Finite Group Databasebeta

← #10695924

Subgroups — #10695924

Order 768.

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

normal subgroups only

This group has 622 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×#2536C211yesmaximal, characteristic, complemented
2#82441C211yesmaximal, characteristic, complemented
3#82308C211yesmaximal, characteristic
4#71621C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9997—31nomaximal, Sylow
9C2×C3×C22⋊Q8—21no—
10C2×C3×C4×Q8C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#942—21no—
13#5477C2211yescharacteristic
14C3×#398—21no—
15C3×#394—21no—
16C3×#388—22no—
17C3×#382C2211yescharacteristic
18C3×#380—21no—
19#5217—21no—
20#5031—21no—
21#5028—22no—
22#5027—21no—
23#5015—21no—
24C4×#213—22no—
25#5007—21no—
26#4967—21no—
27#4966—21no—
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4535—22no—
32D4⋊(C3⋊C8)—22no—
33#4451—22no—
34#3100—31no—
35#3037—31no—
36#2536D311yescharacteristic, complemented
37#1517—31no—
38#1486—31no—
39#1479—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×#235—21no—
45C3×C22⋊Q8—42no—
46C3×C22⋊Q8—42no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C42—21no—
64C2×C3×C42C2311yescharacteristic
65C2×#216—22no—
66#942—42no—
67#942—42no—
68C2×#215—22no—
69C2×#213—22no—
70#930—21no—
71#930—21no—
72C2×#208—21no—
73#875—21no—
74#874D411yescharacteristic
75C4×D12—21no—
76C4×D12—21no—
77#869D411yescharacteristic
78C3×#95—22no—
79C3×#95—21no—
80C3×#95D412yes—
81#832—22no—
82#830—21no—
83#830—21no—
84#810—22no—
85#809—22no—
86#809—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)D412yes—
92C4×C3⋊C8—21no—
93C4×C3⋊C8D412yes—
94C2×#137—61no—
95#532—31no—
96C2×C22⋊Q8—21no—
97C2×C4×Q8C2×D311yescharacteristic
98C2×C4×D4C2×D311yescharacteristic
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#431—31no—
109#423—31no—
110#417—61no—
111#398—21no—
112#394—21no—
113#388—22no—
114#382C2×D311yescharacteristic
115#380—21no—
116#327—62no—
117Q8⋊C8—62no—
118D4⋊C8—62no—
119C24×C3—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8D812yescomplemented
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C2×#67—41no—
143#235—41no—
144#235—41no—
145#235—41no—
146#235—41no—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12—21no—
150C2×C4×D3—41no—
151C2×Dic6—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—44no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C22⋊C4—44no—
162C3×C22⋊C4—44no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C42—41no—
166C3×C42—41no—
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42#4111yescharacteristic
171#216—44no—
172#215—44no—
173#213—44no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—41no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178#208—41no—
179#208—41no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—41no—
183C2×#41—61no—
184C22×Q8#6711yescharacteristic
185C22×D4#6711yescharacteristic
186C23×C4—21no—
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—42no—
197C22⋊Q8—42no—
198#121—61no—
199#121—61no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—61no—
209C4×D4—61no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4#6711yescharacteristic
214C2×C4⋊C4#6711yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C42—21no—
219C2×C42C22×D311yescharacteristic
220#107—62no—
221C4⋊C8—32no—
222C4⋊C8—61no—
223C4⋊C8—61no—
224C4⋊C8—61no—
225#103—62no—
226#102—62no—
227#102—62no—
228#102—62no—
229C22⋊C8—61no—
230C22⋊C8—61no—
231C4×C8—32no—
232C4×C8—61no—
233#95—22no—
234#95—21no—
235#95#6712yes—
236C23×C3—22no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3#12011yescharacteristic
244C22×D3—41no—
245C3×Q8—22no—
246C3×Q8—22no—
247C3×Q8—41no—
248C3×Q8—41no—
249C3×D4—22no—
250C3×D4—22no—
251C3×D4—41no—
252C3×D4—41no—
253C2×C3×C4—22no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—81no—
267C2×C3×C4—81no—
268C2×C3×C4—81no—
269C2×C3×C4—81no—
270C2×C3×C4#12011yescharacteristic
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12111yescharacteristic
275C2×C3×C4#12711yescharacteristic
276C2×C3×C4C2×D811yescharacteristic
277C2×C3×C4#13511yescharacteristic
278C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
279C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
280#67—81no—
281#67—81no—
282C2×Dic3—41no—
283D12—42no—
284D12—41no—
285D12—41no—
286D12—81no—
287C4×D3—81no—
288C4×D3—81no—
289Dic6—41no—
290Dic6—41no—
291C3⋊C8—42no—
292C3⋊C8—42no—
293C3⋊C8—81no—
294C24—21no—
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×Q8#21312yescomplemented
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—42no—
309C2×D4—61no—
310C2×D4—61no—
311C2×D4—61no—
312C2×D4—121no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—121no—
324C22×C4C2×#6711yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327Q16—124no—
328SD16—124no—
329D8—124no—
330M16—121no—
331M16—121no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—121no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—44no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—121no—
351C22⋊C4—121no—
352C42—41no—
353C42—41no—
354C42D3×D411yescharacteristic
355C42D3×D411yescharacteristic
356C42D3×D411yescharacteristic
357C42#23711yescharacteristic
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3—81no—
366C22×C3—81no—
367C22×C3#39211yescharacteristic
368C22×C3#44711yescharacteristic
369C22×C3#44911yescharacteristic
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—82no—
379C3×C4—81no—
380C3×C4—81no—
381C3×C4#44711yescharacteristic
382C3×C4#44911yescharacteristic
383C3×C4#46611yescharacteristic
384C3×C4#46911yescharacteristic
385Dic3—81no—
386C23—22no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—121no—
394C23#95311yescharacteristic
395Q8—22no—
396Q8—22no—
397Q8—41no—
398Q8—41no—
399Q8—121no—
400Q8—121no—
401D4—22no—
402D4—22no—
403D4—41no—
404D4—41no—
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—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—121no—
429C2×C4—241no—
430C2×C4—241no—
431C2×C4C2×#21311yescharacteristic
432C2×C4#93211yescharacteristic
433C2×C4#93711yescharacteristic
434C2×C4#93711yescharacteristic
435C2×C4#93811yescharacteristic
436C2×C4#93911yescharacteristic
437C2×C4#93911yescharacteristic
438C2×C4#94011yescharacteristic
439C2×C4#94911yescharacteristic
440C2×C4#95011yescharacteristic
441C8—122no—
442C8—122no—
443C8—241no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—82no—
447C2×C3#199911yescharacteristic
448C2×C3#200911yescharacteristic
449C2×C3#201011yescharacteristic
450D3—161no—
451C22—21no—
452C22—21no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—241no—
461C22—241no—
462C22—241no—
463C22#521711yescharacteristic
464C22#522311yescharacteristic
465C22#523811yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—42no—
471C4—82no—
472C4—81no—
473C4—81no—
474C4—241no—
475C4#503611yescharacteristic
476C4#503711yescharacteristic
477C4#503811yescharacteristic
478C4#503911yescharacteristic
479C3#999711yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—82no—
483C2—481no—
484C2#7168411yescharacteristic
485C2#7171511yescharacteristic
486C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2536C2
socleC22×C3#392
prime radical#82308C2
hypercenter#2536D3
hypocenterC3#9997
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. C3×#2536C2semidirect
2. #82441C2semidirect
4. #71621C2semidirect
5. #71598C2semidirect
6. #69545C2semidirect
7. #67631C2semidirect
36. #2536D3semidirect
125. C2×C3×Q8D8semidirect
305. C2×Q8#213semidirect
479. C3#9997semidirect

Maximal subgroups

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

subgroupindex
C3×#25362
#824412
#823082
#716212
#715982
#695452
#676312
#99973