Finite Group Databasebeta

← #11788147

Subgroups — #11788147

Order 1152.

1037 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1037 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C2×#64147C211yesmaximal, characteristic
2C22×#3712C211yesmaximal, characteristic, complemented
3C2×C3×#832C212yesmaximal, complemented
4#10591325C211yesmaximal, characteristic, complemented
5#10589164C212yesmaximal, complemented
6#70203—32nomaximal
7C22×C32×D4C2211yescharacteristic
8C22×#638—21no—
9C22×C4×C3⋊D3C2211yescharacteristic
10C2×#3757—21no—
11#64147—22no—
12#64147C414yescomplemented
13C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C4⋊Dic3C2212yes—
15C22×C3×C3⋊C8C2212yes—
16C2×#3722—21no—
17C2×#3712—22no—
18C2×#3712—24no—
19C3×#832—24no—
20#63579—21no—
21C2×#938—32no—
22C2×#938—61no—
23C22×ΓL(1,𝔽25)—32no—
24C2×C3×#102—32no—
25C2×#832D312yescomplemented
26#4987—32no—
27#4551—32no—
28#4529—32no—
29C24×C32—21no—
30C23×C3⋊D3D411yescharacteristic
31C2×C32×D4—22no—
32C2×C32×D4—21no—
33C2×C32×D4—22no—
34C2×C32×D4C2×C412yes—
35C22×C32×C4C2311yescharacteristic
36C2×#638—24no—
37C2×#638—44no—
38C22×C3⋊Dic3—21no—
39C22×C3⋊Dic3D411yescharacteristic
40C2×C4×C3⋊D3—22no—
41C2×C4×C3⋊D3—22no—
42C2×C4×C3⋊D3C2×C412yes—
43C2×C4×C3⋊D3D412yes—
44C22×C3×Dic3—22no—
45#3757—24no—
46C4⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊Dic3)C2×C412yes—
48C3×C4⋊Dic3—22no—
49C3×C4⋊Dic3—22no—
50C2×C3×C3⋊C8—22no—
51C2×C3×C3⋊C8—22no—
52C2×C3×C3⋊C8D414yescomplemented
53#3722—42no—
54#3712—28no—
55#1895—91noSylow
56C22×C3×D4—21no—
57C22×C3×D4C2×D312yes—
58C22×#67—62no—
59C22×#67—62no—
60C22×C4×D3—32no—
61C22×C4×D3—61no—
62C22×C3×C8—32no—
63C2×C3×C4⋊C4—32no—
64C2×#217—62no—
65C2×#217—62no—
66#938—38no—
67#938—64no—
68#938—64no—
69#938—64no—
70C2×(C2×D3)⋊C4—62no—
71C2×C4⋊Dic3—32no—
72C2×C4⋊Dic3—61no—
73C2×C4⋊Dic3C2×D312yes—
74C22×C3⋊C8C2×D312yes—
75C2×ΓL(1,𝔽25)—64no—
76C2×ΓL(1,𝔽25)—64no—
77C2×ΓL(1,𝔽25)—64no—
78C3×#102—64no—
79#832—24no—
80C3⋊#95—62no—
81C23×C32—22no—
82C23×C32—42no—
83C23×C32—41no—
84C23×C32—42no—
85C23×C32—41no—
86C23×C32C2×D411yescharacteristic
87C22×C3⋊D3—22no—
88C22×C3⋊D3—22no—
89C22×C3⋊D3—21no—
90C22×C3⋊D3—21no—
91C22×C3⋊D3C22⋊C412yes—
92C32×D4—24no—
93C32×D4—41no—
94C32×D4—41no—
95C2×C32×C4C22×C411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D412yes—
98C2×C32×C4#4112yes—
99#638—48no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—42no—
103C2×C3⋊Dic3C22⋊C412yes—
104C4×C3⋊D3—24no—
105C4×C3⋊D3—22no—
106C4×C3⋊D3—22no—
107C2×C3×Dic3—42no—
108C2×C3×Dic3—42no—
109C2×C3×Dic3—42no—
110C3×C3⋊C8—24no—
111C22×M16—91no—
112C2×#121—91no—
113C2×#102—32no—
114#380—91no—
115#340—92no—
116C24×C3—22no—
117C24×C3—22no—
118C23×D3—32no—
119C23×D3—61no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—42no—
128C2×C3×D4—42no—
129C2×C3×D4C4×D314yes—
130C22×C3×C4—21no—
131C22×C3×C4—62no—
132C22×C3×C4C22×D312yes—
133C2×#67—68no—
134C2×#67—68no—
135C2×#67—128no—
136C2×#67—124no—
137C2×#67—124no—
138C22×Dic3—22no—
139C22×Dic3—32no—
140C22×Dic3—61no—
141C22×Dic3—62no—
142C22×Dic3—62no—
143C2×C4×D3—34no—
144C2×C4×D3—34no—
145C2×C4×D3—62no—
146C2×C4×D3—62no—
147C2×C4×D3—62no—
148C2×C4×D3—62no—
149C2×C4×D3—62no—
150C2×C4×D3—62no—
151C2×C4×D3—122no—
152C2×C4×D3—122no—
153C2×C3×C8—34no—
154C2×C3×C8—62no—
155C2×C3×C8—62no—
156C3×C4⋊C4—62no—
157C3×C4⋊C4—62no—
158#217—68no—
159#217—68no—
160(C2×D3)⋊C4—124no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—34no—
164C4⋊Dic3—62no—
165C4⋊Dic3—62no—
166C4⋊Dic3—62no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8#6714yescomplemented
170ΓL(1,𝔽25)—616no—
171C22×C32—42no—
172C22×C32—42no—
173C22×C32—42no—
174C22×C32—82no—
175C22×C32—82no—
176C22×C32C2×C22⋊C411yescharacteristic
177C22×C32#12112yes—
178C22×C32AGL(1,ℤ/8)11yescharacteristic
179C22×C32AGL(1,ℤ/8)12yes—
180C22×C32#13711yescharacteristic
181C2×C3⋊D3—22no—
182C2×C3⋊D3—22no—
183C2×C3⋊D3—22no—
184C2×C3⋊D3—42no—
185C2×C3⋊D3—42no—
186C32×C4C4×D412yes—
187C32×C4#12011yescharacteristic
188C32×C4#12311yescharacteristic
189C3⋊Dic3—22no—
190C3⋊Dic3—42no—
191C3×Dic3—82no—
192C22×D4—181no—
193C22×D4D3211yescharacteristic
194C23×C4—91no—
195C2×M16—184no—
196C2×M16—182no—
197C22×C8—32no—
198#121—94no—
199#121—182no—
200C2×C4⋊C4—32no—
201C2×C4⋊C4—91no—
202C2×C22⋊C4—181no—
203C2×C22⋊C4—181no—
204#102—64no—
205#95—181no—
206C23×C3—24no—
207C23×C3—24no—
208C23×C3—21no—
209C23×C3—44no—
210C23×C3—44no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3D3×D412yes—
220C22×D3—34no—
221C22×D3—62no—
222C22×D3—64no—
223C22×D3—62no—
224C22×D3—62no—
225C22×D3—62no—
226C22×D3—62no—
227C22×D3—62no—
228C22×D3—62no—
229C22×D3—122no—
230C22×D3—121no—
231C22×D3—121no—
232C3×D4—24no—
233C3×D4—24no—
234C3×D4—44no—
235C3×D4—42no—
236C3×D4—42no—
237C3×D4—42no—
238C3×D4—42no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—21no—
243C2×C3×C4—122no—
244C2×C3×C4—122no—
245C2×C3×C4—122no—
246C2×C3×C4C2×C4×D312yes—
247C2×C3×C4#23512yes—
248C2×C3×C4D3×D412yes—
249C2×C3×C4D3×D412yes—
250C2×C3×C4#23712yes—
251C2×C3×C4C2×#6712yes—
252#67—1216no—
253#67—1216no—
254C2×Dic3—34no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—62no—
259C2×Dic3—62no—
260C2×Dic3—62no—
261C2×Dic3—64no—
262C2×Dic3—64no—
263C2×Dic3—122no—
264C2×Dic3—124no—
265C2×Dic3—122no—
266C2×Dic3—122no—
267C4×D3—64no—
268C4×D3—64no—
269C4×D3—64no—
270C4×D3—64no—
271C4×D3—124no—
272C4×D3—122no—
273C4×D3—122no—
274C3×C8—64no—
275C3⋊C8—24no—
276C2×C32—82no—
277C2×C32#38611yescharacteristic
278C2×C32#41711yescharacteristic
279C2×C32#41811yescharacteristic
280C2×C32#46812yes—
281C2×C32#46912yes—
282C3⋊D3—42no—
283C24—21no—
284C24—91no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—21no—
288C2×D4—184no—
289C2×D4—364no—
290C2×D4#62412yes—
291C22×C4—62no—
292C22×C4—91no—
293C22×C4—92no—
294C22×C4—92no—
295C22×C4—182no—
296C22×C4—182no—
297C22×C4—181no—
298C22×C4C2×D3211yescharacteristic
299M16—188no—
300C2×C8—34no—
301C2×C8—62no—
302C2×C8—62no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—92no—
306C4⋊C4—181no—
307C22⋊C4—184no—
308C22⋊C4—362no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—21no—
314C22×C3—44no—
315C22×C3—44no—
316C22×C3—44no—
317C22×C3—44no—
318C22×C3—44no—
319C22×C3—44no—
320C22×C3—84no—
321C22×C3—84no—
322C22×C3—82no—
323C22×C3—82no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3D3×C22⋊C412yes—
327C22×C3#88412yes—
328C22×C3#91112yes—
329C22×C3#91112yes—
330C22×C3#91412yes—
331C22×C3#91512yes—
332C22×C3#93812yes—
333C2×D3—64no—
334C2×D3—64no—
335C2×D3—64no—
336C2×D3—64no—
337C2×D3—122no—
338C2×D3—122no—
339C2×D3—124no—
340C2×D3—124no—
341C2×D3—122no—
342C2×D3—122no—
343C2×D3—122no—
344C2×D3—122no—
345C3×C4—22no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—242no—
349C3×C4#67⋊C412yes—
350C3×C4#88312yes—
351C3×C4C4×#6712yes—
352C3×C4#93712yes—
353Dic3—64no—
354Dic3—82no—
355Dic3—122no—
356Dic3—124no—
357Dic3—124no—
358C32#189511yescharacteristic, complemented, Sylow
359C23—22no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—92no—
365C23—181no—
366C23—181no—
367C23—182no—
368C23—182no—
369C23#381111yescharacteristic
370D4—24no—
371D4—41no—
372D4—41no—
373D4—368no—
374C2×C4—92no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—182no—
379C2×C4—184no—
380C2×C4—182no—
381C2×C4—182no—
382C2×C4—182no—
383C2×C4—362no—
384C2×C4#380412yes—
385C2×C4C2×#62411yescharacteristic
386C2×C4D3×#6712yes—
387C2×C4#381111yescharacteristic
388C8—64no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—22no—
392C2×C3—22no—
393C2×C3—21no—
394C2×C3—84no—
395C2×C3—84no—
396C2×C3#477012yes—
397C2×C3#477112yes—
398C2×C3#478012yes—
399C2×C3#478212yes—
400C2×C3#495612yes—
401C2×C3#515912yes—
402C2×C3#517412yes—
403D3—124no—
404D3—124no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—82no—
409C22—82no—
410C22—182no—
411C22—182no—
412C22—182no—
413C22—362no—
414C22—362no—
415C22#6392711yescharacteristic
416C22#6393111yescharacteristic
417C22#6393712yes—
418C22#6395812yes—
419C22#6397411yescharacteristic
420C4—182no—
421C4—242no—
422C4—362no—
423C4#6395212yes—
424C4#6397511yescharacteristic
425C4#6398311yescharacteristic
426C3—21no—
427C3#7020312yescomplemented
428C2—82no—
429C2—362no—
430C2#1059141811yescharacteristic
431C2#1059142611yescharacteristic
432C2#1059145412yes—
433C2#1059146412yes—
434C2#1059172211yescharacteristic
subgroupG/H
centerC23#3811
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×D4C22
socleC23×C32C2×D4
prime radicalC2×#64147C2
cosocleC22×C32×C4C23
hypercenterC22×D4D32
hypocenterC32#1895
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C22×#3712C2semidirect
3. C2×C3×#832C2semidirect
4. #10591325C2semidirect
5. #10589164C2semidirect
12. #64147C4semidirect
25. C2×#832D3semidirect
52. C2×C3×C3⋊C8D4semidirect
169. C2×C3⋊C8#67semidirect
358. C32#1895semidirect
427. C3#70203semidirect

Maximal subgroups

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

subgroupindex
C2×#641472
C22×#37122
C2×C3×#8322
#105913252
#105891642
#702033