Finite Group Databasebeta

← #11897674

Subgroups — #11897674

Order 1152.

728 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5730C211yesmaximal, characteristic
2#10593856C211yesmaximal, characteristic, complemented
3#10592790C211yesmaximal, characteristic, complemented
4C3×#5078C211yesmaximal, characteristic, complemented
5#10590590C212yesmaximal
6#10590555C212yesmaximal
7#10590492C212yesmaximal, complemented
8#10590418C212yesmaximal, complemented
9C4×#3716C211yesmaximal, characteristic
10#10590266C211yesmaximal, characteristic
11#10590258C211yesmaximal, characteristic, complemented
12#83631—31nomaximal
13#81954—31nomaximal
14C32×C4⋊Q8C2211yescharacteristic
15C4×#634C2211yescharacteristic
16C3×#946—22no—
17C3×#897—21no—
18C3×#893C2212yes—
19C3×D3×C4⋊C4—21no—
20C3×#890C2212yes—
21C3×C4×D12C2211yescharacteristic
22C3×#869C2211yescharacteristic
23C3×C4×Dic6C2211yescharacteristic
24#63919—21no—
25#63918C2212yes—
26#63844C2212yes—
27#63843—21no—
28#63839—22no—
29C2×#3716C2211yescharacteristic
30#63637C2212yes—
31#63634C2212yes—
32C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
33C3×#807C2212yes—
34C3×#807C2212yes—
35C3×C4×C3⋊C8C2211yescharacteristic
36#63574C2212yes—
37#63573C2211yescharacteristic
38Dic3⋊(C3⋊C8)C2212yes—
39#63567C2211yescharacteristic
40#63563C2211yescharacteristic
41#63562C2212yes—
42#63554C2212yes—
43C3×#554—31no—
44#5730D311yescharacteristic
45#5722—31no—
46Q8×Dic6—31no—
47C3×#500—31no—
48#5091—31no—
49#5091—31no—
50#5091—61no—
51#5084—31no—
52#5078D311yescharacteristic, complemented
53#5064—32no—
54#5063—32no—
55#5024—31no—
56#5018—32no—
57C4×#214—31no—
58#5008—31no—
59#5007—32no—
60#4890—32no—
61#4889—32no—
62#4877—32no—
63#4876—32no—
64C4×#204—31no—
65#4679—31no—
66C2×C32×Q8—21no—
67C2×#634C2311yescharacteristic
68C2×C3×D12C2311yescharacteristic
69C2×C3×C4×D3—21no—
70C2×C3×C4×D3—22no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6D411yescharacteristic
73C2×#627—22no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4C2312yes—
76C32×C42C2311yescharacteristic
77C4⋊(C3⋊Dic3)C2311yescharacteristic
78#3750—21no—
79C4×C3⋊Dic3—21no—
80C2×C3⋊(C3⋊C8)C2312yes—
81C3×(C2×D3)⋊C4—21no—
82C3×(C2×D3)⋊C4—22no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—22no—
85C3×C4⋊Dic3D411yescharacteristic
86C3×C4⋊Dic3C2311yescharacteristic
87C3×C4⋊Dic3C2312yes—
88C3×Dic3⋊C4—22no—
89C3×Dic3⋊C4D412yes—
90C3×C4×Dic3C2312yes—
91C2×C3×C3⋊C8C2311yescharacteristic
92C2×C3×C3⋊C8C2311yescharacteristic
93#3724—22no—
94Dic3⋊Dic3—22no—
95Dic32—22no—
96#3716Q814yescomplemented
97#3385—91noSylow
98C3×C4⋊Q8—21no—
99C3×C4⋊Q8—31no—
100C3×C4⋊Q8C2×D311yescharacteristic
101C3×C4⋊Q8C2×D311yescharacteristic
102C3×#125—32no—
103C3×C22⋊Q8—61no—
104C3×C22⋊Q8—62no—
105C3×C4×Q8—31no—
106C3×C4×D4—32no—
107C3×C4×D4—31no—
108C2×C3×C4⋊C4—61no—
109#946—22no—
110Q8×Dic3—61no—
111#944—62no—
112C2×#214—31no—
113C2×#204—31no—
114#897—21no—
115#893C2×D312yes—
116D3×C4⋊C4—21no—
117#890C2×D312yes—
118#888—32no—
119#888—61no—
120#888—62no—
121#888—61no—
122#887—32no—
123#887—61no—
124C4×D12C2×D311yescharacteristic
125#869—32no—
126#869C2×D311yescharacteristic
127C4×Dic6—31no—
128C4×Dic6—31no—
129C4×Dic6—32no—
130C4×Dic6—61no—
131C4×Dic6C2×D311yescharacteristic
132C3×#107—32no—
133C3×#107—32no—
134C3×C4×C8—31no—
135#832—32no—
136#832—31no—
137#821—31no—
138#817—31no—
139#816—31no—
140#815—32no—
141Dic3⋊C8—32no—
142#810—32no—
143#810—32no—
144#810—31no—
145#810—62no—
146#809—32no—
147#808—32no—
148#808—32no—
149#808—31no—
150#807—32no—
151#807—32no—
152#807—62no—
153#807C2×D312yes—
154#807C2×D312yes—
155C4⋊(C3⋊C8)—31no—
156C4⋊(C3⋊C8)—31no—
157C4⋊(C3⋊C8)—32no—
158C4⋊(C3⋊C8)—61no—
159#803—32no—
160C4×C3⋊C8C2×D311yescharacteristic
161C22×C3×D3—21no—
162C32×Q8—81no—
163C2×C32×C4—22no—
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2411yescharacteristic
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169#634—21no—
170#634C2×Q812yes—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3C2×D412yes—
174C2×C3×Dic3C2×D412yes—
175C3×D12—21no—
176C3×D12C2×Q812yes—
177C3×C4×D3—42no—
178C3×C4×D3—82no—
179C3×Dic6—22no—
180C3×Dic6—81no—
181#627—82no—
182C3⋊(C3⋊C8)—22no—
183C3×C3⋊C8—21no—
184C3×C3⋊C8C2×Q812yes—
185Q82—91no—
186#554—31no—
187#501—92no—
188#500—31no—
189#479—92no—
190#477—91no—
191#476—91no—
192#474—92no—
193#446—91no—
194#440—92no—
195C4×SD16—91no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—31no—
199C2×C3×Q8—41no—
200C2×C3×Q8—61no—
201C2×C3×D4—31no—
202C22×C3×C4—62no—
203C22×C3×C4—61no—
204C2×D12C22×D311yescharacteristic
205C2×C4×D3—21no—
206C2×C4×D3—22no—
207C2×Dic6—21no—
208C2×Dic6—31no—
209C2×Dic6—31no—
210C2×Dic6—61no—
211C2×Dic6—62no—
212C2×Dic6—62no—
213C2×Dic6#6711yescharacteristic
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—32no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—32no—
223C3×C4⋊C4—41no—
224C3×C4⋊C4—62no—
225C3×C4⋊C4—62no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4C22×D312yes—
228C3×C4⋊C4C22×D312yes—
229C3×C22⋊C4—62no—
230C3×C22⋊C4—61no—
231C3×C42—21no—
232C3×C42—32no—
233C3×C42C22×D311yescharacteristic
234C3×C42C22×D311yescharacteristic
235#214—34no—
236(C2×D3)⋊C4—21no—
237(C2×D3)⋊C4—22no—
238C4⋊Dic3—21no—
239C4⋊Dic3—22no—
240C4⋊Dic3—31no—
241C4⋊Dic3—31no—
242C4⋊Dic3—61no—
243C4⋊Dic3—62no—
244C4⋊Dic3#6711yescharacteristic
245C4⋊Dic3C22×D312yes—
246C4⋊Dic3C22×D311yescharacteristic
247Dic3⋊C4—22no—
248Dic3⋊C4—62no—
249Dic3⋊C4—61no—
250Dic3⋊C4—61no—
251Dic3⋊C4—62no—
252Dic3⋊C4—62no—
253Dic3⋊C4—121no—
254Dic3⋊C4#6712yes—
255C4×Dic3—62no—
256C4×Dic3—61no—
257C4×Dic3—61no—
258C4×Dic3—62no—
259C4×Dic3—121no—
260C4×Dic3C22×D312yes—
261C2×C3⋊C8—32no—
262C2×C3⋊C8—32no—
263C2×C3⋊C8—62no—
264C2×C3⋊C8C22×D311yescharacteristic
265C2×C3⋊C8C22×D311yescharacteristic
266#204—34no—
267C22×C32C22×D411yescharacteristic
268C2×C3×D3—22no—
269C2×C3×D3—41no—
270C2×C3×D3—41no—
271C32×C4—21no—
272C32×C4—42no—
273C32×C4#13712yes—
274C32×C4C22×Q811yescharacteristic
275C32×C42−1+411yescharacteristic
276C3⋊Dic3—22no—
277C3⋊Dic3—41no—
278C3×Dic3—22no—
279C3×Dic3—22no—
280C3×Dic3—41no—
281C3×Dic3—42no—
282C2×SD16—91no—
283C4⋊Q8—31no—
284C4⋊Q8—92no—
285C4⋊Q8—181no—
286C4⋊Q8—182no—
287C4⋊Q8D3211yescharacteristic
288#125—32no—
289C22⋊Q8—61no—
290C22⋊Q8—62no—
291C4×Q8—31no—
292C4×Q8—92no—
293C4×Q8—91no—
294C4×Q8—181no—
295C4×D4—32no—
296C4×D4—31no—
297C2×C4⋊C4—61no—
298#107—32no—
299#107—32no—
300#107—92no—
301#106—92no—
302#106—91no—
303C4⋊C8—92no—
304C4⋊C8—91no—
305#103—91no—
306#103—92no—
307#102—92no—
308#102—91no—
309#97—92no—
310C4×C8—31no—
311C23×C3—61no—
312C22×D3—21no—
313C3×Q8—62no—
314C3×Q8—81no—
315C3×Q8—81no—
316C3×Q8—82no—
317C3×Q8—241no—
318C3×D4—32no—
319C3×D4—61no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—32no—
326C2×C3×C4—32no—
327C2×C3×C4—42no—
328C2×C3×C4—61no—
329C2×C3×C4—62no—
330C2×C3×C4—122no—
331C2×C3×C4—242no—
332C2×C3×C4D3×D411yescharacteristic
333C2×C3×C4D3×D411yescharacteristic
334C2×C3×C4C2×#6711yescharacteristic
335C2×C3×C4C2×#6711yescharacteristic
336C2×C3×C4C23×D311yescharacteristic
337C2×C3×C4C23×D311yescharacteristic
338C2×Dic3—22no—
339C2×Dic3—21no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—121no—
345C2×Dic3—121no—
346C2×Dic3C2×#6712yes—
347C2×Dic3C2×#6712yes—
348D12—21no—
349D12D3×Q812yes—
350C4×D3—42no—
351C4×D3—82no—
352Dic6—22no—
353Dic6—32no—
354Dic6—32no—
355Dic6—62no—
356Dic6—61no—
357Dic6—61no—
358Dic6—81no—
359Dic6—121no—
360Dic6—242no—
361Dic6—242no—
362C3×C8—32no—
363C3×C8—61no—
364C3⋊C8—21no—
365C3⋊C8—62no—
366C3⋊C8—62no—
367C3⋊C8—122no—
368C3⋊C8D3×Q812yes—
369C2×C32D4×Q811yescharacteristic
370C2×C32C2×#13711yescharacteristic
371C2×C32#57611yescharacteristic
372C3×D3—42no—
373C2×Q8—21no—
374C2×Q8—31no—
375C2×Q8—61no—
376C2×Q8—91no—
377C2×Q8—182no—
378C2×D4—31no—
379C22×C4—62no—
380C22×C4—61no—
381SD16—94no—
382C2×C8—31no—
383C2×C8—31no—
384C2×C8—92no—
385C4⋊C4—21no—
386C4⋊C4—31no—
387C4⋊C4—32no—
388C4⋊C4—32no—
389C4⋊C4—31no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C4⋊C4—91no—
394C4⋊C4—182no—
395C4⋊C4—182no—
396C4⋊C4—181no—
397C4⋊C4C2×D3212yes—
398C22⋊C4—62no—
399C22⋊C4—61no—
400C42—32no—
401C42—182no—
402C42—181no—
403C42C2×D3211yescharacteristic
404C22×C3—21no—
405C22×C3—62no—
406C22×C3—121no—
407C22×C3—121no—
408C22×C3C2×D3×D411yescharacteristic
409C22×C3C22×#6711yescharacteristic
410C2×D3—22no—
411C2×D3—41no—
412C2×D3—41no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—22no—
418C3×C4—41no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—62no—
422C3×C4—62no—
423C3×C4—82no—
424C3×C4—121no—
425C3×C4—122no—
426C3×C4#91812yes—
427C3×C4#95112yes—
428C3×C4C2×D3×Q811yescharacteristic
429C3×C4C2×D3×Q811yescharacteristic
430C3×C4#100711yescharacteristic
431C3×C4#100711yescharacteristic
432Dic3—22no—
433Dic3—22no—
434Dic3—42no—
435Dic3—41no—
436Dic3—62no—
437Dic3—62no—
438Dic3—122no—
439Dic3—121no—
440Dic3—121no—
441Dic3—241no—
442C32#338511yescharacteristic, complemented, Sylow
443C23—61no—
444Q8—62no—
445Q8—81no—
446Q8—92no—
447Q8—181no—
448Q8—241no—
449Q8—722no—
450D4—32no—
451D4—61no—
452C2×C4—22no—
453C2×C4—32no—
454C2×C4—32no—
455C2×C4—62no—
456C2×C4—61no—
457C2×C4—122no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—242no—
461C2×C4D3×#6711yescharacteristic
462C2×C4D3×#6711yescharacteristic
463C2×C4C22×D3211yescharacteristic
464C8—32no—
465C8—61no—
466C8—182no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—122no—
471C2×C3#572711yescharacteristic
472C2×C3#575711yescharacteristic
473C2×C3C2×#91811yescharacteristic
474C2×C3Q8×#6711yescharacteristic
475C2×C3C2×#95111yescharacteristic
476C2×C3#583511yescharacteristic
477D3—42no—
478C22—62no—
479C22—121no—
480C22—121no—
481C22C2×D3×#6711yescharacteristic
482C4—21no—
483C4—42no—
484C4—62no—
485C4—62no—
486C4—122no—
487C4—121no—
488C4—182no—
489C4—361no—
490C4#6394312yes—
491C4#6431611yescharacteristic
492C4D32×Q811yescharacteristic
493C3—21no—
494C3#8195411yescharacteristic, complemented
495C3#8363111yescharacteristic, complemented
496C2—122no—
497C2#1059509611yescharacteristic
498C2C2×#6394311yescharacteristic
499C2#1059540311yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radicalC2×C3×D12C23
hypercenterC4⋊Q8D32
hypocenterC32#3385
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10593856C2semidirect
3. #10592790C2semidirect
4. C3×#5078C2semidirect
7. #10590492C2semidirect
8. #10590418C2semidirect
11. #10590258C2semidirect
52. #5078D3semidirect
96. #3716Q8semidirect
442. C32#3385semidirect
494. C3#81954semidirect
495. C3#83631semidirect

Maximal subgroups

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

subgroupindex
C3×#57302
#105938562
#105927902
C3×#50782
#105905902
#105905552
#105904922
#105904182
C4×#37162
#105902662
#105902582
#836313
#819543