Finite Group Databasebeta

← #10695954

Subgroups — #10695954

Order 768.

568 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#82499C211yesmaximal, characteristic
2#82431C211yesmaximal, characteristic, complemented
3#79298C211yesmaximal, characteristic
4C3×#1479C211yesmaximal, characteristic, complemented
5#70793C211yesmaximal, characteristic, complemented
6#70762C211yesmaximal, characteristic
7#67738C211yesmaximal, characteristic, complemented
8#10001—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#946—21no—
11C2×C4×D12C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15#5225—21no—
16#5022—22no—
17#5021—21no—
18#5020—21no—
19#4991—21no—
20#4987—21no—
21#4981—21no—
22#4964—21no—
23#4940—22no—
24#4936C2211yescharacteristic
25#4924C2211yescharacteristic
26#4842—21no—
27#4841—21no—
28#4731—21no—
29#4721—21no—
30C4×#204—22no—
31C3×Q8⋊C8—22no—
32#4467—22no—
33#4459—22no—
34#3101—31no—
35#3078—31no—
36#2536—31no—
37#1518—31no—
38#1486—31no—
39#1479D311yescharacteristic, complemented
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52#946—42no—
53#946—42no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—21no—
57#926—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D411yescharacteristic
60#924—21no—
61#924—21no—
62C2×Dic3⋊C4—22no—
63C2×#208—21no—
64C2×#204—22no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69#870—21no—
70#869D411yescharacteristic
71#869D411yescharacteristic
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×C4×C8—21no—
79C3×C4×C8D412yes—
80#835—22no—
81#835—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84C3⋊#95D412yes—
85#821—22no—
86#817—22no—
87#816—22no—
88#808—22no—
89#807—22no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)D412yes—
92#533—31no—
93C2×C22⋊Q8—61no—
94C2×C4×Q8C2×D311yescharacteristic
95C2×C4×D4—31no—
96#483—61no—
97#479—62no—
98#477—61no—
99#476—61no—
100#475—61no—
101#461—61no—
102#450—61no—
103C4×SD16—62no—
104#431C2×D311yescharacteristic
105#431C2×D311yescharacteristic
106#428—61no—
107#423—31no—
108#419—61no—
109#398—61no—
110#394—61no—
111#388—62no—
112#382—31no—
113#380—61no—
114#328—62no—
115Q8⋊C8—22no—
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×Q8SD1612yescomplemented
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—22no—
130C22×Dic3—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—42no—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C42—41no—
149C3×C42—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#217—41no—
155(C2×D3)⋊C4—44no—
156(C2×D3)⋊C4—44no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—41no—
166C4⋊Dic3—41no—
167Dic3⋊C4—44no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170C4×Dic3—41no—
171#208—41no—
172#208—41no—
173C2×C3⋊C8—22no—
174#204—44no—
175C22×Q8D1211yescharacteristic
176C22×D4—31no—
177C23×C4—61no—
178C2×SD16—62no—
179C2×M16—61no—
180C4⋊Q8—31no—
181C4⋊Q8—31no—
182#125—61no—
183C22⋊Q8—61no—
184C22⋊Q8—61no—
185C22⋊Q8—122no—
186C22⋊Q8—122no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—61no—
192C4×Q8—61no—
193C4×D4—61no—
194C4×D4—61no—
195C4×D4—61no—
196C4×D4—61no—
197#117—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4D1211yescharacteristic
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—61no—
206C2×C42—21no—
207C2×C42C22×D311yescharacteristic
208#107—62no—
209#106—62no—
210#106—62no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—32no—
214C4⋊C8—61no—
215#103—62no—
216#103—62no—
217#103—62no—
218#102—62no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C4×C8—21no—
222C4×C8#6712yes—
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—41no—
237C3×Q8—41no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12711yescharacteristic
251C2×C3×C4C2×SD1611yescharacteristic
252C2×C3×C4#13511yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254C2×C3×C4#13711yescharacteristic
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—41no—
263D12—22no—
264D12—22no—
265D12—41no—
266D12—41no—
267C4×D3—81no—
268C4×D3—81no—
269C4×D3—81no—
270C4×D3—81no—
271Dic6—42no—
272Dic6—81no—
273C3×C8—42no—
274C3×C8—81no—
275C3⋊C8—42no—
276C24—61no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8#20412yescomplemented
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×C4C2×D1211yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301SD16—124no—
302M16—121no—
303M16—121no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—62no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—61no—
314C4⋊C4—61no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—124no—
318C4⋊C4—121no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C22⋊C4—124no—
323C22⋊C4—124no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C42—41no—
328C42—41no—
329C42—121no—
330C42#23511yescharacteristic
331C42D3×D411yescharacteristic
332C42D3×D411yescharacteristic
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#44811yescharacteristic
353C3×C4#44911yescharacteristic
354C3×C4#46511yescharacteristic
355C3×C4#46811yescharacteristic
356Dic3—82no—
357Dic3—81no—
358Dic3—81no—
359Dic3—81no—
360C23—62no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23#88211yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372Q8—122no—
373Q8—241no—
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—122no—
386C2×C4—122no—
387C2×C4—121no—
388C2×C4—121no—
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#88211yescharacteristic
398C2×C4#88411yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4C2×#20411yescharacteristic
401C2×C4#90411yescharacteristic
402C2×C4#90811yescharacteristic
403C2×C4#90911yescharacteristic
404C2×C4#93011yescharacteristic
405C2×C4#93711yescharacteristic
406C2×C4#94011yescharacteristic
407C8—42no—
408C8—81no—
409C8—122no—
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#473111yescharacteristic
426C22#473311yescharacteristic
427C22#496011yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—42no—
433C4—81no—
434C4—242no—
435C4—241no—
436C4—241no—
437C4—241no—
438C4#473411yescharacteristic
439C4#473811yescharacteristic
440C4#511411yescharacteristic
441C4#513411yescharacteristic
442C3#1000111yescharacteristic, complemented, Sylow
443C2—21no—
444C2—21no—
445C2—242no—
446C2#6996211yescharacteristic
447C2#7035111yescharacteristic
448C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1479D3
hypocenterC3#10001
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. #82431C2semidirect
4. C3×#1479C2semidirect
5. #70793C2semidirect
7. #67738C2semidirect
39. #1479D3semidirect
123. C2×C3×Q8SD16semidirect
284. C2×Q8#204semidirect
442. C3#10001semidirect

Maximal subgroups

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

subgroupindex
#824992
#824312
#792982
C3×#14792
#707932
#707622
#677382
#100013