4×C8—81no—
230C4×C8—81no—
231C4×C8C22⋊C812yes—
232C4×C8#10412yes—
233C4×C8#12111yescharacteristic
234C4×C8C22⋊Q811yescharacteristic
235C4×C8AGL(1,ℤ/8)11yescharacteristic
236C4×C8#13711yescharacteristic
237C32—44no—
238#41—42no—
239#41—42no—
240#41—81no—
241#41—81no—
242C2×Q8—41no—
243C2×D4—41no—
244C2×D4—81no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—81no—
248C22×C4—82no—
249C22×C4#36012yes—
250C22×C4C2×#10211yescharacteristic
251Q16—22no—
252Q16—41no—
253SD16—42no—
254SD16—42no—
255D8—22no—
256D8—41no—
257M16—42no—
258M16—42no—
259M16—82no—
260M16—82no—
261M16—84no—
262M16—81no—
263M16—81no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—82no—
271C2×C8—82no—
272C2×C8—84no—
273C2×C8—81no—
274C2×C8—81no—
275C2×C8—162no—
276C2×C8—164no—
277C2×C8D4⋊C812yes—
278C2×C8#36012yes—
279C2×C8#38011yescharacteristic
280C2×C8C2×C22⋊C811yescharacteristic
281C2×C8#40711yescharacteristic
282C2×C8#41711yescharacteristic
283C2×C8C2×#10411yescharacteristic
284C2×C8#42111yescharacteristic
285C2×C8#46111yescharacteristic
286C2×C8#47611yescharacteristic
287C4⋊C4—42no—
288C4⋊C4—82no—
289C22⋊C4—81no—
290C22⋊C4—82no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—44no—
295C42—82no—
296C42—81no—
297C42D4⋊C812yes—
298C42#36111yescharacteristic
299C42#36111yescharacteristic
300C42#45911yescharacteristic
301C42#47411yescharacteristic
302C16—21no—
303C16—21no—
304C16—22no—
305C16—41no—
306C16C2×C4⋊C811yescharacteristic
307C16#42311yescharacteristic
308C16#43111yescharacteristic
309C16#43211yescharacteristic
310C23—81no—
311C23C2×#36011yescharacteristic
312Q8—42no—
313Q8—81no—
314D4—42no—
315D4—82no—
316D4—82no—
317D4—81no—
318C2×C4—24no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—164no—
329C2×C4#133312yes—
330C2×C4#133312yes—
331C2×C4C2×D4⋊C811yescharacteristic
332C2×C4#148611yescharacteristic
333C2×C4#180211yescharacteristic
334C2×C4C2×#36011yescharacteristic
335C2×C4C2×#36111yescharacteristic
336C2×C4#214811yescharacteristic
337C8—21no—
338C8—21no—
339C8—22no—
340C8—22no—
341C8—82no—
342C8—82no—
343C8—322no—
344C8#147711yescharacteristic
345C8#148911yescharacteristic
346C8#179711yescharacteristic
347C8#180611yescharacteristic
348C22—22no—
349C22—82no—
350C22—162no—
351C22C2×#133311yescharacteristic
352C22#1166611yescharacteristic
353C22#1281611yescharacteristic
354C4—22no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—82no—
359C4—162no—
360C4#738914yescomplemented
361C4#856311yescharacteristic
362C4#1159811yescharacteristic
363C4#1166311yescharacteristic
364C4#1281811yescharacteristic
365C2—21no—
366C2—21no—
367C2—162no—
368C2#13436211yescharacteristic
369C2C2×#738912yes—
subgroupG/H
center, socleC22C2×#1333
commutatorC2×C16C22×C8
Frattini, cosocleC2×#346C23
prime radicalC2×#508C2×C4
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ C4×#346 ◁ G
LECSG ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #151979C2semidirect
4. #135715C2semidirect
5. #94057C2semidirect
36. C4×Q32C8semidirect
38. C4×D16C8semidirect
184. Q32C4⋊C8semidirect
188. D16C4⋊C8semidirect
360. C4#7389semidirect

Maximal subgroups

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

subgroupindex
C4×#21792
#1519792
C2×#73892
#1357152
#940572