Finite Group Databasebeta

← #10696380

Subgroups — #10696380

Order 768.

634 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#500C211yesmaximal, characteristic, complemented
2#83240C211yesmaximal, characteristic
3C2×#5083C211yesmaximal, characteristic
4#71780C211yesmaximal, characteristic, complemented
5C2×#4547C211yesmaximal, characteristic, complemented
6C2×#4459C211yesmaximal, characteristic
7#67655C211yesmaximal, characteristic, complemented
8#10066—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#947—21no—
12C22×#215—21no—
13C2×C4×D12C2211yescharacteristic
14C3×#500—22no—
15C3×#500—22no—
16C2×C3×#107—22no—
17C2×C3×C4×C8C2211yescharacteristic
18#5229—21no—
19C2×#835—21no—
20#5083—22no—
21#5083—22no—
22#4992—22no—
23#4987—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27#4547—22no—
28#4529—22no—
29#4459—22no—
30C2×#500D311yescharacteristic, complemented
31C2×#461—31no—
32#2547—31no—
33#1988—31no—
34C2×#332—31no—
35C2×D4⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8D412yes—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D412yes—
54C2×C3×C42C2311yescharacteristic
55#947—42no—
56#947—42no—
57C2×#215—24no—
58C2×#215—42no—
59C2×#215—42no—
60C2×(C2×D3)⋊C4—24no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3D412yes—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4×Dic3—21no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C3×#107—28no—
72C3×C4×C8—21no—
73C3×C4×C8D412yes—
74#835—24no—
75C3⋊#95—24no—
76C3⋊#95—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C22×SD16—61no—
82C2×C4⋊Q8C2×D311yescharacteristic
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×#124—61no—
85C2×C4×D4—31no—
86#500—22no—
87#500—22no—
88#461—62no—
89#461—62no—
90C2×#107—22no—
91C2×C4⋊C8—31no—
92C2×C4⋊C8—31no—
93C2×#103—61no—
94C2×C4×C8C2×D311yescharacteristic
95#397—62no—
96#391—61no—
97#380—62no—
98C4×C4⋊C4—31no—
99#340—62no—
100#332—62no—
101D4⋊C8—62no—
102C23×D3—21no—
103C2×C3×Q8—22no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C22×C3×C4—22no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—22no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12D812yes—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C4×D3—42no—
131C2×C4×D3—42no—
132C2×C3×C8—24no—
133C2×C3×C8—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146#215—48no—
147(C2×D3)⋊C4—48no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—42no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×D4—31no—
164C23×C4—61no—
165C2×SD16—64no—
166C2×SD16—122no—
167C2×SD16—122no—
168C22×C8—21no—
169C22×C8—61no—
170C22×C8—61no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8#6712yes—
179#124—122no—
180#124—122no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C4×D4—61no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4#6712yes—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—61no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#107—28no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#103—64no—
201C4×C8—21no—
202C4×C8#6712yes—
203#95—64no—
204#95—61no—
205C23×C3#12011yescharacteristic
206C22×D3—22no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—42no—
210C22×D3—42no—
211C3×Q8—42no—
212C3×Q8—42no—
213C3×Q8—81no—
214C3×Q8—81no—
215C3×Q8—81no—
216C3×Q8—81no—
217C3×Q8—81no—
218C3×Q8—81no—
219C2×C3×C4—22no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232C2×C3×C4C2×D811yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4#13512yes—
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4#13712yes—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243D12—22no—
244D12—22no—
245D12—41no—
246D12—41no—
247C4×D3—82no—
248C4×D3—81no—
249C4×D3—81no—
250C3×C8—24no—
251C3⋊C8—42no—
252C3⋊C8—81no—
253C24—61no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—32no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—61no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—122no—
277C22×C4—122no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4C2×#6711yescharacteristic
283SD16—128no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—62no—
287C2×C8—121no—
288C2×C8—121no—
289C2×C8—122no—
290C2×C8—121no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—62no—
300C4⋊C4—62no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C22⋊C4—128no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C42—121no—
307C42—121no—
308C42D3×D411yescharacteristic
309C42#23711yescharacteristic
310C42C2×#6711yescharacteristic
311C42C2×#6711yescharacteristic
312C22×C3#39211yescharacteristic
313C22×C3#44711yescharacteristic
314C22×C3#44911yescharacteristic
315C22×C3#45212yes—
316C22×C3#45612yes—
317C2×D3—44no—
318C2×D3—42no—
319C2×D3—82no—
320C2×D3—81no—
321C2×D3—81no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—82no—
325C3×C4—82no—
326C3×C4#44711yescharacteristic
327C3×C4#44911yescharacteristic
328C3×C4#46412yes—
329C3×C4#46611yescharacteristic
330C3×C4#46911yescharacteristic
331C3×C4#49511yescharacteristic
332C3×C4#49711yescharacteristic
333Dic3—82no—
334Dic3—81no—
335C23—62no—
336C23—121no—
337C23—122no—
338C23—121no—
339C23—122no—
340C23#93711yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345Q8—81no—
346Q8—81no—
347Q8—81no—
348Q8—81no—
349D4—62no—
350D4—62no—
351D4—121no—
352D4—121no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—121no—
364C2×C4—122no—
365C2×C4—242no—
366C2×C4—241no—
367C2×C4—241no—
368C2×C4D3×D811yescharacteristic
369C2×C4#91111yescharacteristic
370C2×C4#91111yescharacteristic
371C2×C4#91211yescharacteristic
372C2×C4#91812yes—
373C2×C4#91912yes—
374C2×C4#93711yescharacteristic
375C2×C4#93811yescharacteristic
376C2×C4#93911yescharacteristic
377C2×C4#93911yescharacteristic
378C2×C4#94011yescharacteristic
379C2×C4#95311yescharacteristic
380C8—24no—
381C8—122no—
382C8—241no—
383C2×C3#164712yes—
384C2×C3#165012yes—
385C2×C3#199911yescharacteristic
386C2×C3#200511yescharacteristic
387C2×C3#201511yescharacteristic
388D3—82no—
389C22—122no—
390C22—124no—
391C22—241no—
392C22—241no—
393C22—242no—
394C22#509512yes—
395C22#515611yescharacteristic
396C22#515811yescharacteristic
397C22#518712yes—
398C22#522311yescharacteristic
399C4—21no—
400C4—21no—
401C4—82no—
402C4—82no—
403C4—241no—
404C4—242no—
405C4#515611yescharacteristic
406C4#515711yescharacteristic
407C4#515811yescharacteristic
408C4#515911yescharacteristic
409C4#518512yes—
410C4#519011yescharacteristic
411C4#519111yescharacteristic
412C3#1006611yescharacteristic, complemented, Sylow
413C2—242no—
414C2#7136912yes—
415C2#7137012yes—
416C2#7179511yescharacteristic
417C2#7204311yescharacteristic
418C2#7207211yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#500C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#500D3
hypocenterC3#10066
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
1. C2×C3×#500C2semidirect
4. #71780C2semidirect
5. C2×#4547C2semidirect
7. #67655C2semidirect
30. C2×#500D3semidirect
412. C3#10066semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5002
#832402
C2×#50832
#717802
C2×#45472
C2×#44592
#676552
#100663