Finite Group Databasebeta

← #10824872

Subgroups — #10824872

Order 768.

752 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
2C4×#942C212yesmaximal
3#78455C211yesmaximal, characteristic
4#72074C211yesmaximal, characteristic
5#72070C211yesmaximal, characteristic, complemented
6C4×#836C212yesmaximal, complemented
7#70647C211yesmaximal, characteristic
8#70626C211yesmaximal, characteristic, complemented
9#70265C212yesmaximal, complemented
10#70264C212yesmaximal
11C4×#819C211yesmaximal, characteristic, complemented
12#23717—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2212yes—
15C2×#942C2212yes—
16C4×#235C2211yescharacteristic
17C3×C4×Q16—22no—
18C3×C4×Q16—22no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#103C2212yes—
21C2×C3×C4×C8C2211yescharacteristic
22C2×#836C2212yes—
23C2×#836C2212yes—
24#5225C2212yes—
25#5110C2211yescharacteristic
26#5110C2211yescharacteristic
27C2×#819C2212yes—
28#5102C2211yescharacteristic
29#5102C2211yescharacteristic
30#5030—22no—
31C4×#216—22no—
32#5026—22no—
33C4×#215—22no—
34#4966C2212yes—
35#4964C2212yes—
36C4×#208C2211yescharacteristic
37C4×#208C2212yes—
38#4690—22no—
39#4685—22no—
40#4552C2211yescharacteristic
41#4552C2211yescharacteristic
42#4530C2212yes—
43#4530C2212yes—
44C4×#137—32no—
45C2×C4×Q16D311yescharacteristic, complemented
46#2922—31no—
47#2089—31no—
48#2078—31no—
49#2064—32no—
50#2052—32no—
51#2047—31no—
52#2020—31no—
53C4×#108—31no—
54C4×#101—32no—
55C22×C3×Q8C2312yes—
56C2×#235C2311yescharacteristic
57C2×C3×Q16—24no—
58C2×C3×Q16—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C42—22no—
68C2×C3×C42C2311yescharacteristic
69C2×#216—22no—
70#942—24no—
71#942—24no—
72C2×#215—22no—
73C4×#67—21no—
74C4×#67—21no—
75#926C2311yescharacteristic
76C2×#208C2312yes—
77C2×#208C2311yescharacteristic
78C2×#208C2311yescharacteristic
79C2×#208C2312yes—
80C4×D12D411yescharacteristic
81C4×D12D411yescharacteristic
82#872—21no—
83C42×D3—21no—
84C4×Dic6D411yescharacteristic
85C4×Dic6D411yescharacteristic
86C3×#107—22no—
87C3×#103—24no—
88C3×C4×C8D412yes—
89C3×C4×C8D412yes—
90#836—24no—
91#836—24no—
92#835—22no—
93#835—22no—
94(C2×D3)⋊C8—22no—
95(C2×D3)⋊C8—22no—
96#819—24no—
97Dic3⋊C8—22no—
98Dic3⋊C8—22no—
99#810—22no—
100#809—22no—
101#808—22no—
102#807—22no—
103#803—21no—
104#803—22no—
105C4×C3⋊C8—21no—
106C4×C3⋊C8—22no—
107C2×#137—32no—
108C22×Q16C2×D311yescharacteristic
109C4×#41—31no—
110C2×C4×Q8C2×D312yes—
111#441—62no—
112#440—62no—
113C4×Q16—22no—
114C4×Q16—22no—
115C4×Q16—62no—
116C4×SD16—62no—
117#433—62no—
118#431—62no—
119C2×#108—32no—
120#428—32no—
121C2×#107C2×D311yescharacteristic
122#424—31no—
123#424—31no—
124#419—32no—
125#417—32no—
126C2×#103C2×D312yes—
127C2×#101—32no—
128C2×#101—32no—
129#408—31no—
130#408—31no—
131C4×M16—31no—
132C4×M16—32no—
133C2×C4×C8C2×D311yescharacteristic
134#343—32no—
135#343—32no—
136#341—31no—
137#341—31no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8#4114yes—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C2×#67—21no—
150#235—22no—
151#235—22no—
152C2×D12C2×D411yescharacteristic
153C2×C4×D3—21no—
154C2×C4×D3—21no—
155C2×C4×D3—21no—
156C2×Dic6C2×D411yescharacteristic
157C3×Q16—28no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8C2×D412yes—
161C2×C3×C8C2×D412yes—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—42no—
166C3×C42—42no—
167C3×C42—42no—
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D411yescharacteristic
171C3×C42C2×D411yescharacteristic
172#217—21no—
173#216—28no—
174#215—28no—
175(C2×D3)⋊C4—21no—
176(C2×D3)⋊C4—21no—
177C4⋊Dic3C2×D411yescharacteristic
178C4⋊Dic3C2×D411yescharacteristic
179Dic3⋊C4—21no—
180Dic3⋊C4—21no—
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184#208—22no—
185#208—22no—
186#208—22no—
187#208—22no—
188#208—22no—
189#208#4114yes—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×#41—31no—
195C22×Q8C22×D312yes—
196#137—64no—
197#137—64no—
198C2×Q16—24no—
199C2×Q16—22no—
200C2×Q16—62no—
201C2×SD16—62no—
202C2×M16—32no—
203C2×M16—32no—
204C2×M16—31no—
205C2×M16—31no—
206C22×C8C22×D311yescharacteristic
207C22×C8C22×D311yescharacteristic
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—31no—
213C4×Q8—31no—
214C4×D4—31no—
215C4×D4—31no—
216C4×D4—61no—
217C4×D4—61no—
218#117—31no—
219#117—61no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C22×D312yes—
222C2×C42—22no—
223C2×C42—61no—
224C2×C42C22×D311yescharacteristic
225#108—64no—
226#107—22no—
227#107—62no—
228#106—62no—
229C4⋊C8—62no—
230C4⋊C8—62no—
231#103—24no—
232#103—62no—
233#103—62no—
234#103—62no—
235#102—62no—
236#101—64no—
237#101—64no—
238C22⋊C8—62no—
239C22⋊C8—62no—
240#97—62no—
241#97—61no—
242C4×C8—62no—
243C4×C8—61no—
244C4×C8#6712yes—
245C4×C8#6712yes—
246C23×C3C2×#4111yescharacteristic
247C22×D3—21no—
248C3×Q8—24no—
249C3×Q8—24no—
250C3×Q8—42no—
251C3×Q8—42no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—82no—
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C2×#4112yes—
266C2×C3×C4C2×#4112yes—
267#67—42no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271D12—22no—
272C4×D3—42no—
273C4×D3—41no—
274C4×D3—41no—
275C4×D3—42no—
276Dic6—22no—
277C3×C8—22no—
278C3×C8—22no—
279C3⋊C8—24no—
280C3⋊C8—41no—
281C3⋊C8—42no—
282C3⋊C8—41no—
283#41—62no—
284#41—62no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—31no—
291C2×Q8#23514yes—
292C2×D4—31no—
293C2×D4—61no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4#23711yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C23×D311yescharacteristic
302Q16—28no—
303Q16—68no—
304SD16—68no—
305M16—34no—
306M16—62no—
307M16—62no—
308M16—62no—
309M16—62no—
310M16—62no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—62no—
314C2×C8—62no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8C2×#6712yes—
318C2×C8C2×#6712yes—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—31no—
322C4⋊C4—31no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C22⋊C4—61no—
328C22⋊C4—61no—
329C22⋊C4—61no—
330C42—42no—
331C42—42no—
332C42—61no—
333C42—61no—
334C42—61no—
335C42D3×D411yescharacteristic
336C42D3×D411yescharacteristic
337C42C2×#6711yescharacteristic
338C42C2×#6711yescharacteristic
339C22×C3—21no—
340C22×C3—21no—
341C22×C3C2×#12111yescharacteristic
342C22×C3#52912yes—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—44no—
350C3×C4—82no—
351C3×C4#47312yes—
352C3×C4#52511yescharacteristic
353C3×C4#52512yes—
354C3×C4#53211yescharacteristic
355Dic3—41no—
356Dic3—41no—
357Dic3—41no—
358C23—61no—
359C23C2×#23711yescharacteristic
360Q8—24no—
361Q8—24no—
362Q8—42no—
363Q8—42no—
364Q8—62no—
365D4—62no—
366D4—122no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—82no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4#93811yescharacteristic
382C2×C4#93811yescharacteristic
383C2×C4#93811yescharacteristic
384C2×C4#93811yescharacteristic
385C2×C4C2×#23512yes—
386C2×C4C2×D3×D411yescharacteristic
387C2×C4D3×#4112yes—
388C2×C4C22×#6711yescharacteristic
389C8—22no—
390C8—22no—
391C8—64no—
392C8—121no—
393C8—122no—
394C8—121no—
395C2×C3—22no—
396C2×C3#239211yescharacteristic
397C2×C3C2×#47311yescharacteristic
398C2×C3#306711yescharacteristic
399D3—81no—
400C22—21no—
401C22—21no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22#553812yes—
406C22C2×#93811yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—44no—
411C4—82no—
412C4—121no—
413C4—121no—
414C4—121no—
415C4#519212yes—
416C4#554712yes—
417C4#579111yescharacteristic
418C4#583211yescharacteristic
419C3#2371711yescharacteristic, complemented, Sylow
420C2—22no—
421C2—241no—
422C2#8240111yescharacteristic
423C2C2×#519211yescharacteristic
424C2#8392811yescharacteristic
subgroupG/H
centerC2×C4#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×Q16C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenterC2×C4×Q16D3
hypocenterC3#23717
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×Q16C2semidirect
5. #72070C2semidirect
6. C4×#836C2semidirect
8. #70626C2semidirect
9. #70265C2semidirect
11. C4×#819C2semidirect
45. C2×C4×Q16D3semidirect
419. C3#23717semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q162
C4×#9422
#784552
#720742
#720702
C4×#8362
#706472
#706262
#702652
#702642
C4×#8192
#237173