Finite Group Databasebeta

← #10844989

Subgroups — #10844989

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5027C211yesmaximal, characteristic
2#82431C211yesmaximal, characteristic, complemented
3#78526C211yesmaximal, characteristic
4#78508C211yesmaximal, characteristic, complemented
5C3×C4×#101C211yesmaximal, characteristic, complemented
6#70118C212yesmaximal, complemented
7#70107C212yesmaximal, complemented
8#69876C211yesmaximal, characteristic
9#69875C211yesmaximal, characteristic, complemented
10#68996C212yesmaximal, complemented
11#68983C212yesmaximal, complemented
12#25315—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C22×#215C2212yes—
15C2×#874C2211yescharacteristic
16#5470C2211yescharacteristic
17C2×C3×#101C2211yescharacteristic
18C2×C3×#101C2211yescharacteristic
19C3×C4×M16C2211yescharacteristic
20C3×C4×M16C2211yescharacteristic
21#5225C2212yes—
22#5131C2212yes—
23#5131C2212yes—
24C2×#823C2212yes—
25C2×#823C2212yes—
26C2×#822C2212yes—
27#5122C2212yes—
28#5119C2212yes—
29#5027—21no—
30#5027—21no—
31#5027—22no—
32#5022—22no—
33#5021—21no—
34#5020—21no—
35C2×#809C2212yes—
36#4964C2212yes—
37#4924C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4713—22no—
40#4701—22no—
41#4694—21no—
42#4694—21no—
43#4679—21no—
44C8×Dic6—21no—
45C3×#343C2211yescharacteristic
46C3×#343C2211yescharacteristic
47#4558C2212yes—
48#3138—31no—
49#3078—31no—
50C2×#460—31no—
51#2962—31no—
52#2063—32no—
53#2046—32no—
54#1977—32no—
55#1965—31no—
56#1950—31no—
57#1931—32no—
58C4×#101D311yescharacteristic, complemented
59C22×C3×Q8C2311yescharacteristic
60C22×D12—21no—
61C22×D12—21no—
62C22×D12C2312yes—
63C2×C3×M16C2311yescharacteristic
64C2×C3×M16C2311yescharacteristic
65C2×C3×M16C2311yescharacteristic
66C2×C3×M16C2311yescharacteristic
67C3×C4×Q8—21no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C3×C4×Q8—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#215—24no—
76C2×#215—22no—
77C2×#215—22no—
78C2×#215—42no—
79#926C2312yes—
80#924—21no—
81#924—21no—
82C2×#208C2312yes—
83C22×C3⋊C8C2312yes—
84#874—21no—
85#874—21no—
86#874—21no—
87#874—21no—
88#870D412yes—
89#869D411yescharacteristic
90#869D411yescharacteristic
91C4×Dic6—21no—
92C4×Dic6—21no—
93C3×#101—22no—
94C3×#101—22no—
95C3×#97—21no—
96C3×#97—21no—
97C3×C4×C8—21no—
98C3×C4×C8—21no—
99#835—22no—
100#835—22no—
101#823—24no—
102#823—42no—
103#822—42no—
104#821—22no—
105#821—22no—
106#821—22no—
107#821—22no—
108#815—22no—
109Dic3⋊C8—22no—
110Dic3⋊C8—22no—
111C8×Dic3—22no—
112#809—22no—
113#809—22no—
114#808—22no—
115#807—22no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118C4⋊(C3⋊C8)D411yescharacteristic
119C4⋊(C3⋊C8)D411yescharacteristic
120C4⋊(C3⋊C8)D412yes—
121C22×SD16—32no—
122#533—31no—
123C2×#127—31no—
124C2×C4×Q8C2×D311yescharacteristic
125#490—62no—
126#486—61no—
127#486—61no—
128#479—62no—
129#477—61no—
130#475—61no—
131#460—61no—
132#460—62no—
133#460—61no—
134#446—61no—
135#446—62no—
136C8×Q8—61no—
137C2×#108—32no—
138#428—32no—
139#424—32no—
140#423—31no—
141C2×C4⋊C8—31no—
142#419—32no—
143#418—32no—
144#418—32no—
145C2×#102—32no—
146C2×#101C2×D311yescharacteristic
147C2×#101C2×D311yescharacteristic
148C2×#100—32no—
149C2×#100—32no—
150#405—32no—
151C4×M16C2×D311yescharacteristic
152C4×M16C2×D311yescharacteristic
153#343C2×D311yescharacteristic
154#343C2×D311yescharacteristic
155#341—32no—
156C23×D3—22no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8#4112yes—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4#4111yescharacteristic
166C22×C3×C4#4111yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—41no—
175C2×D12—41no—
176C2×D12—41no—
177C2×D12—41no—
178C2×D12—82no—
179C2×D12—82no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C3×M16—21no—
183C3×M16—21no—
184C3×M16—21no—
185C3×M16—21no—
186C3×M16—21no—
187C3×M16—21no—
188C3×M16—21no—
189C3×M16#4112yes—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—41no—
198C3×C42—41no—
199C3×C42—41no—
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204#217—22no—
205#215—44no—
206#215—44no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3C2×D412yes—
210C4⋊Dic3C2×D412yes—
211Dic3⋊C4—22no—
212Dic3⋊C4—22no—
213C4×Dic3—22no—
214#208—22no—
215#208—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8C2×D412yes—
220C2×C3⋊C8C2×D412yes—
221C22×Q8C22×D311yescharacteristic
222C22×D4—32no—
223C22×D4—61no—
224C22×D4—61no—
225C2×SD16—62no—
226C2×SD16—62no—
227C2×SD16—64no—
228C2×SD16—122no—
229C2×M16—32no—
230C2×M16C22×D311yescharacteristic
231C2×M16C22×D311yescharacteristic
232C2×M16C22×D311yescharacteristic
233C2×M16C22×D311yescharacteristic
234C22×C8—32no—
235C4⋊Q8—31no—
236C4⋊Q8—31no—
237#127—61no—
238#127—61no—
239#127—61no—
240#127—61no—
241#125—32no—
242C22⋊Q8—61no—
243C22⋊Q8—61no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—61no—
249C4×Q8—61no—
250#117—32no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C42—21no—
254C2×C42C22×D311yescharacteristic
255#108—122no—
256#107—62no—
257#106—62no—
258C4⋊C8—31no—
259C4⋊C8—31no—
260C4⋊C8—32no—
261C4⋊C8—62no—
262C4⋊C8—62no—
263C4⋊C8—61no—
264C4⋊C8—61no—
265#103—62no—
266#103—62no—
267#102—62no—
268#102—62no—
269#102—62no—
270#102—62no—
271#102—62no—
272#102—62no—
273#101—22no—
274#101—22no—
275#100—64no—
276#100—122no—
277#97—21no—
278#97—21no—
279#97—62no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—62no—
283C23×C3C2×#4111yescharacteristic
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—82no—
288C22×D3—82no—
289C3×Q8—22no—
290C3×Q8—22no—
291C3×Q8—41no—
292C3×Q8—41no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—81no—
300C2×C3×C4#12111yescharacteristic
301C2×C3×C4#12111yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×C3×C4#12111yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C2×C3×C42−1+411yescharacteristic
310C2×Dic3—22no—
311C2×Dic3—22no—
312D12—42no—
313D12—42no—
314D12—42no—
315D12—42no—
316D12—82no—
317D12—82no—
318D12—82no—
319D12—82no—
320Dic6—42no—
321Dic6—81no—
322C3×C8—22no—
323C3×C8—41no—
324C3×C8—41no—
325C3×C8—41no—
326C3⋊C8—22no—
327C3⋊C8—22no—
328C3⋊C8—42no—
329C24—62no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×Q8#23512yes—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—121no—
345C2×D4—121no—
346C2×D4—121no—
347C2×D4—121no—
348C2×D4—242no—
349C2×D4—242no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4#23511yescharacteristic
353C22×C4#23511yescharacteristic
354C22×C4C23×D311yescharacteristic
355SD16—124no—
356SD16—124no—
357M16—21no—
358M16—21no—
359M16—21no—
360M16—21no—
361M16—21no—
362M16—21no—
363M16—21no—
364M16—62no—
365M16—62no—
366M16#23512yes—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—32no—
372C2×C8—32no—
373C2×C8—62no—
374C2×C8—62no—
375C2×C8—62no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—32no—
379C4⋊C4—32no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C22⋊C4—62no—
387C42—41no—
388C42—41no—
389C42—62no—
390C42D3×D411yescharacteristic
391C42D3×D411yescharacteristic
392C42D3×D411yescharacteristic
393C42D3×D411yescharacteristic
394C22×C3—21no—
395C22×C3—21no—
396C22×C3C2×#12111yescharacteristic
397C22×C3#52911yescharacteristic
398C22×C3#54211yescharacteristic
399C2×D3—82no—
400C2×D3—82no—
401C2×D3—82no—
402C2×D3—162no—
403C2×D3—162no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—42no—
408C3×C4—81no—
409C3×C4#47112yes—
410C3×C4#54611yescharacteristic
411C3×C4#54811yescharacteristic
412C3×C4#54811yescharacteristic
413C3×C4#55011yescharacteristic
414Dic3—42no—
415Dic3—42no—
416C23—122no—
417C23—122no—
418C23—122no—
419C23—242no—
420C23—242no—
421C23C2×#23511yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—41no—
425Q8—41no—
426Q8—122no—
427Q8—241no—
428D4—122no—
429D4—122no—
430D4—122no—
431D4—122no—
432D4—242no—
433D4—242no—
434D4—242no—
435D4—242no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—81no—
445C2×C4#88411yescharacteristic
446C2×C4#88411yescharacteristic
447C2×C4#88411yescharacteristic
448C2×C4#88411yescharacteristic
449C2×C4C2×#23511yescharacteristic
450C2×C4C2×#23511yescharacteristic
451C2×C4C2×D3×D411yescharacteristic
452C2×C4C2×D3×D411yescharacteristic
453C2×C4#100911yescharacteristic
454C2×C4#101011yescharacteristic
455C8—22no—
456C8—41no—
457C8—41no—
458C8—41no—
459C8—62no—
460C8—62no—
461C8—122no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3#242611yescharacteristic
465C2×C3C2×#47111yescharacteristic
466C2×C3#306811yescharacteristic
467D3—162no—
468C22—21no—
469C22—21no—
470C22—242no—
471C22—242no—
472C22—242no—
473C22—482no—
474C22—482no—
475C22#547911yescharacteristic
476C22C2×#88411yescharacteristic
477C22#553411yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—81no—
483C4—122no—
484C4—122no—
485C4#475412yes—
486C4#555011yescharacteristic
487C4#555211yescharacteristic
488C4#555211yescharacteristic
489C4#555311yescharacteristic
490C3#2531511yescharacteristic, complemented, Sylow
491C2—21no—
492C2—21no—
493C2—482no—
494C2#7348511yescharacteristic
495C2C2×#475411yescharacteristic
496C2#7872011yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#101C2
socleC22×C3C2×#121
prime radicalC2×#874C22
hypercenterC4×#101D3
hypocenterC3#25315
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #82431C2semidirect
4. #78508C2semidirect
5. C3×C4×#101C2semidirect
6. #70118C2semidirect
7. #70107C2semidirect
9. #69875C2semidirect
10. #68996C2semidirect
11. #68983C2semidirect
58. C4×#101D3semidirect
490. C3#25315semidirect

Maximal subgroups

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

subgroupindex
C2×#50272
#824312
#785262
#785082
C3×C4×#1012
#701182
#701072
#698762
#698752
#689962
#689832
#253153