Finite Group Databasebeta

← #10695950

Subgroups — #10695950

Order 768.

568 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3078C211yesmaximal, characteristic, complemented
2#83351C211yesmaximal, characteristic
3#82499C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic, complemented
5#71621C211yesmaximal, characteristic, complemented
6#70496C211yesmaximal, characteristic
7#67713C211yesmaximal, characteristic, complemented
8#10001—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#946—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#479—22no—
14C3×#477—21no—
15C3×#475—21no—
16C3×#428—21no—
17C3×#423C2211yescharacteristic
18C3×#419—21no—
19#5084—21no—
20#5083—21no—
21#5046—21no—
22#5043—21no—
23C4×#215—22no—
24#4991—21no—
25#4987—21no—
26#4981—21no—
27#4940—22no—
28#4936C2211yescharacteristic
29#4925C2211yescharacteristic
30#4925C2211yescharacteristic
31#4538—22no—
32#4535—22no—
33#4483—22no—
34#3101—31no—
35#3078D311yescharacteristic, complemented
36#2536—31no—
37#1518—31no—
38#1486—31no—
39#1479—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×#117—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61#946—42no—
62#946—42no—
63C2×#215—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×Dic3⋊C4—22no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C3×#107—22no—
75C3×#106—22no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8D412yes—
78C3×#103—22no—
79C3×#103—22no—
80#835—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86#809—22no—
87#808—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8D412yes—
92#533C2×D311yescharacteristic
93C2×C22⋊Q8—61no—
94C2×C4×Q8C2×D311yescharacteristic
95C2×C4×D4—31no—
96#483—61no—
97#479—22no—
98#477—21no—
99#476—61no—
100#475—21no—
101#461—61no—
102#450—61no—
103C4×SD16—62no—
104#431—31no—
105#431—31no—
106#428—21no—
107#423C2×D311yescharacteristic
108#419—21no—
109#398—61no—
110#394—61no—
111#388—62no—
112#382—31no—
113#380—61no—
114#328—62no—
115Q8⋊C8—62no—
116D4⋊C8—62no—
117C23×D3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8SD1612yescomplemented
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×Dic3—22no—
132C22×Dic3—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—42no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C3×M16—41no—
143C3×M16—41no—
144C2×C3×C8—22no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C42—41no—
157C3×C42—41no—
158C3×C42—41no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163#215—44no—
164(C2×D3)⋊C4—44no—
165(C2×D3)⋊C4—44no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3—41no—
171C4⋊Dic3—41no—
172Dic3⋊C4—44no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175C22×Q8#6711yescharacteristic
176C22×D4—31no—
177C23×C4—61no—
178C2×SD16—62no—
179C2×M16—21no—
180C4⋊Q8#6711yescharacteristic
181C4⋊Q8#6711yescharacteristic
182#125—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—122no—
186C22⋊Q8—122no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×D4—61no—
194C4×D4—61no—
195C4×D4—61no—
196C4×D4—61no—
197#117—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4#6711yescharacteristic
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—61no—
206C2×C42—21no—
207C2×C42C22×D311yescharacteristic
208#107—22no—
209#106—22no—
210#106—62no—
211C4⋊C8—21no—
212C4⋊C8—61no—
213C4⋊C8—61no—
214C4⋊C8#6712yes—
215#103—22no—
216#103—22no—
217#103—62no—
218#102—62no—
219C22⋊C8—61no—
220C22⋊C8—61no—
221C4×C8—32no—
222C4×C8—61no—
223#95—32no—
224#95—62no—
225#95—61no—
226C23×C3#12011yescharacteristic
227C22×D3—22no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C3×Q8—22no—
235C3×Q8—22no—
236C3×Q8—42no—
237C3×Q8—41no—
238C3×Q8—41no—
239C3×Q8—81no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12711yescharacteristic
255C2×C3×C4C2×SD1611yescharacteristic
256C2×C3×C4#13511yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C2×C3×C4#13711yescharacteristic
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—42no—
264C2×Dic3—41no—
265D12—22no—
266D12—22no—
267D12—41no—
268D12—41no—
269C4×D3—81no—
270C4×D3—81no—
271C4×D3—81no—
272C4×D3—81no—
273C3×C8—42no—
274C3⋊C8—42no—
275C3⋊C8—81no—
276C24—61no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8#21512yescomplemented
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—122no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—62no—
291C22×C4—61no—
292C22×C4—121no—
293C22×C4—121no—
294C22×C4—121no—
295C22×C4—121no—
296C22×C4—121no—
297C22×C4—121no—
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4C2×#6711yescharacteristic
301SD16—124no—
302M16—41no—
303M16—41no—
304C2×C8—22no—
305C2×C8—62no—
306C2×C8—121no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—124no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C22⋊C4—41no—
323C22⋊C4—124no—
324C22⋊C4—124no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42D3×D411yescharacteristic
331C42#23711yescharacteristic
332C42C2×#6711yescharacteristic
333C42C2×#6711yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3#39211yescharacteristic
337C22×C3#45011yescharacteristic
338C22×C3#45211yescharacteristic
339C2×D3—42no—
340C2×D3—42no—
341C2×D3—42no—
342C2×D3—81no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—42no—
351C3×C4—81no—
352C3×C4—81no—
353C3×C4—81no—
354C3×C4#44811yescharacteristic
355C3×C4#44911yescharacteristic
356C3×C4#46511yescharacteristic
357C3×C4#46811yescharacteristic
358Dic3—82no—
359Dic3—81no—
360C23—62no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23#93711yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—42no—
371Q8—41no—
372Q8—41no—
373Q8—81no—
374D4—62no—
375D4—62no—
376D4—121no—
377D4—121no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—121no—
390C2×C4—122no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—241no—
394C2×C4—241no—
395C2×C4—241no—
396C2×C4—241no—
397C2×C4#93711yescharacteristic
398C2×C4#93811yescharacteristic
399C2×C4#93911yescharacteristic
400C2×C4#93911yescharacteristic
401C2×C4#94011yescharacteristic
402C2×C4C2×#21511yescharacteristic
403C2×C4#94211yescharacteristic
404C2×C4#94911yescharacteristic
405C2×C4#95011yescharacteristic
406C2×C4#95311yescharacteristic
407C8—42no—
408C8—122no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3#200011yescharacteristic
413C2×C3#200911yescharacteristic
414C2×C3#201011yescharacteristic
415D3—82no—
416C22—21no—
417C22—21no—
418C22—122no—
419C22—122no—
420C22—122no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22—241no—
425C22#504611yescharacteristic
426C22#504711yescharacteristic
427C22#522311yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—42no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4—242no—
437C4—241no—
438C4#504811yescharacteristic
439C4#504911yescharacteristic
440C4#522711yescharacteristic
441C4#523811yescharacteristic
442C3#1000111yescharacteristic, complemented, Sylow
443C2—21no—
444C2—21no—
445C2—242no—
446C2#7169911yescharacteristic
447C2#7211111yescharacteristic
448C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3078C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3078D3
hypocenterC3#10001
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
1. C3×#3078C2semidirect
4. #71838C2semidirect
5. #71621C2semidirect
7. #67713C2semidirect
35. #3078D3semidirect
125. C2×C3×Q8SD16semidirect
284. C2×Q8#215semidirect
442. C3#10001semidirect

Maximal subgroups

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

subgroupindex
C3×#30782
#833512
#824992
#718382
#716212
#704962
#677132
#100013