Finite Group Databasebeta

← #10696382

Subgroups — #10696382

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×#5083C211yesmaximal, characteristic
2C2×#5078C211yesmaximal, characteristic, complemented
3#73802C211yesmaximal, characteristic
4C3×#1988C211yesmaximal, characteristic, complemented
5C2×#4486C211yesmaximal, characteristic, complemented
6C2×#4483C211yesmaximal, characteristic
7#67619C211yesmaximal, characteristic, complemented
8#10066—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—21no—
11C2×#875—21no—
12C2×C4×D12C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16C2×#835—21no—
17#5083—22no—
18#5083—22no—
19#5078—22no—
20#5078—22no—
21#4987—22no—
22C2×#807—22no—
23#4929—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4674—22no—
27C3×#340—22no—
28#4486—22no—
29#4483—22no—
30C2×#500—31no—
31C2×#461—31no—
32#2547—31no—
33#1988D311yescharacteristic, complemented
34C2×#332—31no—
35C2×D4⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×Dic6—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#215—24no—
52C2×#215—42no—
53C2×#215—42no—
54C2×(C2×D3)⋊C4—24no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D412yes—
59C22×C3⋊C8—21no—
60C22×C3⋊C8—21no—
61#875—42no—
62#875—42no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×#95—21no—
74#835—24no—
75C3⋊#95—24no—
76#807—28no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79C4×C3⋊C8—21no—
80C4×C3⋊C8D412yes—
81C22×SD16—61no—
82C2×C4⋊Q8—31no—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×#124—61no—
85C2×C4×D4—31no—
86#500—62no—
87#500—62no—
88#461—62no—
89#461—62no—
90C2×#107—62no—
91C2×C4⋊C8—31no—
92C2×C4⋊C8C2×D311yescharacteristic
93C2×#103—61no—
94C2×C4×C8—31no—
95#397—62no—
96#391—61no—
97#380—62no—
98C4×C4⋊C4C2×D311yescharacteristic
99#340—22no—
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—
109C22×C3×C4—22no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—22no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12D812yes—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×C4×D3—42no—
125C2×C4×D3—42no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C2×C3×C8—41no—
133C2×C3×C8—42no—
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×C42—41no—
141C3×C42—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—22no—
151C4⋊Dic3—21no—
152C4⋊Dic3—22no—
153C4⋊Dic3—21no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156C2×C3⋊C8—24no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C22×Q8—21no—
162C22×Q8—61no—
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—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178C4⋊Q8D1212yes—
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—31no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4#6712yes—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—61no—
193C2×C42—21no—
194C2×C42C22×D311yescharacteristic
195#107—68no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200#103—64no—
201C4×C8—32no—
202C4×C8—61no—
203#95—21no—
204#95—64no—
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—
215C2×C3×C4—22no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×C3×C4C2×D811yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4#13512yes—
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4#13712yes—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239D12—22no—
240D12—22no—
241D12—41no—
242D12—41no—
243C4×D3—82no—
244C4×D3—81no—
245C4×D3—81no—
246Dic6—81no—
247Dic6—81no—
248Dic6—81no—
249Dic6—81no—
250C3×C8—81no—
251C3⋊C8—24no—
252C3⋊C8—42no—
253C24—61no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—121no—
261C2×Q8—121no—
262C2×Q8—121no—
263C2×Q8—121no—
264C2×Q8—121no—
265C2×Q8—121no—
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—121no—
277C22×C4—121no—
278C22×C4—122no—
279C22×C4—122no—
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4C2×#6711yescharacteristic
283SD16—128no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—64no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—121no—
290C2×C8—121no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—61no—
298C4⋊C4—62no—
299C4⋊C4—61no—
300C4⋊C4—62no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C22⋊C4—128no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C42—41no—
307C42—41no—
308C42C2×D1211yescharacteristic
309C42#23511yescharacteristic
310C42D3×D411yescharacteristic
311C42D3×D411yescharacteristic
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—241no—
346Q8—241no—
347Q8—241no—
348Q8—241no—
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—121no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—242no—
366C2×C4—241no—
367C2×C4—241no—
368C2×C4#88211yescharacteristic
369C2×C4#88211yescharacteristic
370C2×C4#88411yescharacteristic
371C2×C4#88411yescharacteristic
372C2×C4D3×D811yescharacteristic
373C2×C4#91111yescharacteristic
374C2×C4#91411yescharacteristic
375C2×C4#91611yescharacteristic
376C2×C4#93011yescharacteristic
377C2×C4#94011yescharacteristic
378C2×C4#94212yes—
379C2×C4#95012yes—
380C8—64no—
381C8—81no—
382C8—122no—
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#496011yescharacteristic
395C22#504712yes—
396C22#515611yescharacteristic
397C22#517211yescharacteristic
398C22#520512yes—
399C4—21no—
400C4—21no—
401C4—82no—
402C4—82no—
403C4—241no—
404C4—242no—
405C4#477311yescharacteristic
406C4#477511yescharacteristic
407C4#478111yescharacteristic
408C4#478211yescharacteristic
409C4#502812yes—
410C4#505911yescharacteristic
411C4#508611yescharacteristic
412C3#1006611yescharacteristic, complemented, Sylow
413C2—242no—
414C2#6968911yescharacteristic
415C2#7020411yescharacteristic
416C2#7034611yescharacteristic
417C2#7111012yes—
418C2#7112612yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1988C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1988D3
hypocenterC3#10066
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. C2×#5078C2semidirect
4. C3×#1988C2semidirect
5. C2×#4486C2semidirect
7. #67619C2semidirect
33. #1988D3semidirect
412. C3#10066semidirect

Maximal subgroups

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

subgroupindex
C2×#50832
C2×#50782
#738022
C3×#19882
C2×#44862
C2×#44832
#676192
#100663