Finite Group Databasebeta

← #11778305

Subgroups — #11778305

Order 1152.

528 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 84 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×#5645C211yesmaximal, characteristic
2C3×#5050C211yesmaximal, characteristic, complemented
3#10591344C211yesmaximal, characteristic, complemented
4#10591171C211yesmaximal, characteristic
5#10589404C211yesmaximal, characteristic, complemented
6#10589150C211yesmaximal, characteristic
7#10589147C211yesmaximal, characteristic, complemented
8#71889—31nomaximal
9#70982—31nomaximal
10C22×C3×Dic6C2211yescharacteristic
11C32×C22⋊Q8C2211yescharacteristic
12C3×#946—21no—
13C2×C3×#216—21no—
14C3×#930C2211yescharacteristic
15C2×C3×Dic3⋊C4C2211yescharacteristic
16C3×#898—21no—
17C3×#894—21no—
18C3×#890—21no—
19C3×#886—21no—
20C3×#67⋊C4—21no—
21C2×#3724—21no—
22C2×#3715—21no—
23#63658C2211yescharacteristic
24#63657C2211yescharacteristic
25C3×#830C2211yescharacteristic
26C3×#810—21no—
27#63579—21no—
28#63565—21no—
29C3×#539—31no—
30#5645D311yescharacteristic
31C3×#451—31no—
32#5230—31no—
33#5227—31no—
34#5050D311yescharacteristic, complemented
35#5040—31no—
36#4979—31no—
37#4538—31no—
38#4538—31no—
39#4538—61no—
40#4537—31no—
41#4528—31no—
42#4470—31no—
43#4469—31no—
44C2×C32×Q8D411yescharacteristic
45C22×C32×C4C2311yescharacteristic
46C22×C3⋊Dic3—21no—
47C2×C3×#67—21no—
48C22×C3×Dic3—21no—
49C22×C3×Dic3D411yescharacteristic
50C22×C3×Dic3D411yescharacteristic
51C2×C3×D12D411yescharacteristic
52C2×C3×C4×D3—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—42no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4D411yescharacteristic
58C32×C22⋊C4—21no—
59C2×C3⋊(C3⋊C8)—21no—
60C3×#216—42no—
61C3×(C2×D3)⋊C4—21no—
62C3×(C2×D3)⋊C4—21no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3D411yescharacteristic
68C3×Dic3⋊C4—21no—
69C3×Dic3⋊C4—21no—
70C3×Dic3⋊C4—21no—
71C3×C4×Dic3—21no—
72C2×C3×C3⋊C8—21no—
73#3724—42no—
74#3715—42no—
75#1603—91noSylow
76C22×C3×Q8—31no—
77C22×Dic6C2×D311yescharacteristic
78C2×C3×Q16—61no—
79C3×#126—61no—
80C3×#125—61no—
81C3×#123—61no—
82C3×#123—61no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—61no—
85C3×C22⋊Q8C2×D311yescharacteristic
86C3×C22⋊Q8C2×D311yescharacteristic
87C3×#121—31no—
88C3×C4×D4—61no—
89C2×C3×C4⋊C4—31no—
90#946—21no—
91C2×#216—21no—
92C2×#215—61no—
93C2×#214—61no—
94#930C2×D311yescharacteristic
95C2×Dic3⋊C4—61no—
96C2×Dic3⋊C4—61no—
97C2×Dic3⋊C4C2×D311yescharacteristic
98#898—21no—
99#894—21no—
100#890—21no—
101#886—21no—
102#67⋊C4—21no—
103C3×#103—61no—
104C3×C22⋊C8—31no—
105#835—61no—
106C3⋊#95—31no—
107C3⋊#95—31no—
108C3⋊#95—61no—
109C3⋊#95—61no—
110C3⋊#95—61no—
111#830—31no—
112#830—31no—
113#830—61no—
114#830C2×D311yescharacteristic
115#810—21no—
116#810—61no—
117C23×C32C2×D411yescharacteristic
118C22×C3×D3—21no—
119C32×Q8—41no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—41no—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3—41no—
127C2×C3⋊Dic3—41no—
128C3×#67—41no—
129C3×#67—41no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C3×D12—41no—
140C3×C4×D3—81no—
141C3×Dic6—22no—
142C3×Dic6—22no—
143C3×Dic6—41no—
144C3×Dic6—41no—
145C3⋊(C3⋊C8)—41no—
146C3×C3⋊C8—41no—
147#539—31no—
148#451—31no—
149#448—91no—
150#395—91no—
151#328—91no—
152#327—91no—
153#324—91no—
154C2×C3×Q8—21no—
155C2×C3×Q8—61no—
156C2×C3×Q8—61no—
157C2×C3×Q8—122no—
158C2×C3×Q8#6711yescharacteristic
159C2×C3×Q8#6711yescharacteristic
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C22×C3×C4—21no—
163C22×C3×C4—31no—
164C22×C3×C4—31no—
165C22×C3×C4—61no—
166C22×C3×C4—61no—
167C22×C3×C4C22×D311yescharacteristic
168C22×C3×C4C22×D311yescharacteristic
169C2×#67—21no—
170C22×Dic3—21no—
171C22×Dic3—61no—
172C22×Dic3—61no—
173C22×Dic3—121no—
174C22×Dic3#6711yescharacteristic
175C22×Dic3#6711yescharacteristic
176C2×D12#6711yescharacteristic
177C2×C4×D3—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—42no—
181C3×Q16—122no—
182C2×C3×C8—61no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—31no—
187C3×C4⋊C4—41no—
188C3×C4⋊C4—61no—
189C3×C4⋊C4—61no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4#6711yescharacteristic
194C3×C4⋊C4#6711yescharacteristic
195C3×C22⋊C4—21no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4—41no—
198C3×C22⋊C4—61no—
199C3×C22⋊C4—61no—
200C3×C22⋊C4—61no—
201C3×C22⋊C4—61no—
202C3×C42—61no—
203#216—42no—
204#215—122no—
205#214—122no—
206(C2×D3)⋊C4—21no—
207(C2×D3)⋊C4—21no—
208(C2×D3)⋊C4—21no—
209(C2×D3)⋊C4—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3#6711yescharacteristic
213Dic3⋊C4—21no—
214Dic3⋊C4—21no—
215Dic3⋊C4—21no—
216Dic3⋊C4—122no—
217Dic3⋊C4—122no—
218C4×Dic3—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—61no—
221C2×C3⋊C8—61no—
222C2×C3⋊C8—121no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32#12011yescharacteristic
226C22×C32#13511yescharacteristic
227C22×C32#13711yescharacteristic
228C2×C3×D3—41no—
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—41no—
234C32×C4—41no—
235C3⋊Dic3—42no—
236C3×Dic3—41no—
237C3×Dic3—41no—
238C3×Dic3—42no—
239C3×Dic3—41no—
240C22×Q8—31no—
241C2×Q16—61no—
242C2×SD16—181no—
243#126—61no—
244#125—61no—
245#123—61no—
246#123—61no—
247C22⋊Q8—61no—
248C22⋊Q8D3211yescharacteristic
249#121—31no—
250C4×D4—61no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—181no—
253#103—61no—
254#103—181no—
255C22⋊C8—31no—
256C22⋊C8—91no—
257#95—91no—
258#95—181no—
259C23×C3—21no—
260C23×C3—61no—
261C23×C3D3×D411yescharacteristic
262C23×C3C2×#6711yescharacteristic
263C22×D3—21no—
264C3×Q8—41no—
265C3×Q8—41no—
266C3×Q8—62no—
267C3×Q8—62no—
268C3×Q8—81no—
269C3×Q8—121no—
270C3×Q8—121no—
271C3×D4—121no—
272C3×D4—121no—
273C3×D4—121no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—62no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—81no—
291C2×C3×C4—121no—
292C2×C3×C4—121no—
293C2×C3×C4—121no—
294C2×C3×C4—241no—
295C2×C3×C4D3×D411yescharacteristic
296C2×C3×C4C2×#6711yescharacteristic
297C2×C3×C4C2×#6711yescharacteristic
298C2×C3×C4C2×#6711yescharacteristic
299#67—41no—
300#67—41no—
301C2×Dic3—22no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—62no—
311C2×Dic3—62no—
312C2×Dic3—122no—
313C2×Dic3—121no—
314C2×Dic3—121no—
315C2×Dic3—121no—
316C2×Dic3—121no—
317C2×Dic3—241no—
318C2×Dic3—241no—
319D12—41no—
320C4×D3—81no—
321Dic6—22no—
322Dic6—22no—
323Dic6—41no—
324Dic6—41no—
325C3×C8—121no—
326C3⋊C8—41no—
327C3⋊C8—121no—
328C3⋊C8—121no—
329C3⋊C8—241no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32#45211yescharacteristic
333C2×C32#45611yescharacteristic
334C2×C32B(4,𝔽2)11yescharacteristic
335C3×D3—81no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×Q8—122no—
339C2×Q8#62511yescharacteristic
340C2×D4—31no—
341C2×D4—61no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—181no—
347C22×C4C2×D3211yescharacteristic
348Q16—122no—
349SD16—362no—
350C2×C8—61no—
351C2×C8—181no—
352C4⋊C4—21no—
353C4⋊C4—31no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C4⋊C4—61no—
357C4⋊C4—61no—
358C4⋊C4—61no—
359C4⋊C4—362no—
360C4⋊C4#62511yescharacteristic
361C22⋊C4—21no—
362C22⋊C4—61no—
363C22⋊C4—61no—
364C22⋊C4—61no—
365C22⋊C4—61no—
366C42—61no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—121no—
377C22×C3—121no—
378C22×C3—121no—
379C22×C3#91811yescharacteristic
380C22×C3#91911yescharacteristic
381C22×C3#93711yescharacteristic
382C22×C3#95011yescharacteristic
383C22×C3#95111yescharacteristic
384C22×C3#95311yescharacteristic
385C2×D3—41no—
386C2×D3—41no—
387C2×D3—41no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—41no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—41no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4—121no—
401C3×C4—121no—
402C3×C4—122no—
403C3×C4—121no—
404Dic3—41no—
405Dic3—41no—
406Dic3—41no—
407Dic3—42no—
408Dic3—122no—
409Dic3—122no—
410Dic3—242no—
411C32#160311yescharacteristic, complemented, Sylow
412C23—61no—
413C23D3×#6711yescharacteristic
414Q8—41no—
415Q8—62no—
416Q8—62no—
417Q8—121no—
418Q8—121no—
419D4—121no—
420D4—121no—
421D4—121no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—41no—
425C2×C4—62no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—182no—
435C2×C4—241no—
436C2×C4—361no—
437C2×C4—361no—
438C2×C4C2×#62511yescharacteristic
439C2×C4D3×#6711yescharacteristic
440C8—121no—
441C8—361no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—241no—
452C2×C3#503211yescharacteristic
453C2×C3#518711yescharacteristic
454C2×C3#520511yescharacteristic
455C2×C3#523911yescharacteristic
456C2×C3#524411yescharacteristic
457C2×C3#524611yescharacteristic
458D3—81no—
459C22—21no—
460C22—21no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22#6394311yescharacteristic
465C22#6394911yescharacteristic
466C22#6397611yescharacteristic
467C4—21no—
468C4—21no—
469C4—41no—
470C4—41no—
471C4—122no—
472C4—121no—
473C4—121no—
474C4—121no—
475C4—362no—
476C3—21no—
477C3#7098211yescharacteristic, complemented
478C3#7188911yescharacteristic, complemented
479C2—21no—
480C2—21no—
481C2—241no—
482C2#1059153011yescharacteristic
483C2#1059159711yescharacteristic
484C2#1059168711yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
prime radicalC3×#930C22
hypercenterC22⋊Q8D32
hypocenterC32#1603
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5050C2semidirect
3. #10591344C2semidirect
5. #10589404C2semidirect
7. #10589147C2semidirect
34. #5050D3semidirect
411. C32#1603semidirect
477. C3#70982semidirect
478. C3#71889semidirect

Maximal subgroups

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

subgroupindex
C3×#56452
C3×#50502
#105913442
#105911712
#105894042
#105891502
#105891472
#718893
#709823