Finite Group Databasebeta

← #11775210

Subgroups — #11775210

Order 1152.

905 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 905 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C3⋊D3×#121C211yesmaximal, characteristic
2#10591260C211yesmaximal, characteristic, complemented
3#10591159C211yesmaximal, characteristic, complemented
4C3×#4537C212yesmaximal, complemented
5#10589151C212yesmaximal, complemented
6#68303—32nomaximal
7C2×D4×C3⋊D3—21no—
8C2×D4×C3⋊D3C411yescharacteristic, complemented
9C22×C4×C3⋊D3C2211yescharacteristic
10C32×#121C2211yescharacteristic
11#64148—21no—
12#64147C411yescharacteristic, complemented
13#64139C2211yescharacteristic
14#64104C411yescharacteristic, complemented
15C4⋊C4×C3⋊D3C411yescharacteristic, complemented
16#64093—21no—
17C22⋊C4×C3⋊D3—21no—
18C2×#3722—21no—
19C2×Dic32—21no—
20C2×#3712—21no—
21C3×C3⋊#95C2212yescomplemented
22C3×#830C2212yescomplemented
23#63557—21no—
24D3×#121—32no—
25D3×#121—61no—
26#4726—32no—
27#4666—32no—
28C3×#327—32no—
29#4537D312yescomplemented
30#4470—32no—
31#4451—32no—
32C23×C3⋊D3—21no—
33C23×C3⋊D3D411yescharacteristic
34C2×C32×D4—21no—
35C2×C32×D4C2×C411yescharacteristic
36C22×C32×C4C2311yescharacteristic
37C2×#638—21no—
38C2×#638—21no—
39C2×#638—21no—
40C22×C3⋊Dic3D411yescharacteristic
41D4×C3⋊D3—42no—
42D4×C3⋊D3—44no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
45C2×C4×C3⋊D3—21no—
46C2×C4×C3⋊D3—42no—
47C2×C4×C3⋊D3C2×C411yescharacteristic
48C2×C4×C3⋊D3C2×C411yescharacteristic
49C2×C4×C3⋊D3D411yescharacteristic
50C2×C4×C3⋊D3D411yescharacteristic
51C22×C3×Dic3—22no—
52C32×C4⋊C4C2×C411yescharacteristic
53C32×C22⋊C4—21no—
54#3757—21no—
55#3752—21no—
56#3752—21no—
57C4⋊(C3⋊Dic3)C2×C411yescharacteristic
58#3750—21no—
59C2×C3×C3⋊C8—22no—
60#3722—24no—
61Dic32—42no—
62#3712—42no—
63#1507—91noSylow
64C2×D3×D4—32no—
65C2×D3×D4—61no—
66C2×D3×D4—62no—
67C2×D3×D4—62no—
68C22×C4×D3—32no—
69C22×C4×D3—61no—
70C3×#121—21no—
71C3×#121C2×D312yes—
72#939—62no—
73#939—62no—
74#938—32no—
75#938—61no—
76#930—32no—
77#930—61no—
78C2×(C2×D3)⋊C4—62no—
79C2×C4×Dic3—62no—
80C2×ΓL(1,𝔽25)—62no—
81#895—32no—
82#895—61no—
83D3×C4⋊C4—32no—
84D3×C4⋊C4—61no—
85#884—62no—
86#884—62no—
87D3×C22⋊C4—62no—
88D3×C22⋊C4—62no—
89C3×C22⋊C8—32no—
90C3×#95—32no—
91C3⋊#95C2×D312yescomplemented
92#830C2×D312yescomplemented
93(C2×D3)⋊C8—62no—
94C23×C32—21no—
95C23×C32C2×D411yescharacteristic
96C22×C3⋊D3—22no—
97C22×C3⋊D3—21no—
98C22×C3⋊D3—22no—
99C22×C3⋊D3—41no—
100C22×C3⋊D3—42no—
101C22×C3⋊D3—42no—
102C22×C3⋊D3—42no—
103C22×C3⋊D3C22⋊C411yescharacteristic
104C22×C3⋊D3C22⋊C411yescharacteristic
105C32×D4—41no—
106C32×D4—42no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4C22×C411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111#638—42no—
112#638—42no—
113#638—42no—
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3C22⋊C411yescharacteristic
117C2×C3⋊Dic3C22⋊C411yescharacteristic
118C4⋊(C3⋊D3)—41no—
119C4⋊(C3⋊D3)—42no—
120C4×C3⋊D3—22no—
121C4×C3⋊D3—22no—
122C4×C3⋊D3—41no—
123C4×C3⋊D3—41no—
124C4×C3⋊D3—42no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—42no—
127C2×C3×Dic3—42no—
128C3×C3⋊C8—42no—
129C2×#121—91no—
130#407—91no—
131C4×C22⋊C4—91no—
132#327—32no—
133#327—92no—
134C23×D3—32no—
135C23×D3—61no—
136C23×D3—62no—
137C23×D3—62no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—21no—
141C2×C3×D4C4×D312yes—
142C22×C3×C4—21no—
143C22×C3×C4—62no—
144C22×C3×C4C22×D312yes—
145C2×#67—62no—
146C2×#67—62no—
147C2×#67—62no—
148C2×#67—62no—
149C2×#67—62no—
150C2×#67—62no—
151C22×Dic3—22no—
152C22×Dic3—32no—
153C22×Dic3—61no—
154D3×D4—124no—
155D3×D4—124no—
156D3×D4—128no—
157D3×D4—124no—
158D3×D4—124no—
159C2×D12—32no—
160C2×D12—61no—
161C2×D12—62no—
162C2×D12—62no—
163C2×C4×D3—32no—
164C2×C4×D3—32no—
165C2×C4×D3—32no—
166C2×C4×D3—32no—
167C2×C4×D3—61no—
168C2×C4×D3—61no—
169C2×C4×D3—61no—
170C2×C4×D3—61no—
171C2×C4×D3—62no—
172C2×C4×D3—62no—
173C2×C4×D3—124no—
174C2×C4×D3—124no—
175C2×C3×C8—62no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4C4×D312yes—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—22no—
180#217—62no—
181#217—62no—
182(C2×D3)⋊C4—68no—
183(C2×D3)⋊C4—62no—
184(C2×D3)⋊C4—62no—
185(C2×D3)⋊C4—62no—
186(C2×D3)⋊C4—62no—
187C4⋊Dic3—32no—
188C4⋊Dic3—61no—
189Dic3⋊C4—62no—
190Dic3⋊C4—62no—
191C4×Dic3—124no—
192C2×C3⋊C8—22no—
193ΓL(1,𝔽25)—124no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32C2×C22⋊C411yescharacteristic
200C22×C32AGL(1,ℤ/8)12yes—
201C2×C3⋊D3—21no—
202C2×C3⋊D3—21no—
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3—44no—
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3⋊D3—84no—
211C2×C3⋊D3#10412yes—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—41no—
215C3⋊Dic3—21no—
216C3⋊Dic3—41no—
217C3⋊Dic3#9912yes—
218C3×Dic3—44no—
219C22×D4—91no—
220C22×D4—181no—
221C23×C4—91no—
222C2×M16—181no—
223#121—91no—
224#121—91no—
225#121—91no—
226#121—181no—
227#121—181no—
228#121D3211yescharacteristic
229C2×C4⋊C4—91no—
230C2×C22⋊C4—181no—
231C2×C22⋊C4—181no—
232C2×C42—181no—
233C22⋊C8—32no—
234C22⋊C8—181no—
235#95—32no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—21no—
239C23×C3D3×D412yes—
240C22×D3—32no—
241C22×D3—32no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—64no—
245C22×D3—64no—
246C22×D3—62no—
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—62no—
250C22×D3—122no—
251C22×D3—124no—
252C22×D3—122no—
253C22×D3—122no—
254C22×D3—124no—
255C22×D3—124no—
256C22×D3—124no—
257C22×D3—124no—
258C22×D3—124no—
259C3×D4—42no—
260C3×D4—42no—
261C3×D4—44no—
262C3×D4—42no—
263C3×D4—42no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—64no—
271C2×C3×C4—122no—
272C2×C3×C4—122no—
273C2×C3×C4C2×C4×D312yes—
274C2×C3×C4D3×D412yes—
275#67—124no—
276#67—124no—
277#67—124no—
278#67—124no—
279#67—124no—
280#67—122no—
281#67—122no—
282C2×Dic3—24no—
283C2×Dic3—32no—
284C2×Dic3—32no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—64no—
290C2×Dic3—62no—
291C2×Dic3—62no—
292C2×Dic3—122no—
293D12—122no—
294D12—122no—
295D12—124no—
296D12—124no—
297C4×D3—64no—
298C4×D3—64no—
299C4×D3—64no—
300C4×D3—122no—
301C4×D3—122no—
302C4×D3—122no—
303C4×D3—122no—
304C4×D3—122no—
305C4×D3—124no—
306C4×D3—124no—
307C3×C8—122no—
308C3⋊C8—42no—
309C2×C32—22no—
310C2×C32—81no—
311C2×C32C2×#9911yescharacteristic
312C2×C32#41811yescharacteristic
313C2×C32C2×#10411yescharacteristic
314C3⋊D3—22no—
315C3⋊D3—41no—
316C3⋊D3—81no—
317C24—91no—
318C24—181no—
319C2×D4—21no—
320C2×D4—91no—
321C2×D4—181no—
322C2×D4—181no—
323C2×D4—181no—
324C2×D4—181no—
325C2×D4—362no—
326C2×D4—364no—
327C2×D4#62411yescharacteristic
328C22×C4—62no—
329C22×C4—91no—
330C22×C4—91no—
331C22×C4—91no—
332C22×C4—91no—
333C22×C4—91no—
334C22×C4—181no—
335C22×C4—362no—
336C22×C4C2×D3211yescharacteristic
337M16—362no—
338C2×C8—62no—
339C4⋊C4—91no—
340C4⋊C4—181no—
341C4⋊C4#62411yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—181no—
344C22⋊C4—184no—
345C22⋊C4—181no—
346C22⋊C4—181no—
347C42—362no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—21no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—42no—
360C22×C3D3×C22⋊C412yes—
361C22×C3#91112yes—
362C22×C3#91412yes—
363C2×D3—34no—
364C2×D3—62no—
365C2×D3—62no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—62no—
372C2×D3—62no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—124no—
376C2×D3—124no—
377C2×D3—128no—
378C2×D3—122no—
379C2×D3—122no—
380C2×D3—122no—
381C2×D3—122no—
382C2×D3—122no—
383C2×D3—122no—
384C2×D3—248no—
385C2×D3—248no—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—41no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—124no—
393Dic3—34no—
394Dic3—44no—
395Dic3—62no—
396Dic3—62no—
397Dic3—121no—
398Dic3—122no—
399Dic3—122no—
400C32#150711yescharacteristic, complemented, Sylow
401C23—21no—
402C23—91no—
403C23—91no—
404C23—182no—
405C23—182no—
406C23—181no—
407C23—361no—
408C23—362no—
409C23—362no—
410C23—362no—
411C23#381111yescharacteristic
412D4—41no—
413D4—42no—
414D4—361no—
415D4—362no—
416D4—362no—
417D4—362no—
418D4—362no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—64no—
422C2×C4—91no—
423C2×C4—91no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—182no—
427C2×C4—182no—
428C2×C4—182no—
429C2×C4—181no—
430C2×C4—361no—
431C2×C4—361no—
432C2×C4—362no—
433C2×C4C2×#62411yescharacteristic
434C2×C4#381111yescharacteristic
435C8—122no—
436C2×C3—22no—
437C2×C3—22no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—42no—
442C2×C3—82no—
443C2×C3—82no—
444C2×C3D3×#9912yes—
445C2×C3#477012yes—
446C2×C3D3×#10412yes—
447D3—64no—
448D3—64no—
449D3—122no—
450D3—122no—
451D3—242no—
452D3—242no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—92no—
459C22—181no—
460C22—181no—
461C22—181no—
462C22—181no—
463C22—181no—
464C22—364no—
465C22—361no—
466C22—361no—
467C22—361no—
468C22—724no—
469C22#6393712yes—
470C22#6397411yescharacteristic
471C4—21no—
472C4—21no—
473C4—41no—
474C4—92no—
475C4—124no—
476C4—181no—
477C4—361no—
478C3—21no—
479C3#6830312yescomplemented
480C2—22no—
481C2—81no—
482C2—182no—
483C2—361no—
484C2—721no—
485C2#1059142611yescharacteristic
486C2#1059152711yescharacteristic
487C2#1059167111yescharacteristic
subgroupG/H
centerC22#63974
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#121C22
socleC22×C32C2×C22⋊C4
prime radicalC3⋊D3×#121C2
cosocleC22×C32×C4C23
hypercenter#121D32
hypocenterC32#1507
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #10591260C2semidirect
3. #10591159C2semidirect
4. C3×#4537C2semidirect
5. #10589151C2semidirect
8. C2×D4×C3⋊D3C4semidirect
12. #64147C4semidirect
14. #64104C4semidirect
15. C4⋊C4×C3⋊D3C4semidirect
21. C3×C3⋊#95C22semidirect
22. C3×#830C22semidirect
29. #4537D3semidirect
91. C3⋊#95C2×D3semidirect
92. #830C2×D3semidirect
400. C32#1507semidirect
479. C3#68303semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#1212
#105912602
#105911592
C3×#45372
#105891512
#683033