Finite Group Databasebeta

← #10714024

Subgroups — #10714024

Order 768.

614 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#495C211yesmaximal, characteristic, complemented
2C22×#828C211yesmaximal, characteristic, complemented
3C22×#826C211yesmaximal, characteristic
4C2×#4564C211yesmaximal, characteristic, complemented
5C2×#4562C211yesmaximal, characteristic, complemented
6#69776C211yesmaximal, characteristic, complemented
7#67657C211yesmaximal, characteristic, complemented
8#13326—31nomaximal, Sylow
9C22×C3×Q16C2211yescharacteristic
10C22×C3×SD16—21no—
11C22×C3×D8C2211yescharacteristic
12C2×C3×#124—21no—
13C22×D24C2211yescharacteristic
14C3×#495—22no—
15C3×#495—22no—
16C2×C3×C4×C8C2211yescharacteristic
17C2×#828—22no—
18C2×#828—22no—
19C2×#828—22no—
20C2×#826—22no—
21C2×#826—22no—
22C2×#826—22no—
23C2×#821—21no—
24C2×#818C2211yescharacteristic
25C22×C3⋊C16C2211yescharacteristic
26C22×C3⋊C16C2211yescharacteristic
27#4939—21no—
28#4564—22no—
29#4562—22no—
30C22×SD32—31no—
31C22×D16—31no—
32C2×#495D311yescharacteristic, complemented
33C2×#358—31no—
34C2×#357—31no—
35#1970—31no—
36#1374—31no—
37C22×C3×Q8—21no—
38C22×C3×D4—21no—
39C22×D12—21no—
40C2×C3×Q16—21no—
41C2×C3×Q16—21no—
42C2×C3×Q16—21no—
43C2×C3×Q16—21no—
44C2×C3×Q16—21no—
45C2×C3×Q16D412yes—
46C2×C3×SD16—24no—
47C2×C3×SD16—42no—
48C2×C3×SD16—42no—
49C2×C3×D8—21no—
50C2×C3×D8—21no—
51C2×C3×D8—21no—
52C2×C3×D8—21no—
53C2×C3×D8—21no—
54C2×C3×D8D412yes—
55C22×C3×C8D411yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C3×#124—24no—
58C3×#124—42no—
59C2×C3×C22⋊C4—42no—
60C2×C3×C42D411yescharacteristic
61C2×(C2×D3)⋊C4—42no—
62C2×C4⋊Dic3—21no—
63C2×D24—21no—
64C2×D24—21no—
65C2×D24—21no—
66C2×D24—21no—
67C2×D24—21no—
68C2×D24D412yes—
69C3×C4×C8—21no—
70C3×C4×C8—21no—
71#828—28no—
72#826—28no—
73#821—42no—
74#818—21no—
75#818—21no—
76C2×C3⋊C16—21no—
77C2×C3⋊C16—21no—
78C2×C3⋊C16—21no—
79C2×C3⋊C16—21no—
80C2×C3⋊C16D412yes—
81C2×C3⋊C16D412yes—
82C22×Q16C2×D311yescharacteristic
83C22×SD16—21no—
84C22×D8—31no—
85C22×D8C2×D311yescharacteristic
86C2×#124—21no—
87C2×SD32—62no—
88C2×SD32—62no—
89C2×SD32—62no—
90C2×D16—62no—
91C2×D16—62no—
92C2×D16—62no—
93C22×C16—31no—
94C22×C16—31no—
95#495—22no—
96#495—22no—
97C2×#107—31no—
98C2×#102—61no—
99C2×C4×C8C2×D311yescharacteristic
100#390—61no—
101#358—62no—
102#357—62no—
103C24×C3—41no—
104C23×D3—41no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×D4—22no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C22×C3×C4—41no—
118C22×C3×C4D812yes—
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—41no—
121C2×D12—22no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C3×Q16—22no—
128C3×Q16—22no—
129C3×Q16—41no—
130C3×Q16—41no—
131C3×SD16—48no—
132C3×D8—22no—
133C3×D8—22no—
134C3×D8—41no—
135C3×D8—41no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8D812yes—
139C2×C3×C8C2×D411yescharacteristic
140C2×C3×C8C2×D411yescharacteristic
141C2×C3×C8C2×D411yescharacteristic
142C2×C3×C8C2×D411yescharacteristic
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8#4111yescharacteristic
145C3×C22⋊C4—48no—
146C3×C42—21no—
147C3×C42—21no—
148(C2×D3)⋊C4—84no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151D24—22no—
152D24—22no—
153D24—41no—
154D24—41no—
155C3⋊C16—22no—
156C3⋊C16—22no—
157C22×Q8—21no—
158C22×D4—21no—
159C22×D4—61no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16#6712yes—
166C2×SD16—24no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—32no—
175C2×D8—61no—
176C2×D8—61no—
177C2×D8—61no—
178C2×D8—61no—
179C2×D8—61no—
180C2×D8#6712yes—
181C22×C8#6711yescharacteristic
182C22×C8C22×D311yescharacteristic
183#124—24no—
184#124—42no—
185C2×C4⋊C4—61no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—122no—
188C2×C42#6711yescharacteristic
189SD32—68no—
190D16—68no—
191C2×C16—32no—
192C2×C16—32no—
193C2×C16—61no—
194C2×C16—61no—
195C2×C16—61no—
196C2×C16—61no—
197#107—61no—
198#107—61no—
199#102—122no—
200C4×C8—21no—
201C4×C8—21no—
202C23×C3—42no—
203C23×C3—81no—
204C23×C3—81no—
205C23×C3—82no—
206C23×C3—82no—
207C23×C3C2×D811yescharacteristic
208C22×D3—42no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—82no—
212C22×D3—82no—
213C3×Q8—42no—
214C3×Q8—42no—
215C3×Q8—81no—
216C3×Q8—81no—
217C3×D4—42no—
218C3×D4—42no—
219C3×D4—81no—
220C3×D4—81no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—42no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×C3×C4C2×D811yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×Dic3—81no—
234C2×Dic3—82no—
235D12—42no—
236D12—42no—
237D12—81no—
238D12—81no—
239C3×C8—22no—
240C3×C8#12011yescharacteristic
241C3×C8#12011yescharacteristic
242C3×C8#12111yescharacteristic
243C3×C8#12111yescharacteristic
244C24—41no—
245C24—121no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—62no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C22×C4—41no—
265C22×C4—121no—
266C22×C4#21312yes—
267C22×C4C2×#6711yescharacteristic
268Q16—22no—
269Q16—22no—
270Q16—41no—
271Q16—41no—
272SD16—48no—
273D8—22no—
274D8—22no—
275D8—41no—
276D8—41no—
277D8—62no—
278D8—62no—
279D8—121no—
280D8—121no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8#21312yes—
284C2×C8D3×D411yescharacteristic
285C2×C8D3×D411yescharacteristic
286C2×C8D3×D411yescharacteristic
287C2×C8#23711yescharacteristic
288C2×C8C2×#6711yescharacteristic
289C2×C8C2×#6711yescharacteristic
290C4⋊C4—121no—
291C4⋊C4—121no—
292C22⋊C4—48no—
293C22⋊C4—244no—
294C42—21no—
295C42—21no—
296C16—62no—
297C16—62no—
298C22×C3—84no—
299C22×C3—82no—
300C22×C3—162no—
301C22×C3—161no—
302C22×C3—161no—
303C22×C3#44711yescharacteristic
304C22×C3#45911yescharacteristic
305C22×C3#49311yescharacteristic
306C22×C3#50812yes—
307C22×C3#50912yes—
308C2×D3—84no—
309C2×D3—82no—
310C2×D3—162no—
311C2×D3—161no—
312C2×D3—161no—
313C3×C4—42no—
314C3×C4—82no—
315C3×C4#39211yescharacteristic
316C3×C4#44711yescharacteristic
317C3×C4#45911yescharacteristic
318C3×C4#49611yescharacteristic
319Dic3—161no—
320C23—42no—
321C23—82no—
322C23—82no—
323C23—81no—
324C23—81no—
325C23—122no—
326C23—242no—
327C23—242no—
328C23—241no—
329C23—241no—
330C23C2×#21311yescharacteristic
331Q8—42no—
332Q8—42no—
333Q8—81no—
334Q8—81no—
335D4—42no—
336D4—42no—
337D4—81no—
338D4—81no—
339D4—122no—
340D4—122no—
341D4—241no—
342D4—241no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—242no—
350C2×C4—241no—
351C2×C4C2×#21311yescharacteristic
352C2×C4#93211yescharacteristic
353C2×C4#93211yescharacteristic
354C2×C4#93811yescharacteristic
355C2×C4#94011yescharacteristic
356C2×C4#95311yescharacteristic
357C8—22no—
358C8#93711yescharacteristic
359C8#93711yescharacteristic
360C8#93911yescharacteristic
361C8#93911yescharacteristic
362C2×C3—162no—
363C2×C3#201111yescharacteristic
364C2×C3#218612yes—
365C2×C3#221012yes—
366C2×C3#224911yescharacteristic
367C2×C3#225011yescharacteristic
368D3—162no—
369C22—82no—
370C22—84no—
371C22—161no—
372C22—161no—
373C22—162no—
374C22—242no—
375C22—244no—
376C22—482no—
377C22—481no—
378C22—481no—
379C22#507211yescharacteristic
380C22#507311yescharacteristic
381C22#519312yes—
382C22#519412yes—
383C22#521711yescharacteristic
384C4—42no—
385C4—82no—
386C4—481no—
387C4#507211yescharacteristic
388C4#507411yescharacteristic
389C4#521711yescharacteristic
390C4#522311yescharacteristic
391C3#1332611yescharacteristic, complemented, Sylow
392C2—162no—
393C2—482no—
394C2#7132011yescharacteristic
395C2#7132111yescharacteristic
396C2#7132312yes—
397C2#7176711yescharacteristic
398C2#7208212yes—
subgroupG/H
centerC23C2×#213
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC2×C3×#495C2
socleC23×C3C2×D8
prime radicalC22×#826C2
hypercenterC2×#495D3
hypocenterC3#13326
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×#495

Complements

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

NH ≅ G/Ntype
1. C2×C3×#495C2semidirect
2. C22×#828C2semidirect
4. C2×#4564C2semidirect
5. C2×#4562C2semidirect
6. #69776C2semidirect
7. #67657C2semidirect
32. C2×#495D3semidirect
391. C3#13326semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4952
C22×#8282
C22×#8262
C2×#45642
C2×#45622
#697762
#676572
#133263