Finite Group Databasebeta

← #10695923

Subgroups — #10695923

Order 768.

622 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 97 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#83314C211yesmaximal, characteristic, complemented
2#83298C211yesmaximal, characteristic
3#74240C211yesmaximal, characteristic, complemented
4C3×#1517C211yesmaximal, characteristic, complemented
5#70766C211yesmaximal, characteristic, complemented
6#70762C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9997—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#951—21no—
11C2×#897—21no—
12C2×C4×D12C2211yescharacteristic
13C2×C4×Dic6C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#382C2211yescharacteristic
16#5232—21no—
17#5090—21no—
18#5087—21no—
19#5075—21no—
20#5072—21no—
21#5054—22no—
22#5036—21no—
23#5033—21no—
24C4×#213—22no—
25#4987—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4672—21no—
29#4670—22no—
30#4661—21no—
31C3×#327—22no—
32#4484—22no—
33#4459—22no—
34#3100—31no—
35#3037—31no—
36#2536—31no—
37#1517D311yescharacteristic, complemented
38#1486—31no—
39#1479—31no—
40#1286—31no—
41C2×C3×#41—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C22×Dic6D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54#951—42no—
55#951—42no—
56C2×#216—22no—
57C2×#214—22no—
58C2×#213—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C2×#208—21no—
68#897—42no—
69#897—42no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×Dic6—21no—
75C4×Dic6—21no—
76C4×Dic6—21no—
77C4×Dic6—21no—
78C3×C4⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C4×C8—21no—
81C3×#95—21no—
82C3×#95D412yes—
83#835—22no—
84#832—22no—
85#832—22no—
86C3⋊#95—22no—
87#830—21no—
88#809—22no—
89#807—22no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)—21no—
92C4⋊(C3⋊C8)D412yes—
93C4×C3⋊C8D412yes—
94C2×#137—61no—
95#532C2×D311yescharacteristic
96C2×C22⋊Q8—61no—
97C2×C4×Q8—31no—
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#431—31no—
108#431C2×D311yescharacteristic
109#423—31no—
110#417—61no—
111#398—61no—
112#394—61no—
113#388—62no—
114#382C2×D311yescharacteristic
115#380—61no—
116#327—22no—
117Q8⋊C8—62no—
118D4⋊C8—62no—
119C23×D3—21no—
120C3×#41—41no—
121C3×#41—41no—
122C3×#41—41no—
123C3×#41—41no—
124C2×C3×Q8—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—41no—
129C22×C3×C4—22no—
130C22×C3×C4—41no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—42no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C4×D3—41no—
143C2×C4×D3—41no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×Dic6—21no—
147C2×Dic6—21no—
148C2×Dic6—21no—
149C2×Dic6—21no—
150C2×Dic6—21no—
151C2×Dic6D812yescomplemented
152C2×C3×C8—41no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163#216—44no—
164#214—44no—
165#213—44no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—44no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3—44no—
175Dic3⋊C4—41no—
176Dic3⋊C4—41no—
177C4×Dic3—41no—
178C4×Dic3—41no—
179#208—41no—
180#208—41no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×#41—21no—
184C22×Q8—31no—
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⋊Q8D1211yescharacteristic
194#127D1211yescharacteristic
195#124—21no—
196C22⋊Q8—122no—
197C22⋊Q8—122no—
198#121—21no—
199#121—21no—
200C4×Q8—61no—
201C4×Q8—61no—
202C4×Q8—61no—
203C4×Q8—61no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—61no—
207C4×D4—61no—
208C4×D4—61no—
209C4×D4—61no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—31no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4—61no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—61no—
218C2×C42—61no—
219C2×C42C22×D311yescharacteristic
220#107—62no—
221C4⋊C8—21no—
222C4⋊C8—32no—
223C4⋊C8—61no—
224C4⋊C8—61no—
225#103—62no—
226#102—62no—
227#102—62no—
228#102—62no—
229C22⋊C8—21no—
230C22⋊C8—61no—
231C4×C8—21no—
232C4×C8—32no—
233#95—21no—
234#95—62no—
235#95#6712yes—
236C23×C3—41no—
237C23×C3#12011yescharacteristic
238C22×D3—22no—
239C22×D3—41no—
240C22×D3—41no—
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C3×Q8—41no—
246C3×Q8—41no—
247C3×D4—42no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—81no—
251C3×D4—81no—
252C3×D4—81no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—81no—
260C2×C3×C4—81no—
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12711yescharacteristic
267C2×C3×C4C2×D811yescharacteristic
268C2×C3×C4#13511yescharacteristic
269C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
270C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
271C2×Dic3—22no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279D12—22no—
280D12—22no—
281D12—41no—
282D12—41no—
283C4×D3—81no—
284C4×D3—81no—
285C4×D3—81no—
286C4×D3—81no—
287Dic6—22no—
288Dic6—22no—
289Dic6—41no—
290Dic6—41no—
291C3×C8—81no—
292C3⋊C8—42no—
293C3⋊C8—42no—
294C24—61no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299C2×Q8—21no—
300C2×Q8—32no—
301C2×Q8—61no—
302C2×Q8—61no—
303C2×Q8—61no—
304C2×Q8—61no—
305C2×Q8—61no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—61no—
311C2×D4—61no—
312C2×D4—122no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4—121no—
324C22×C4D3×D411yescharacteristic
325C22×C4D3×D411yescharacteristic
326C22×C4C2×#6711yescharacteristic
327Q16—124no—
328SD16—124no—
329D8—124no—
330M16—121no—
331M16—121no—
332C2×C8—41no—
333C2×C8—62no—
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—41no—
347C22⋊C4—41no—
348C22⋊C4—124no—
349C22⋊C4—124no—
350C22⋊C4—121no—
351C22⋊C4—121no—
352C42—121no—
353C42—121no—
354C42C2×D1211yescharacteristic
355C42#23511yescharacteristic
356C42D3×D411yescharacteristic
357C42D3×D411yescharacteristic
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3#39211yescharacteristic
364C22×C3#44711yescharacteristic
365C22×C3#44911yescharacteristic
366C2×D3—42no—
367C2×D3—42no—
368C2×D3—42no—
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—82no—
378C3×C4—81no—
379C3×C4#44711yescharacteristic
380C3×C4#44911yescharacteristic
381C3×C4#46611yescharacteristic
382C3×C4#46911yescharacteristic
383Dic3—42no—
384Dic3—81no—
385Dic3—81no—
386C23—41no—
387C23—62no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23—121no—
394C23#93711yescharacteristic
395Q8—41no—
396Q8—41no—
397Q8—62no—
398Q8—62no—
399Q8—121no—
400Q8—121no—
401D4—42no—
402D4—41no—
403D4—41no—
404D4—62no—
405D4—62no—
406D4—81no—
407D4—81no—
408D4—81no—
409D4—121no—
410D4—121no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—62no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—122no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—241no—
428C2×C4—241no—
429C2×C4—241no—
430C2×C4—241no—
431C2×C4#88211yescharacteristic
432C2×C4#88211yescharacteristic
433C2×C4#88411yescharacteristic
434C2×C4#88411yescharacteristic
435C2×C4D3×D811yescharacteristic
436C2×C4#91111yescharacteristic
437C2×C4#91411yescharacteristic
438C2×C4#91611yescharacteristic
439C2×C4#93011yescharacteristic
440C2×C4#94011yescharacteristic
441C8—81no—
442C8—122no—
443C8—122no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—161no—
447C2×C3#199911yescharacteristic
448C2×C3#200911yescharacteristic
449C2×C3#201011yescharacteristic
450D3—82no—
451C22—21no—
452C22—21no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—122no—
457C22—122no—
458C22—122no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22—241no—
463C22#496011yescharacteristic
464C22#515611yescharacteristic
465C22#517211yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—82no—
471C4—81no—
472C4—122no—
473C4—241no—
474C4—241no—
475C4#477311yescharacteristic
476C4#477511yescharacteristic
477C4#478111yescharacteristic
478C4#478211yescharacteristic
479C3#999711yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—161no—
483C2—242no—
484C2#7005311yescharacteristic
485C2#7020411yescharacteristic
486C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1517C2
socleC22×C3#392
prime radical#83298C2
hypercenter#1517D3
hypocenterC3#9997
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
1. #83314C2semidirect
3. #74240C2semidirect
4. C3×#1517C2semidirect
5. #70766C2semidirect
6. #70762C2semidirect
7. #67631C2semidirect
37. #1517D3semidirect
151. C2×Dic6D8semidirect
479. C3#9997semidirect

Maximal subgroups

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

subgroupindex
#833142
#832982
#742402
C3×#15172
#707662
#707622
#676312
#99973