Finite Group Databasebeta

← #11793157

Subgroups — #11793157

Order 1152.

1104 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64139C211yesmaximal, characteristic
2C2×C32×C22⋊C8C211yesmaximal, characteristic, complemented
3#10592911C211yesmaximal, characteristic, complemented
4C2×#63648C211yesmaximal, characteristic, complemented
5C2×#63648C211yesmaximal, characteristic, complemented
6C2×#63645C211yesmaximal, characteristic, complemented
7#10589417C211yesmaximal, characteristic, complemented
8#69934—34nomaximal
9C23×C32×C4C2211yescharacteristic
10C22×#638—21no—
11C22×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C2×#3757—21no—
15#64139—22no—
16#64139—22no—
17C2×#3752—21no—
18C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C32×C22⋊C8—22no—
22#63658—21no—
23#63648—22no—
24#63648—22no—
25#63645—22no—
26C2×#930—34no—
27C2×C3×C22⋊C8D314yescomplemented
28#5212—34no—
29C2×#821—34no—
30C2×#821—34no—
31C2×#818—34no—
32#4551—34no—
33C24×C32D411yescharacteristic
34C23×C3⋊D3—21no—
35C22×C32×C4—21no—
36C22×C32×C4—21no—
37C22×C32×C4—21no—
38C22×C32×C4—21no—
39C22×C32×C4—21no—
40C22×C32×C4—21no—
41C22×C32×C4D411yescharacteristic
42C22×C32×C4C2311yescharacteristic
43C2×#638—24no—
44C2×#638—42no—
45C2×#638—42no—
46C22×C3⋊Dic3—21no—
47C22×C3⋊Dic3—21no—
48C22×C3⋊Dic3—21no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4⋊(C3⋊D3)—21no—
51C2×C4⋊(C3⋊D3)—21no—
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)D412yes—
55C2×C32×C8—21no—
56C2×C32×C8—21no—
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C2×C32×C8D412yescomplemented
61#3757—42no—
62#3752—24no—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)—21no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)—21no—
67C4⋊(C3⋊Dic3)—21no—
68C4⋊(C3⋊Dic3)D412yes—
69#2033—91noSylow
70C23×C3×C4C2×D314yes—
71C22×#67—64no—
72C22×D12—34no—
73C22×C3×C8C2×D314yes—
74C22×C3×C8C2×D314yes—
75C2×#217—64no—
76#930—68no—
77#930—68no—
78C2×(C2×D3)⋊C4—64no—
79C2×C4⋊Dic3—34no—
80C2×C4⋊Dic3—34no—
81C2×C4⋊Dic3—34no—
82C3×C22⋊C8—28no—
83C3⋊#95—64no—
84#821—68no—
85#821—68no—
86#818—68no—
87C23×C32—21no—
88C23×C32—21no—
89C23×C32—21no—
90C23×C32—21no—
91C23×C32—21no—
92C23×C32—21no—
93C23×C32D812yes—
94C23×C32C2×D411yescharacteristic
95C22×C3⋊D3—22no—
96C22×C3⋊D3—41no—
97C22×C3⋊D3—41no—
98C22×C3⋊D3—41no—
99C22×C3⋊D3—41no—
100C22×C3⋊D3—41no—
101C22×C3⋊D3—41no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—41no—
106C2×C32×C4—41no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4#4111yescharacteristic
113C2×C32×C4#4111yescharacteristic
114C2×C32×C4#4111yescharacteristic
115#638—48no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—41no—
118C2×C3⋊Dic3—41no—
119C2×C3⋊Dic3—41no—
120C2×C3⋊Dic3—41no—
121C2×C3⋊Dic3—41no—
122C2×C3⋊Dic3—41no—
123C2×C3⋊Dic3—41no—
124C2×C3⋊Dic3—41no—
125C4⋊(C3⋊D3)—22no—
126C4⋊(C3⋊D3)—22no—
127C4⋊(C3⋊D3)—41no—
128C4⋊(C3⋊D3)—41no—
129C32×C8—22no—
130C32×C8—41no—
131C2×#121—91no—
132C2×#107—91no—
133C2×#102—91no—
134C2×#102—91no—
135C2×C22⋊C8C3⋊D311yescharacteristic, complemented
136#380—91no—
137#340—91no—
138C24×C3D1214yes—
139C23×D3—64no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4D1214yes—
147C22×C3×C4C22×D314yes—
148C2×#67—616no—
149C2×#67—128no—
150C2×#67—128no—
151C22×Dic3—64no—
152C22×Dic3—64no—
153C22×Dic3—64no—
154C2×D12—38no—
155C2×D12—64no—
156C2×D12—64no—
157C2×D12—64no—
158C2×D12—64no—
159C2×D12—64no—
160C2×C3×C8—24no—
161C2×C3×C8—24no—
162C2×C3×C8—24no—
163C2×C3×C8—24no—
164C2×C3×C8—24no—
165C2×C3×C8#6718yescomplemented
166#217—128no—
167(C2×D3)⋊C4—616no—
168C4⋊Dic3—38no—
169C4⋊Dic3—64no—
170C4⋊Dic3—64no—
171C4⋊Dic3—64no—
172C4⋊Dic3—64no—
173C4⋊Dic3—64no—
174C22×C32—22no—
175C22×C32—22no—
176C22×C32—22no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32—41no—
181C22×C32#12011yescharacteristic
182C22×C32#12111yescharacteristic
183C22×C32#12311yescharacteristic
184C22×C32C2×D811yescharacteristic
185C22×C32#13511yescharacteristic
186C22×C32AGL(1,ℤ/8)11yescharacteristic
187C22×C32#13711yescharacteristic
188C2×C3⋊D3—42no—
189C2×C3⋊D3—42no—
190C2×C3⋊D3—42no—
191C2×C3⋊D3—81no—
192C2×C3⋊D3—81no—
193C2×C3⋊D3—81no—
194C2×C3⋊D3—81no—
195C32×C4—41no—
196C32×C4#12111yescharacteristic
197C32×C4#12111yescharacteristic
198C32×C4#12311yescharacteristic
199C32×C4#12411yescharacteristic
200C3⋊Dic3—42no—
201C3⋊Dic3—81no—
202C3⋊Dic3—81no—
203C22×D4—91no—
204C22×D4—181no—
205C23×C4C2×C3⋊D311yescharacteristic
206C22×C8C2×C3⋊D311yescharacteristic
207C22×C8C2×C3⋊D311yescharacteristic
208#121—182no—
209#121—182no—
210C2×C4⋊C4—91no—
211C2×C4⋊C4—91no—
212C2×C4⋊C4—91no—
213C2×C22⋊C4—181no—
214C2×C22⋊C4—181no—
215#107—182no—
216#102—182no—
217#102—182no—
218C22⋊C8—22no—
219#95—181no—
220C23×C3—24no—
221C23×C3—24no—
222C23×C3—24no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—24no—
226C23×C3D2418yes—
227C23×C3C2×D1214yes—
228C22×D3—68no—
229C22×D3—124no—
230C22×D3—124no—
231C22×D3—124no—
232C22×D3—124no—
233C22×D3—124no—
234C22×D3—124no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4—44no—
241C2×C3×C4—44no—
242C2×C3×C4#23514yes—
243C2×C3×C4D3×D414yes—
244C2×C3×C4D3×D414yes—
245C2×C3×C4#23714yes—
246C2×C3×C4#23714yes—
247C2×C3×C4C2×#6714yes—
248#67—1232no—
249C2×Dic3—68no—
250C2×Dic3—124no—
251C2×Dic3—124no—
252C2×Dic3—124no—
253C2×Dic3—124no—
254C2×Dic3—124no—
255C2×Dic3—124no—
256C2×Dic3—124no—
257C2×Dic3—124no—
258D12—68no—
259D12—68no—
260D12—124no—
261D12—124no—
262C3×C8—28no—
263C3×C8—44no—
264C2×C32—42no—
265C2×C32#39411yescharacteristic
266C2×C32#44711yescharacteristic
267C2×C32#45211yescharacteristic
268C2×C32#46611yescharacteristic
269C2×C32#46811yescharacteristic
270C2×C32#48011yescharacteristic
271C2×C32#48211yescharacteristic
272C3⋊D3—82no—
273C24—181no—
274C24C4⋊(C3⋊D3)11yescharacteristic
275C2×D4—92no—
276C2×D4—184no—
277C2×D4—181no—
278C2×D4—181no—
279C2×D4—181no—
280C2×D4—181no—
281C2×D4—181no—
282C2×D4—362no—
283C2×D4—362no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—181no—
291C22×C4—181no—
292C22×C4—181no—
293C22×C4C4⋊(C3⋊D3)11yescharacteristic
294C22×C4C22×C3⋊D311yescharacteristic
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8#63812yescomplemented
301C4⋊C4—92no—
302C4⋊C4—181no—
303C4⋊C4—181no—
304C4⋊C4—181no—
305C4⋊C4—181no—
306C4⋊C4—181no—
307C22⋊C4—184no—
308C22⋊C4—362no—
309C22×C3—28no—
310C22×C3—28no—
311C22×C3—28no—
312C22×C3—44no—
313C22×C3—44no—
314C22×C3—44no—
315C22×C3—44no—
316C22×C3#88214yes—
317C22×C3#88614yes—
318C22×C3C2×D2414yes—
319C22×C3#90414yes—
320C22×C3#90814yes—
321C22×C3#90914yes—
322C22×C3#93014yes—
323C2×D3—128no—
324C2×D3—128no—
325C2×D3—128no—
326C2×D3—244no—
327C2×D3—244no—
328C2×D3—244no—
329C2×D3—244no—
330C3×C4—44no—
331C3×C4#88314yes—
332C3×C4#88514yes—
333C3×C4#93814yes—
334C3×C4#93914yes—
335Dic3—128no—
336Dic3—244no—
337Dic3—244no—
338C32#203311yescharacteristic, complemented, Sylow
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—182no—
346C23—361no—
347C23—361no—
348C23—361no—
349C23—361no—
350C23—361no—
351C23—361no—
352C23#374512yes—
353C23C2×C4⋊(C3⋊D3)11yescharacteristic
354D4—182no—
355D4—182no—
356D4—368no—
357D4—361no—
358D4—361no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—182no—
367C2×C4—361no—
368C2×C4—361no—
369C2×C4—361no—
370C2×C4—361no—
371C2×C4—361no—
372C2×C4—361no—
373C2×C4—361no—
374C2×C4—361no—
375C2×C4#382811yescharacteristic
376C2×C4D4×C3⋊D311yescharacteristic
377C2×C4D4×C3⋊D311yescharacteristic
378C2×C4#383011yescharacteristic
379C2×C4#383011yescharacteristic
380C2×C4C2×#63811yescharacteristic
381C8—22no—
382C8—41no—
383C2×C3—48no—
384C2×C3#473214yes—
385C2×C3#473314yes—
386C2×C3#473614yes—
387C2×C3#473714yes—
388C2×C3#496114yes—
389C2×C3#511514yes—
390C2×C3#513414yes—
391D3—248no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—362no—
400C22—362no—
401C22—362no—
402C22—721no—
403C22—721no—
404C22—721no—
405C22—721no—
406C22#6409111yescharacteristic
407C22#6409511yescharacteristic
408C22C2×#374511yescharacteristic
409C22#6411311yescharacteristic
410C22#6411711yescharacteristic
411C22#6411811yescharacteristic
412C22#6413911yescharacteristic
413C4—41no—
414C4—362no—
415C4—721no—
416C4—721no—
417C4#6409211yescharacteristic
418C4#6409411yescharacteristic
419C4#6414711yescharacteristic
420C4#6414811yescharacteristic
421C3#6993414yescomplemented
422C2—42no—
423C2—722no—
424C2#1059243111yescharacteristic
425C2#1059243211yescharacteristic
426C2#1059243511yescharacteristic
427C2#1059243611yescharacteristic
428C2#1059266011yescharacteristic
429C2#1059281411yescharacteristic
430C2#1059283311yescharacteristic
subgroupG/H
centerC23C2×C4⋊(C3⋊D3)
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC2×C32×C22⋊C8C2
socleC23×C32C2×D4
prime radicalC2×#64139C2
hypercenterC2×C22⋊C8C3⋊D3
hypocenterC32#2033
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C32×C22⋊C8C2semidirect
3. #10592911C2semidirect
4. C2×#63648C2semidirect
5. C2×#63648C2semidirect
6. C2×#63645C2semidirect
7. #10589417C2semidirect
27. C2×C3×C22⋊C8D3semidirect
60. C2×C32×C8D4semidirect
135. C2×C22⋊C8C3⋊D3semidirect
165. C2×C3×C8#67semidirect
300. C2×C8#638semidirect
338. C32#2033semidirect
421. C3#69934semidirect

Maximal subgroups

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

subgroupindex
C2×#641392
C2×C32×C22⋊C82
#105929112
C2×#636482
C2×#636452
#105894172
#699343