Finite Group Databasebeta

← #72227255

Subgroups — #72227255

Order 4096.

703 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72227194C211yesmaximal, characteristic
2#72227190—21no—
3#72227188C412yes—
4#72227185C211yesmaximal, characteristic
5#72227182C411yescharacteristic
6#72225535—21no—
7#72225407—21no—
8#72225325C411yescharacteristic
9#72216542—21no—
10#72216231C211yesmaximal, characteristic
11#72216226—21no—
12C8×#1398—21no—
13#72216220C411yescharacteristic
14C16×M64C411yescharacteristic
15#72216182C2211yescharacteristic
16C4×#1398—22no—
17#151042—22no—
18#149513—41no—
19#149495—22no—
20C8×M64—21no—
21C8×M64—21no—
22C4×#1368—21no—
23#129086—21no—
24#129049—41no—
25C8×#346C2×C411yescharacteristic
26C8×#346D411yescharacteristic
27C8×#346D411yescharacteristic
28C2×#7099—21no—
29C2×#7052C2×C411yescharacteristic
30C2×#7052D411yescharacteristic
31C2×#7052Q811yescharacteristic
32C2×C162C2×C411yescharacteristic
33#96889—21no—
34#94011—41no—
35#93876—22no—
36#93876C814yescomplemented
37#93666—24no—
38#93665—21no—
39#93665C812yescomplemented
40#93664—21no—
41C16×C32C812yescomplemented
42#93269C812yes—
43#93261—41no—
44#92913C812yes—
45#13640—42no—
46C2×#1398—28no—
47C4×M64—22no—
48C4×M64—22no—
49#12653—41no—
50C4×#346—21no—
51C4×#346C2×C811yescharacteristic
52C4×#346M1611yescharacteristic
53C4×#346D811yescharacteristic
54C4×#346SD1611yescharacteristic
55C2×#1368—41no—
56C2×#1367—22no—
57C2×#1313—81no—
58C2×#1311—21no—
59C2×#1311C2×C811yescharacteristic
60C2×#1311M1611yescharacteristic
61C2×#1311SD1611yescharacteristic
62C2×#1311Q1611yescharacteristic
63C2×C8×C16C4211yescharacteristic
64C2×C8×C16C22⋊C411yescharacteristic
65C2×C8×C16C22⋊C411yescharacteristic
66C2×C8×C16C4⋊C411yescharacteristic
67C2×C8×C16C4⋊C411yescharacteristic
68C2×C8×C16C4⋊C411yescharacteristic
69C4×#334—22no—
70#8260—41no—
71C4×C82C22⋊C411yescharacteristic
72#7458—42no—
73#7455—24no—
74#7332—81no—
75#7330—28no—
76#7330—81no—
77#7329—22no—
78#7329—22no—
79C8×C32—22no—
80C8×C32—22no—
81#7156—41no—
82#7151—41no—
83#7137—81no—
84#7099—41no—
85#7099—41no—
86#7052—22no—
87#7052—21no—
88#7052—21no—
89#7052C2×C812yes—
90#7052M1612yes—
91C162C2×C811yescharacteristic
92C162C2×C811yescharacteristic
93C162M1611yescharacteristic
94C162M1611yescharacteristic
95#7015—41no—
96C2×M64—24no—
97C2×M64—24no—
98C2×C4⋊C16—42no—
99C2×#346—24no—
100C2×#346—21no—
101C2×#346—21no—
102C2×#346—21no—
103C2×#346—21no—
104C2×#346—41no—
105C2×#346—41no—
106C2×#346—81no—
107C2×C4×C16—21no—
108C2×C4×C16—21no—
109C2×C4×C16C4×C811yescharacteristic
110C2×C4×C16#9711yescharacteristic
111C2×C4×C16#10211yescharacteristic
112C2×C4×C16#10311yescharacteristic
113C2×C4×C16C4⋊C811yescharacteristic
114C2×C4×C16C4⋊C811yescharacteristic
115C2×C4×C16#10611yescharacteristic
116C2×C4×C16#10711yescharacteristic
117C4×C4⋊C8—42no—
118C42×C8C22⋊C811yescharacteristic
119C42×C8#10211yescharacteristic
120C42×C8#10411yescharacteristic
121C2×#334—42no—
122C2×#322—81no—
123C2×C82#9511yescharacteristic
124C2×C82C22⋊C811yescharacteristic
125C2×C82#10311yescharacteristic
126C2×C82#10411yescharacteristic
127C4⋊C32—44no—
128C22⋊C32—82no—
129#1398—416no—
130#1397—24no—
131#1397—24no—
132#1397—82no—
133C4×C32—24no—
134C4×C32—24no—
135#1374—81no—
136#1368—82no—
137#1367—42no—
138#1367—42no—
139#1313—162no—
140#1311—24no—
141#1311—21no—
142#1311—21no—
143#1311—21no—
144#1311—21no—
145#1311—41no—
146#1311—41no—
147C8×C16—21no—
148C8×C16—21no—
149C8×C16—21no—
150C8×C16—21no—
151C8×C16C4×C811yescharacteristic
152C8×C16C4×C811yescharacteristic
153C8×C16#9711yescharacteristic
154C8×C16#9711yescharacteristic
155C8×C16C22⋊C811yescharacteristic
156C8×C16C22⋊C811yescharacteristic
157C8×C16C22⋊C812yes—
158C8×C16#10412yes—
159C8×C16C4⋊C812yes—
160C8×C16C4⋊C811yescharacteristic
161C8×C16C4⋊C811yescharacteristic
162C8×C16#10811yescharacteristic
163C8×C16#10811yescharacteristic
164#1277—42no—
165#1273—81no—
166C22×C16—22no—
167C22×C16—22no—
168C22×C16—21no—
169C22×C16—21no—
170C22×C16—21no—
171C22×C16—21no—
172C22×C16—21no—
173C22×C16—21no—
174C22×C16—82no—
175C22×C16#34711yescharacteristic
176C22×C16#34711yescharacteristic
177C22×C16#36511yescharacteristic
178C22×C16#36511yescharacteristic
179C2×C4⋊C8—82no—
180C2×C4×C8—21no—
181C2×C4×C8—21no—
182C2×C4×C8—22no—
183C2×C4×C8—21no—
184C2×C4×C8—21no—
185C2×C4×C8—82no—
186C2×C4×C8D4⋊C811yescharacteristic
187C2×C4×C8Q8⋊C811yescharacteristic
188C2×C4×C8Q8⋊C811yescharacteristic
189C2×C4×C8#33611yescharacteristic
190C2×C4×C8#33911yescharacteristic
191C2×C4×C8#34011yescharacteristic
192C43D4⋊C811yescharacteristic
193M64—216no—
194M64—48no—
195C2×C32—28no—
196C2×C32—28no—
197C2×C32—84no—
198C4⋊C16—82no—
199C4⋊C16—82no—
200#346—42no—
201#346—42no—
202#346—48no—
203#346—81no—
204#346—81no—
205#346—82no—
206#346—82no—
207#346—161no—
208#346—161no—
209C4×C16—22no—
210C4×C16—22no—
211C4×C16—21no—
212C4×C16—21no—
213C4×C16—21no—
214C4×C16—21no—
215C4×C16—21no—
216C4×C16—21no—
217C4×C16—21no—
218C4×C16—21no—
219C4×C16—22no—
220C4×C16—21no—
221C4×C16—21no—
222C4×C16C8211yescharacteristic
223C4×C16#32211yescharacteristic
224C4×C16#32211yescharacteristic
225C4×C16#32211yescharacteristic
226C4×C16D4⋊C812yes—
227C4×C16Q8⋊C812yes—
228C4×C16#33411yescharacteristic
229C4×C16#33411yescharacteristic
230C4×C16#33511yescharacteristic
231C4×C16#33511yescharacteristic
232C4×C16#34711yescharacteristic
233C4×C16#34711yescharacteristic
234C4×C16#36511yescharacteristic
235C4×C16#36511yescharacteristic
236#336—82no—
237#334—82no—
238#334—82no—
239#322—161no—
240#322—161no—
241C82—22no—
242C82—21no—
243C82—21no—
244C82—21no—
245C82—21no—
246C82#33611yescharacteristic
247C82#33611yescharacteristic
248C82#33911yescharacteristic
249C82#34111yescharacteristic
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—162no—
255C22×C8#137511yescharacteristic
256C22×C8#137811yescharacteristic
257C22×C8#138311yescharacteristic
258C22×C8#138411yescharacteristic
259C2×C42—21no—
260C2×C42—21no—
261C2×C42#127911yescharacteristic
262C2×C42#133512yes—
263C2×C16—22no—
264C2×C16—22no—
265C2×C16—28no—
266C2×C16—21no—
267C2×C16—21no—
268C2×C16—22no—
269C2×C16—22no—
270C2×C16—24no—
271C2×C16—22no—
272C2×C16—22no—
273C2×C16—21no—
274C2×C16—21no—
275C2×C16—41no—
276C2×C16—41no—
277C2×C16—42no—
278C2×C16—42no—
279C2×C16—44no—
280C2×C16—41no—
281C2×C16—41no—
282C2×C16—41no—
283C2×C16—41no—
284C2×C16—41no—
285C2×C16—41no—
286C2×C16—81no—
287C2×C16—81no—
288C2×C16—162no—
289C2×C16—162no—
290C2×C16—162no—
291C2×C16#131311yescharacteristic
292C2×C16#131311yescharacteristic
293C2×C16#131311yescharacteristic
294C2×C16#131311yescharacteristic
295C2×C16#136811yescharacteristic
296C2×C16#136811yescharacteristic
297C2×C16#136811yescharacteristic
298C2×C16#136811yescharacteristic
299C4⋊C8—162no—
300C4⋊C8—162no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—41no—
308C4×C8—41no—
309C4×C8—41no—
310C4×C8—41no—
311C4×C8—44no—
312C4×C8—81no—
313C4×C8—81no—
314C4×C8—162no—
315C4×C8—162no—
316C4×C8#127311yescharacteristic
317C4×C8#127611yescharacteristic
318C4×C8#127711yescharacteristic
319C4×C8#127811yescharacteristic
320C4×C8#127811yescharacteristic
321C4×C8#127911yescharacteristic
322C4×C8#133512yes—
323C4×C8#137511yescharacteristic
324C4×C8#137811yescharacteristic
325C4×C8#138311yescharacteristic
326C4×C8#138511yescharacteristic
327C32—216no—
328C32—48no—
329C32—164no—
330C22×C4—22no—
331C22×C4#716811yescharacteristic
332C22×C4#717411yescharacteristic
333C22×C4#717911yescharacteristic
334C2×C8—24no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—322no—
352C2×C8—322no—
353C2×C8—322no—
354C2×C8#713711yescharacteristic
355C2×C8#714011yescharacteristic
356C2×C8#715611yescharacteristic
357C2×C8#715711yescharacteristic
358C2×C8#716411yescharacteristic
359C2×C8#716411yescharacteristic
360C2×C8#717311yescharacteristic
361C2×C8#717411yescharacteristic
362C42—42no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42#701511yescharacteristic
368C42#716811yescharacteristic
369C42#717311yescharacteristic
370C42#717911yescharacteristic
371C16—24no—
372C16—24no—
373C16—22no—
374C16—22no—
375C16—21no—
376C16—21no—
377C16—21no—
378C16—21no—
379C16—21no—
380C16—21no—
381C16—42no—
382C16—44no—
383C16—41no—
384C16—42no—
385C16—41no—
386C16—81no—
387C16—81no—
388C16—322no—
389C16#733312yes—
390C16#745712yes—
391C23#9293811yescharacteristic
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4#9293711yescharacteristic
401C2×C4#9293711yescharacteristic
402C2×C4#9293811yescharacteristic
403C2×C4#9326111yescharacteristic
404C2×C4#9326411yescharacteristic
405C2×C4#9326911yescharacteristic
406C8—22no—
407C8—22no—
408C8—21no—
409C8—21no—
410C8—21no—
411C8—21no—
412C8—21no—
413C8—21no—
414C8—42no—
415C8—81no—
416C8—81no—
417C8—642no—
418C8#9399511yescharacteristic
419C8#9399711yescharacteristic
420C8#9403011yescharacteristic
421C8#9403111yescharacteristic
422C22—21no—
423C22—21no—
424C22#7221436112yes—
425C22#7221654211yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4#7222365211yescharacteristic
432C4#7222532311yescharacteristic
433C4#7222552611yescharacteristic
434C4#7222725211yescharacteristic
435C2—22no—
436C2#7221654411yescharacteristic
437C2#7222725111yescharacteristic
438C2#7222725411yescharacteristic
subgroupG/H
centerC2×C4#93269
commutatorC4×C16C82
Frattini, cosocle#72216182C22
socleC22#72216542
prime radicalC23#92938
LCSG ▷ C4×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C16 ◁ C8×#346 ◁ #72216182 ◁ G
LECSG ▷ #72216182 ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
36. #93876C8semidirect
39. #93665C8semidirect
41. C16×C32C8semidirect

Maximal subgroups

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

subgroupindex
#722271942
#722271852
#722162312