Finite Group Databasebeta

← #10695926

Subgroups — #10695926

Order 768.

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

normal subgroups only

This group has 622 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, complemented
2#82441C211yesmaximal, characteristic, complemented
3#79297C211yesmaximal, characteristic
4C3×#1479C211yesmaximal, characteristic, complemented
5#70790C211yesmaximal, characteristic, complemented
6#70762C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9997—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#946—21no—
11C2×#942—21no—
12#5477C2211yescharacteristic
13C2×C4×D12C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#431C2211yescharacteristic
16#5031—21no—
17#5028—22no—
18#5027—21no—
19#4991—21no—
20#4987—21no—
21#4981—21no—
22#4966—21no—
23#4940—22no—
24#4936C2211yescharacteristic
25#4924C2211yescharacteristic
26#4846—21no—
27#4843—21no—
28#4736—21no—
29#4732—21no—
30C4×D24—22no—
31C3×Q8⋊C8—22no—
32#4470—22no—
33#4459—22no—
34#3100—31no—
35#3037—31no—
36#2536—31no—
37#1517—31no—
38#1486—31no—
39#1479D311yescharacteristic, complemented
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C22×D12D411yescharacteristic
44C22×C4×D3—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53#946—42no—
54#946—42no—
55C2×#216—22no—
56#942—42no—
57#942—42no—
58C2×#215—22no—
59#930—21no—
60#930—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×Dic3⋊C4—22no—
67C2×#208—21no—
68C2×D24—22no—
69#875—21no—
70#874D411yescharacteristic
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77#869D411yescharacteristic
78C3×C4⋊C8—21no—
79C3×C4⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C4×C8—21no—
83C3×C4×C8D412yes—
84C3⋊#95—22no—
85C3⋊#95—21no—
86C3⋊#95D412yes—
87#821—22no—
88#821—22no—
89#818—22no—
90#810—22no—
91#809—22no—
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)D412yes—
94C2×#137—61no—
95#532—31no—
96C2×C22⋊Q8—61no—
97C2×C4×Q8C2×D311yescharacteristic
98C2×C4×D4—31no—
99#480—61no—
100#474—61no—
101#464—62no—
102#462—61no—
103#460—61no—
104#459—61no—
105#447—61no—
106C4×D8—62no—
107#431C2×D311yescharacteristic
108#431C2×D311yescharacteristic
109#423—31no—
110#417—61no—
111#398—61no—
112#394—61no—
113#388—62no—
114#382—31no—
115#380—61no—
116#327—62no—
117Q8⋊C8—22no—
118D4⋊C8—62no—
119C23×D3—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8D812yescomplemented
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C2×#67—41no—
132C22×Dic3—22no—
133C22×Dic3—21no—
134#235—41no—
135#235—41no—
136#235—41no—
137#235—41no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12—42no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×C4×D3—41no—
148C2×C4×D3—41no—
149C2×C4×D3—41no—
150C2×C4×D3—41no—
151C2×Dic6—21no—
152C2×C3×C8—22no—
153C2×C3×C8—41no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—41no—
158C3×C42—41no—
159C3×C42—41no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42#4111yescharacteristic
164#216—44no—
165#215—44no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—44no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—41no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176C4⋊Dic3—41no—
177C4⋊Dic3—41no—
178Dic3⋊C4—44no—
179#208—41no—
180#208—41no—
181C2×C3⋊C8—22no—
182D24—44no—
183C2×#41—61no—
184C22×Q8D1211yescharacteristic
185C22×D4—31no—
186C23×C4—61no—
187#137—122no—
188#137—122no—
189C2×Q16—62no—
190C2×SD16—62no—
191C2×D8—62no—
192C2×M16—61no—
193C4⋊Q8—31no—
194#127—31no—
195#124—61no—
196C22⋊Q8—122no—
197C22⋊Q8—122no—
198#121—61no—
199#121—61no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×D4—61no—
205C4×D4—61no—
206C4×D4—61no—
207C4×D4—61no—
208C4×D4—61no—
209C4×D4—61no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—62no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4D1211yescharacteristic
215C2×C22⋊C4—62no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—61no—
218C2×C42—21no—
219C2×C42C22×D311yescharacteristic
220#107—62no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—32no—
224C4⋊C8—61no—
225#103—62no—
226#102—62no—
227#102—62no—
228#102—62no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C4×C8—21no—
232C4×C8#6712yes—
233#95—32no—
234#95—62no—
235#95—61no—
236C23×C3#12011yescharacteristic
237C22×D3—22no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3—41no—
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C3×Q8—22no—
246C3×Q8—22no—
247C3×Q8—41no—
248C3×Q8—41no—
249C2×C3×C4—22no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12711yescharacteristic
262C2×C3×C4C2×D811yescharacteristic
263C2×C3×C4#13511yescharacteristic
264C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
265C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
266#67—81no—
267#67—81no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—42no—
274C2×Dic3—41no—
275D12—22no—
276D12—22no—
277D12—41no—
278D12—41no—
279D12—42no—
280D12—41no—
281D12—41no—
282D12—81no—
283C4×D3—81no—
284C4×D3—81no—
285C4×D3—81no—
286C4×D3—81no—
287C4×D3—81no—
288C4×D3—81no—
289Dic6—41no—
290Dic6—41no—
291C3×C8—42no—
292C3×C8—81no—
293C3⋊C8—42no—
294C24—61no—
295#41—121no—
296#41—121no—
297#41—121no—
298#41—121no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—61no—
305C2×Q8D2412yescomplemented
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—61no—
309C2×D4—61no—
310C2×D4—61no—
311C2×D4—122no—
312C2×D4—121no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—62no—
316C22×C4—61no—
317C22×C4—121no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4—121no—
324C22×C4C2×D1211yescharacteristic
325C22×C4D3×D411yescharacteristic
326C22×C4D3×D411yescharacteristic
327Q16—124no—
328SD16—124no—
329D8—124no—
330M16—121no—
331M16—121no—
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—62no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—61no—
340C4⋊C4—61no—
341C4⋊C4—61no—
342C4⋊C4—61no—
343C4⋊C4—124no—
344C4⋊C4—121no—
345C4⋊C4—121no—
346C22⋊C4—124no—
347C22⋊C4—124no—
348C22⋊C4—121no—
349C22⋊C4—121no—
350C22⋊C4—121no—
351C22⋊C4—121no—
352C42—41no—
353C42—41no—
354C42#23511yescharacteristic
355C42D3×D411yescharacteristic
356C42D3×D411yescharacteristic
357C42C2×#6711yescharacteristic
358C22×C3—21no—
359C22×C3—21no—
360C22×C3#39211yescharacteristic
361C22×C3#44711yescharacteristic
362C22×C3#44911yescharacteristic
363C2×D3—42no—
364C2×D3—42no—
365C2×D3—42no—
366C2×D3—81no—
367C2×D3—81no—
368C2×D3—81no—
369C2×D3—81no—
370C2×D3—81no—
371C2×D3—81no—
372C2×D3—81no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—42no—
378C3×C4—81no—
379C3×C4#44711yescharacteristic
380C3×C4#44911yescharacteristic
381C3×C4#46611yescharacteristic
382C3×C4#46911yescharacteristic
383Dic3—82no—
384Dic3—81no—
385Dic3—81no—
386C23—62no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23—121no—
394C23#88211yescharacteristic
395Q8—22no—
396Q8—22no—
397Q8—41no—
398Q8—41no—
399Q8—121no—
400Q8—121no—
401D4—62no—
402D4—62no—
403D4—121no—
404D4—121no—
405D4—122no—
406D4—121no—
407D4—121no—
408D4—241no—
409D4—241no—
410D4—241no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—122no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—241no—
426C2×C4—241no—
427C2×C4—241no—
428C2×C4—241no—
429C2×C4—241no—
430C2×C4—241no—
431C2×C4#88211yescharacteristic
432C2×C4#88411yescharacteristic
433C2×C4#88411yescharacteristic
434C2×C4C2×D2411yescharacteristic
435C2×C4#90411yescharacteristic
436C2×C4#90811yescharacteristic
437C2×C4#90811yescharacteristic
438C2×C4#93011yescharacteristic
439C2×C4#93711yescharacteristic
440C2×C4#94011yescharacteristic
441C8—42no—
442C8—81no—
443C8—122no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3#199911yescharacteristic
447C2×C3#200911yescharacteristic
448C2×C3#201011yescharacteristic
449D3—82no—
450D3—161no—
451C22—21no—
452C22—21no—
453C22—122no—
454C22—122no—
455C22—122no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22—241no—
463C22#473211yescharacteristic
464C22#473411yescharacteristic
465C22#496011yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—42no—
471C4—81no—
472C4—242no—
473C4—241no—
474C4—241no—
475C4#473211yescharacteristic
476C4#473411yescharacteristic
477C4#511511yescharacteristic
478C4#513511yescharacteristic
479C3#999711yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—242no—
483C2—481no—
484C2#6996111yescharacteristic
485C2#7035111yescharacteristic
486C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#392
prime radical#79297C2
hypercenter#1479D3
hypocenterC3#9997
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
1. #82499C2semidirect
2. #82441C2semidirect
4. C3×#1479C2semidirect
5. #70790C2semidirect
6. #70762C2semidirect
7. #67738C2semidirect
39. #1479D3semidirect
125. C2×C3×Q8D8semidirect
305. C2×Q8D24semidirect
479. C3#9997semidirect

Maximal subgroups

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

subgroupindex
#824992
#824412
#792972
C3×#14792
#707902
#707622
#677382
#99973