Finite Group Databasebeta

← #11791495

Subgroups — #11791495

Order 1152.

853 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64084C211yesmaximal, characteristic
2C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C2×#63567C212yesmaximal, complemented
5C2×#63564C212yesmaximal
6#70458—31nomaximal
7#69488—31nomaximal
8C22×C4⋊(C3⋊D3)C2211yescharacteristic
9C22×#634C2211yescharacteristic
10C2×C32×C42C2211yescharacteristic
11C2×#3752—21no—
12C2×#3752—21no—
13#64084—22no—
14#64084C414yescomplemented
15C2×C3×C4⋊Dic3C2212yes—
16C2×C3×C4×Dic3—21no—
17C22×C3×C3⋊C8C2212yes—
18C3×C3⋊#95—21no—
19C3×C4⋊(C3⋊C8)—22no—
20#63567—24no—
21#63564—24no—
22C2×#875—31no—
23C2×#875—31no—
24C2×#875—61no—
25C2×C3×C4⋊C8—31no—
26C3×C4×C4⋊C4—31no—
27C2×#821—32no—
28C2×#816—32no—
29C2×#810—32no—
30C2×#809—32no—
31C4×C4⋊Dic3D311yescharacteristic, complemented
32C2×C4⋊(C3⋊C8)D311yescharacteristic, complemented
33C23×C3⋊D3—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4D411yescharacteristic
36C22×C32×C4C2311yescharacteristic
37C22×C3⋊Dic3—21no—
38C2×C4⋊(C3⋊D3)—22no—
39C2×C4⋊(C3⋊D3)—22no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)C2×C412yes—
42C2×#634—22no—
43C2×#634—22no—
44C2×#634—21no—
45C2×#634C2×C412yes—
46C22×C3×Dic3—22no—
47C32×C42C2×C412yes—
48C32×C42D412yes—
49#3752—24no—
50#3752—24no—
51C3×C4⋊Dic3—22no—
52C3×C4⋊Dic3D414yes—
53C3×C4×Dic3—42no—
54C2×C3×C3⋊C8—22no—
55C2×C3×C3⋊C8—22no—
56C2×C3×C3⋊C8—22no—
57#1983—91noSylow
58C22×D12—31no—
59C22×D12—31no—
60C22×D12—61no—
61C22×Dic6—31no—
62C22×Dic6—31no—
63C22×Dic6—61no—
64C22×C3×C8—32no—
65C2×C3×C4⋊C4—32no—
66C2×C3×C42—21no—
67C2×C3×C42—61no—
68C2×C3×C42C2×D311yescharacteristic
69C2×C3×C42C2×D311yescharacteristic
70C2×(C2×D3)⋊C4—62no—
71C2×(C2×D3)⋊C4—62no—
72C2×(C2×D3)⋊C4—61no—
73C2×(C2×D3)⋊C4—61no—
74C2×(C2×D3)⋊C4—61no—
75C2×(C2×D3)⋊C4—61no—
76C2×C4⋊Dic3C2×D312yes—
77C2×C4×Dic3—21no—
78C22×C3⋊C8C2×D312yes—
79#875—34no—
80#875—34no—
81#875—64no—
82#875—64no—
83#875—62no—
84#875—62no—
85C3×C4⋊C8—62no—
86C3×#95—61no—
87C3⋊#95—21no—
88#821—64no—
89#816—64no—
90#810—64no—
91#809—64no—
92C4⋊(C3⋊C8)—22no—
93C23×C32C2×D411yescharacteristic
94C22×C3⋊D3—22no—
95C22×C3⋊D3—42no—
96C22×C3⋊D3—42no—
97C22×C3⋊D3—41no—
98C22×C3⋊D3—41no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4C22⋊C412yes—
103C2×C32×C4C22⋊C412yes—
104C2×C32×C4C22×C411yescharacteristic
105C2×C32×C4C2×D412yes—
106C2×C32×C4C2×D411yescharacteristic
107C2×C32×C4#4112yes—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—42no—
110C4⋊(C3⋊D3)—24no—
111C4⋊(C3⋊D3)—41no—
112C4⋊(C3⋊D3)—41no—
113#634—24no—
114#634—41no—
115#634—41no—
116C2×C3×Dic3—24no—
117C2×C3×Dic3—42no—
118C2×C3×Dic3—42no—
119C3×C3⋊C8—42no—
120C2×#124—91no—
121C2×C4⋊C8—31no—
122C2×#103—92no—
123C2×#102—92no—
124C4×C4⋊C4—31no—
125C23×D3—62no—
126C23×D3—61no—
127C23×D3—61no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—62no—
132C22×C3×C4D1211yescharacteristic
133C22×C3×C4D1211yescharacteristic
134C22×C3×C4#6711yescharacteristic
135C22×C3×C4#6711yescharacteristic
136C22×C3×C4C22×D311yescharacteristic
137C22×C3×C4C22×D311yescharacteristic
138C22×Dic3—22no—
139C22×Dic3—62no—
140C22×Dic3—61no—
141C22×Dic3—61no—
142C2×D12—32no—
143C2×D12—32no—
144C2×D12—62no—
145C2×D12—62no—
146C2×D12—62no—
147C2×D12—62no—
148C2×D12—62no—
149C2×D12—62no—
150C2×D12—61no—
151C2×D12—61no—
152C2×D12—122no—
153C2×D12—122no—
154C2×Dic6—32no—
155C2×Dic6—32no—
156C2×Dic6—62no—
157C2×Dic6—62no—
158C2×Dic6—62no—
159C2×Dic6—62no—
160C2×Dic6—62no—
161C2×Dic6—62no—
162C2×Dic6—61no—
163C2×Dic6—61no—
164C2×Dic6—122no—
165C2×Dic6—122no—
166C2×C3×C8—62no—
167C2×C3×C8—62no—
168C2×C3×C8—62no—
169C3×C4⋊C4—34no—
170C3×C4⋊C4—62no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42—122no—
174C3×C42C4×D312yes—
175C3×C42C4×D312yes—
176C3×C42D1212yes—
177C3×C42#6712yes—
178(C2×D3)⋊C4—68no—
179(C2×D3)⋊C4—68no—
180(C2×D3)⋊C4—64no—
181(C2×D3)⋊C4—64no—
182(C2×D3)⋊C4—64no—
183(C2×D3)⋊C4—64no—
184C4⋊Dic3—22no—
185C4⋊Dic3D1214yes—
186C4×Dic3—42no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C22×C32C2×C22⋊C411yescharacteristic
191C22×C32#12112yes—
192C22×C32#13512yes—
193C22×C32AGL(1,ℤ/8)11yescharacteristic
194C22×C32#13711yescharacteristic
195C2×C3⋊D3—42no—
196C2×C3⋊D3—42no—
197C2×C3⋊D3—42no—
198C2×C3⋊D3—82no—
199C2×C3⋊D3—82no—
200C32×C4—22no—
201C32×C4—22no—
202C32×C4C4×D412yes—
203C32×C4#12411yescharacteristic
204C32×C4#12711yescharacteristic
205C3⋊Dic3—42no—
206C3×Dic3—44no—
207C22×Q8—91no—
208C22×D4—91no—
209C22×C8—32no—
210#124—94no—
211#124—182no—
212C2×C4⋊C4—32no—
213C2×C22⋊C4—181no—
214C2×C22⋊C4—181no—
215C2×C42—61no—
216C2×C42D3211yescharacteristic
217C4⋊C8—62no—
218#103—184no—
219#102—184no—
220#95—61no—
221C23×C3—21no—
222C23×C3C2×D1211yescharacteristic
223C23×C3C2×#6711yescharacteristic
224C22×D3—64no—
225C22×D3—62no—
226C22×D3—62no—
227C22×D3—124no—
228C22×D3—124no—
229C22×D3—122no—
230C22×D3—122no—
231C22×D3—122no—
232C22×D3—122no—
233C22×D3—121no—
234C22×D3—121no—
235C22×D3—122no—
236C22×D3—121no—
237C22×D3—121no—
238C22×D3—122no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—64no—
255C2×C3×C4—122no—
256C2×C3×C4—122no—
257C2×C3×C4(C2×D3)⋊C412yes—
258C2×C3×C4(C2×D3)⋊C412yes—
259C2×C3×C4(C2×D3)⋊C412yes—
260C2×C3×C4(C2×D3)⋊C412yes—
261C2×C3×C4C2×C4×D311yescharacteristic
262C2×C3×C4C2×C4×D311yescharacteristic
263C2×C3×C4C2×D1212yes—
264C2×C3×C4C2×D1211yescharacteristic
265C2×C3×C4#23512yes—
266C2×C3×C4D3×D412yes—
267C2×C3×C4#23912yes—
268C2×C3×C4C2×#6711yescharacteristic
269C2×Dic3—24no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—64no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3—124no—
276C2×Dic3—122no—
277C2×Dic3—122no—
278D12—64no—
279D12—64no—
280D12—124no—
281D12—122no—
282D12—122no—
283D12—121no—
284D12—121no—
285D12—121no—
286D12—121no—
287Dic6—64no—
288Dic6—64no—
289Dic6—124no—
290Dic6—122no—
291Dic6—122no—
292Dic6—121no—
293Dic6—121no—
294Dic6—121no—
295Dic6—121no—
296C3×C8—122no—
297C3⋊C8—42no—
298C2×C32#39011yescharacteristic
299C2×C32#41611yescharacteristic
300C2×C32#41711yescharacteristic
301C2×C32#46312yes—
302C2×C32#46412yes—
303C3⋊D3—82no—
304C24—181no—
305C2×Q8—92no—
306C2×Q8—181no—
307C2×Q8—182no—
308C2×Q8—182no—
309C2×D4—92no—
310C2×D4—182no—
311C2×D4—182no—
312C2×D4—181no—
313C22×C4—62no—
314C22×C4—181no—
315C22×C4#62611yescharacteristic
316C22×C4#62611yescharacteristic
317C22×C4C2×D3211yescharacteristic
318C2×C8—62no—
319C2×C8—62no—
320C2×C8—62no—
321C4⋊C4—34no—
322C4⋊C4—62no—
323C22⋊C4—184no—
324C22⋊C4—184no—
325C42—122no—
326C42#62412yes—
327C42#62612yes—
328C22×C3—22no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—22no—
332C22×C3—21no—
333C22×C3#89512yes—
334C22×C3#90412yes—
335C22×C3#90811yescharacteristic
336C22×C3#90911yescharacteristic
337C22×C3C2×(C2×D3)⋊C411yescharacteristic
338C22×C3C2×(C2×D3)⋊C411yescharacteristic
339C22×C3#93012yes—
340C22×C3#93211yescharacteristic
341C22×C3#94211yescharacteristic
342C22×C3#95012yes—
343C2×D3—124no—
344C2×D3—124no—
345C2×D3—124no—
346C2×D3—122no—
347C2×D3—122no—
348C2×D3—122no—
349C2×D3—122no—
350C2×D3—122no—
351C2×D3—122no—
352C2×D3—244no—
353C2×D3—244no—
354C2×D3—242no—
355C2×D3—242no—
356C2×D3—242no—
357C2×D3—242no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—22no—
362C3×C4—21no—
363C3×C4—22no—
364C3×C4—21no—
365C3×C4—42no—
366C3×C4—42no—
367C3×C4—124no—
368C3×C4C4×D1212yes—
369C3×C4#87411yescharacteristic
370C3×C4#87511yescharacteristic
371C3×C4#89312yes—
372C3×C4#94011yescharacteristic
373C3×C4#94711yescharacteristic
374Dic3—44no—
375Dic3—124no—
376Dic3—122no—
377Dic3—122no—
378C32#198311yescharacteristic, complemented, Sylow
379C23—182no—
380C23—362no—
381C23—361no—
382C23—361no—
383C23—362no—
384C23C2×#62611yescharacteristic
385Q8—184no—
386Q8—361no—
387Q8—361no—
388D4—184no—
389D4—361no—
390D4—361no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—64no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—182no—
398C2×C4—362no—
399C2×C4#372212yes—
400C2×C4#372212yes—
401C2×C4#379912yes—
402C2×C4D3×D1212yes—
403C2×C4C2×#62411yescharacteristic
404C2×C4C2×#62611yescharacteristic
405C8—122no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—22no—
410C2×C3—22no—
411C2×C3#484412yes—
412C2×C3#484512yes—
413C2×C3#493911yescharacteristic
414C2×C3#496611yescharacteristic
415C2×C3#498811yescharacteristic
416C2×C3#500211yescharacteristic
417C2×C3#501612yes—
418C2×C3#502812yes—
419C2×C3#511311yescharacteristic
420C2×C3#513311yescharacteristic
421D3—244no—
422D3—242no—
423D3—242no—
424C22—362no—
425C22—362no—
426C22—362no—
427C22—722no—
428C22—722no—
429C22#6382912yes—
430C22#6383011yescharacteristic
431C22#6383311yescharacteristic
432C22#6390912yes—
433C22C2×#372211yescharacteristic
434C4—22no—
435C4—22no—
436C4—124no—
437C4—362no—
438C4#6386411yescharacteristic
439C4#6389412yes—
440C4#6391111yescharacteristic
441C3—21no—
442C3#6948811yescharacteristic, complemented
443C3#7045811yescharacteristic, complemented
444C2—722no—
445C2#1059036612yes—
446C2#1059036812yes—
447C2#1059110511yescharacteristic
448C2#1059110611yescharacteristic
449C2#1059132011yescharacteristic
subgroupG/H
centerC23C2×#626
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socleC23×C32C2×D4
prime radicalC22×C4⋊(C3⋊D3)C22
cosocleC22×C32×C4C23
hypercenterC2×C42D32
hypocenterC32#1983
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C3×C4×C4⋊Dic3C2semidirect
3. C2×C3×C4⋊(C3⋊C8)C2semidirect
4. C2×#63567C2semidirect
14. #64084C4semidirect
31. C4×C4⋊Dic3D3semidirect
32. C2×C4⋊(C3⋊C8)D3semidirect
378. C32#1983semidirect
442. C3#69488semidirect
443. C3#70458semidirect

Maximal subgroups

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

subgroupindex
C2×#640842
C3×C4×C4⋊Dic32
C2×C3×C4⋊(C3⋊C8)2
C2×#635672
C2×#635642
#704583
#694883