Finite Group Databasebeta

← #10847304

Subgroups — #10847304

Order 768.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2962C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3#83341C211yesmaximal, characteristic, complemented
4C2×#4842C211yesmaximal, characteristic
5#70542C212yesmaximal
6#70533C212yesmaximal, complemented
7#70474C211yesmaximal, characteristic, complemented
8#69893C212yesmaximal, complemented
9#69892C212yesmaximal
10#69879C211yesmaximal, characteristic, complemented
11C4×#823C211yesmaximal, characteristic
12#25490—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14C2×C4×D12C2211yescharacteristic
15C2×#869C2211yescharacteristic
16C3×#446—22no—
17C3×#446—21no—
18C3×C8×Q8—21no—
19C3×#424C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C3×#405C2212yes—
22C3×C4×M16C2211yescharacteristic
23C2×#823C2211yescharacteristic
24C2×#823C2211yescharacteristic
25C2×#821C2212yes—
26C2×#819C2212yes—
27C2×#817C2212yes—
28#5087—21no—
29#5084—21no—
30#5080—21no—
31#5078—21no—
32#5066—22no—
33#5064—22no—
34#5001C2212yes—
35#4995C2212yes—
36#4995C2212yes—
37#4995C2212yes—
38#4924C2211yescharacteristic
39C4×#208C2211yescharacteristic
40#4842—21no—
41#4842—22no—
42#4842—21no—
43#4554C2212yes—
44#4554C2211yescharacteristic
45#4554C2211yescharacteristic
46#4554C2212yes—
47#4552C2212yes—
48#3165—31no—
49#3077—31no—
50C2×#476—31no—
51#2962D311yescharacteristic, complemented
52#2090—32no—
53#2063—32no—
54#1997—31no—
55#1951—32no—
56#1950—31no—
57#1931—32no—
58C4×#100—31no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×Dic6—21no—
62C2×C3×M16C2311yescharacteristic
63C2×C3×M16C2311yescharacteristic
64C22×C3×C8C2312yes—
65C3×C4⋊Q8D411yescharacteristic
66C3×C4⋊Q8D411yescharacteristic
67C3×#125D412yes—
68C3×C22⋊Q8—21no—
69C3×C22⋊Q8—21no—
70C3×C4×Q8—21no—
71C3×C4×Q8—21no—
72C3×#117C2312yes—
73C2×C3×C42C2311yescharacteristic
74C2×(C2×D3)⋊C4—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×#208C2312yes—
79C2×#208C2311yescharacteristic
80C2×#208C2311yescharacteristic
81C4×D12—21no—
82C4×D12—21no—
83C4×D12—21no—
84C4×D12—21no—
85#869—21no—
86#869—21no—
87#869—21no—
88#869—21no—
89C3×C4⋊C8—22no—
90C3×C4⋊C8—22no—
91C3×C4⋊C8D411yescharacteristic
92C3×C4⋊C8D411yescharacteristic
93C3×C4⋊C8D412yes—
94C3×#97—22no—
95C3×#97—21no—
96C3×C4×C8—22no—
97C3×C4×C8—21no—
98#823—22no—
99#823Q814yescomplemented
100#821—22no—
101#821—22no—
102#819—24no—
103#817—22no—
104#817—22no—
105#809—22no—
106#809—22no—
107#808—22no—
108#808—22no—
109#808—22no—
110#807—22no—
111#807—22no—
112#807—22no—
113C4⋊(C3⋊C8)—21no—
114C4⋊(C3⋊C8)—21no—
115#803—21no—
116C4×C3⋊C8—21no—
117#533C2×D311yescharacteristic
118C2×C4⋊Q8—31no—
119C2×C4×D4—31no—
120#501—62no—
121#500—61no—
122#498—61no—
123#478—62no—
124#476—61no—
125#476—61no—
126#476—62no—
127#476—61no—
128#474—61no—
129#446—22no—
130#446—21no—
131C8×Q8—21no—
132C2×#108—32no—
133#428—32no—
134#428—32no—
135#428—32no—
136C2×#106—32no—
137#424C2×D312yes—
138#423—31no—
139C2×C4⋊C8C2×D311yescharacteristic
140#418—32no—
141C2×#102—32no—
142C2×#100—31no—
143C2×#100—31no—
144#405C2×D312yes—
145C4×M16—31no—
146C4×M16C2×D311yescharacteristic
147#343—32no—
148#343—31no—
149#343—32no—
150#343—31no—
151#341—32no—
152C23×D3—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C22×C3×C4#4111yescharacteristic
156C22×C3×C4#4111yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—22no—
159C22×Dic3—21no—
160C2×D12—21no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12C2×Q812yes—
166C2×C4×D3—41no—
167C2×C4×D3—41no—
168C2×C4×D3—82no—
169C2×Dic6—41no—
170C2×Dic6—41no—
171C2×Dic6—82no—
172C3×M16—21no—
173C3×M16—21no—
174C3×M16—21no—
175C3×M16C2×Q812yes—
176C2×C3×C8—21no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8C2×D412yes—
181C2×C3×C8C2×D412yes—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4C2×D412yes—
187C3×C4⋊C4C2×D412yes—
188C3×C22⋊C4—22no—
189C3×C42—22no—
190C3×C42C2×D411yescharacteristic
191C3×C42C2×D411yescharacteristic
192C3×C42C2×D411yescharacteristic
193C3×C42C2×D411yescharacteristic
194(C2×D3)⋊C4—41no—
195(C2×D3)⋊C4—41no—
196C4⋊Dic3—22no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—22no—
200C4⋊Dic3—41no—
201C4⋊Dic3—41no—
202#208—21no—
203#208—21no—
204#208—22no—
205#208—22no—
206#208C2×Q812yes—
207#208#4112yes—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C22×Q8—61no—
212C22×D4—31no—
213C23×C4—61no—
214C2×M16—31no—
215C2×M16—31no—
216C2×M16—32no—
217C2×M16C22×D311yescharacteristic
218C2×M16C22×D311yescharacteristic
219C22×C8C22×D312yes—
220C4⋊Q8—61no—
221C4⋊Q8—61no—
222C4⋊Q8—61no—
223C4⋊Q8—61no—
224C4⋊Q8D1211yescharacteristic
225C4⋊Q8D1211yescharacteristic
226#125D1212yes—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×D4—61no—
232C4×D4—61no—
233C4×D4—61no—
234C4×D4—61no—
235#117C22×D312yes—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—32no—
238C2×C4⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C42C22×D311yescharacteristic
241#108—64no—
242#107—62no—
243#107—62no—
244#107—62no—
245#106—62no—
246#106—62no—
247#106—62no—
248#106—62no—
249#106—62no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—61no—
253C4⋊C8—61no—
254C4⋊C8#6711yescharacteristic
255C4⋊C8#6711yescharacteristic
256C4⋊C8#6712yes—
257#102—62no—
258#102—62no—
259#102—62no—
260#102—62no—
261#100—34no—
262#100—62no—
263#97—22no—
264#97—21no—
265#97—61no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—61no—
269C23×C3C2×#4111yescharacteristic
270C22×D3—22no—
271C22×D3—41no—
272C22×D3—41no—
273C22×D3—81no—
274C22×D3—81no—
275C3×Q8—42no—
276C3×Q8—81no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—41no—
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12111yescharacteristic
284C2×C3×C4#12111yescharacteristic
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×Q811yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C42−1+411yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—82no—
294C2×Dic3—81no—
295D12—22no—
296D12—22no—
297D12—41no—
298D12—41no—
299C4×D3—82no—
300C4×D3—82no—
301Dic6—82no—
302Dic6—82no—
303C3×C8—22no—
304C3×C8—22no—
305C3×C8—22no—
306C3×C8—41no—
307C3⋊C8—22no—
308C3⋊C8—22no—
309C3⋊C8—42no—
310C24—61no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—121no—
314C2×Q8—121no—
315C2×Q8—242no—
316C2×D4—32no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—121no—
325C22×C4—121no—
326C22×C4—242no—
327C22×C4#23511yescharacteristic
328C22×C4#23511yescharacteristic
329C22×C4C23×D311yescharacteristic
330M16—21no—
331M16—21no—
332M16—21no—
333M16—32no—
334M16—32no—
335M16—61no—
336M16—61no—
337M16—62no—
338M16—62no—
339M16D3×Q812yes—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—61no—
345C2×C8—61no—
346C2×C8—62no—
347C2×C8C2×#6712yes—
348C2×C8C2×#6712yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—61no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—61no—
357C4⋊C4—121no—
358C4⋊C4—121no—
359C4⋊C4C2×D1212yes—
360C4⋊C4C2×D1212yes—
361C22⋊C4—22no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C42—22no—
365C42C2×D1211yescharacteristic
366C42C2×D1211yescharacteristic
367C42C2×#6711yescharacteristic
368C42C2×#6711yescharacteristic
369C22×C3—21no—
370C22×C3—21no—
371C22×C3C2×#12111yescharacteristic
372C22×C3#53311yescharacteristic
373C22×C3#54111yescharacteristic
374C2×D3—42no—
375C2×D3—82no—
376C2×D3—81no—
377C2×D3—81no—
378C2×D3—82no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—42no—
383C3×C4—42no—
384C3×C4#47112yes—
385C3×C4#54711yescharacteristic
386C3×C4#54811yescharacteristic
387C3×C4D4×Q811yescharacteristic
388C3×C4D4×Q811yescharacteristic
389Dic3—81no—
390Dic3—82no—
391C23—62no—
392C23—121no—
393C23—121no—
394C23—241no—
395C23—241no—
396C23C2×#23511yescharacteristic
397Q8—42no—
398Q8—81no—
399Q8—242no—
400Q8—242no—
401D4—62no—
402D4—62no—
403D4—121no—
404D4—121no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—121no—
410C2×C4—122no—
411C2×C4—242no—
412C2×C4—242no—
413C2×C4—241no—
414C2×C4—242no—
415C2×C4#93011yescharacteristic
416C2×C4#93011yescharacteristic
417C2×C4#93011yescharacteristic
418C2×C4#93011yescharacteristic
419C2×C4C22×D1211yescharacteristic
420C2×C4C2×D3×Q811yescharacteristic
421C2×C4#100711yescharacteristic
422C2×C4D3×#4111yescharacteristic
423C2×C4#101011yescharacteristic
424C2×C4C22×#6711yescharacteristic
425C8—22no—
426C8—22no—
427C8—22no—
428C8—41no—
429C8—62no—
430C8—62no—
431C8—122no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3#242911yescharacteristic
435C2×C3C2×#47111yescharacteristic
436C2×C3#306811yescharacteristic
437D3—82no—
438C22—21no—
439C22—21no—
440C22—122no—
441C22—242no—
442C22—242no—
443C22—241no—
444C22—241no—
445C22#557811yescharacteristic
446C22#558311yescharacteristic
447C22C2×#93011yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—42no—
452C4—42no—
453C4—241no—
454C4—242no—
455C4#514112yes—
456C4Q8×D1211yescharacteristic
457C4#557611yescharacteristic
458C4Q8×#6711yescharacteristic
459C4#582411yescharacteristic
460C3#2549011yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—242no—
464C2#8249611yescharacteristic
465C2C2×#514111yescharacteristic
466C2#8381411yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2962C2
socleC22×C3C2×#121
prime radicalC22×D12C23
hypercenter#2962D3
hypocenterC3#25490
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. C3×#2962C2semidirect
3. #83341C2semidirect
6. #70533C2semidirect
7. #70474C2semidirect
8. #69893C2semidirect
10. #69879C2semidirect
51. #2962D3semidirect
99. #823Q8semidirect
460. C3#25490semidirect

Maximal subgroups

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

subgroupindex
C3×#29622
#833502
#833412
C2×#48422
#705422
#705332
#704742
#698932
#698922
#698792
C4×#8232
#254903