Finite Group Databasebeta

← #10883590

Subgroups — #10883590

Order 768.

764 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2955C211yesmaximal, characteristic
2C2×#5143C211yesmaximal, characteristic, complemented
3#83807C211yesmaximal, characteristic, complemented
4#78573C211yesmaximal, characteristic, complemented
5#72120C212yesmaximal
6#72013C211yesmaximal, characteristic, complemented
7#72012C211yesmaximal, characteristic, complemented
8#70733C212yesmaximal, complemented
9#70411C212yesmaximal, complemented
10#70410C212yesmaximal, complemented
11#69896C211yesmaximal, characteristic, complemented
12#28184—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C2×C3×#131C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16#5825C2212yes—
17C2×#951C2212yes—
18C22×Dic12C2211yescharacteristic
19#5516C2212yes—
20C3×#444—22no—
21C3×#443—22no—
22C3×#430C2211yescharacteristic
23C3×#421C2211yescharacteristic
24C3×#421C2211yescharacteristic
25C3×#405C2211yescharacteristic
26#5240C2212yes—
27#5236C2212yes—
28#5143—22no—
29#5143—22no—
30#5137—22no—
31C2×#824C2212yes—
32#5120C2212yes—
33#5120C2212yes—
34#5116—22no—
35C2×#819C2211yescharacteristic
36#5106C2211yescharacteristic
37#5106C2211yescharacteristic
38#4716—22no—
39C4×Dic12—22no—
40#4559C2212yes—
41#4557C2212yes—
42#4556C2212yes—
43#4531C2212yes—
44C2×#473—31no—
45#3059—31no—
46#2955D311yescharacteristic
47#2950—31no—
48#2082—32no—
49#2009—32no—
50#1912—32no—
51#1905—32no—
52#1833—31no—
53#1814—31no—
54#1812—31no—
55C2×C3×#41C2311yescharacteristic
56C2×C3×#41C2311yescharacteristic
57C22×Dic6C2312yes—
58C3×#137—21no—
59C3×#137—21no—
60C3×AGL(1,ℤ/8)—21no—
61C3×AGL(1,ℤ/8)—21no—
62C3×#135—22no—
63C3×#131—21no—
64C3×#131—21no—
65C3×#131—21no—
66C3×#131—21no—
67C3×#131—21no—
68C3×#131—21no—
69C3×#131—21no—
70C3×#131—21no—
71C2×C3×M16—21no—
72C2×C3×M16—21no—
73C2×C3×M16C2311yescharacteristic
74C2×C3×M16C2311yescharacteristic
75C2×C3×M16C2311yescharacteristic
76C22×C3×C8—21no—
77C22×C3×C8—21no—
78C22×C3×C8C2311yescharacteristic
79C3×#117C2311yescharacteristic
80#951—24no—
81#951—42no—
82#944—22no—
83C2×#216—22no—
84#936—22no—
85#935—42no—
86C2×#214—22no—
87#926C2312yes—
88#926C2312yes—
89#924—24no—
90C2×#208C2312yes—
91C2×Dic12—21no—
92C2×Dic12—21no—
93C2×Dic12—21no—
94C2×Dic12—21no—
95C2×Dic12D412yes—
96C2×Dic12D412yes—
97#887—42no—
98#877—42no—
99C4×Dic6—24no—
100C3×#108—22no—
101C3×#104—22no—
102C3×#104—22no—
103C3×#97D412yes—
104C3×C4×C8D412yes—
105#837—42no—
106#834—42no—
107#824—24no—
108#819—21no—
109#819—21no—
110#818—21no—
111#818—21no—
112#817—21no—
113#817—21no—
114#816—24no—
115#816—24no—
116#575C2×D311yescharacteristic
117C2×#137—32no—
118C22×Q16—31no—
119C2×#131C2×D311yescharacteristic
120C2×#131C2×D311yescharacteristic
121#536—32no—
122#519—32no—
123#473—62no—
124#473—62no—
125#470—62no—
126#467—62no—
127#444—22no—
128#443—22no—
129#441—62no—
130C4×Q16—62no—
131#430C2×D311yescharacteristic
132C2×#108—31no—
133#427—31no—
134#427—31no—
135#421—32no—
136#421C2×D311yescharacteristic
137#421C2×D311yescharacteristic
138#419—32no—
139#419—32no—
140C2×#101—32no—
141#410—32no—
142#405C2×D311yescharacteristic
143#344—32no—
144#344—32no—
145H(3,ℤ/4)—32no—
146H(3,ℤ/4)—32no—
147C3×#41—21no—
148C3×#41—21no—
149C3×#41—21no—
150C3×#41—21no—
151C3×#41—21no—
152C3×#41—21no—
153C3×#41—21no—
154C3×#41—21no—
155C2×C3×Q8#4111yescharacteristic
156C2×C3×Q8#4111yescharacteristic
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4#4111yescharacteristic
160C2×C3×D4#4111yescharacteristic
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—22no—
170C2×Dic6C2×D414yes—
171C3×Q16—22no—
172C3×SD16—22no—
173C3×SD16—22no—
174C3×D8—22no—
175C3×M16—21no—
176C3×M16—21no—
177C3×M16—21no—
178C3×M16—21no—
179C3×M16—21no—
180C3×M16—21no—
181C3×M16—41no—
182C3×M16—41no—
183C3×M16—41no—
184C3×M16—41no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—41no—
188C2×C3×C8—41no—
189C2×C3×C8—41no—
190C2×C3×C8—41no—
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8C2×D412yes—
194C2×C3×C8C2×D411yescharacteristic
195C2×C3×C8C2×D411yescharacteristic
196C2×C3×C8#4111yescharacteristic
197C2×C3×C8#4111yescharacteristic
198C3×C4⋊C4—21no—
199C3×C22⋊C4—21no—
200C3×C42C2×D412yes—
201#217—22no—
202#217—22no—
203#217—44no—
204#216—44no—
205#214—44no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208Dic3⋊C4—44no—
209Dic3⋊C4—44no—
210C4×Dic3—22no—
211C4×Dic3—22no—
212C4×Dic3—44no—
213#208—22no—
214#208—22no—
215C2×C3⋊C8—22no—
216Dic12—22no—
217Dic12—22no—
218Dic12—22no—
219Dic12—22no—
220C2×#41C22×D311yescharacteristic
221C2×#41C22×D311yescharacteristic
222C22×Q8—32no—
223#137—21no—
224#137—21no—
225#137—64no—
226#137—122no—
227AGL(1,ℤ/8)—21no—
228AGL(1,ℤ/8)—21no—
229#135—22no—
230C2×Q16—32no—
231C2×Q16—32no—
232C2×Q16—62no—
233C2×Q16—61no—
234C2×Q16—61no—
235C2×Q16—61no—
236C2×Q16—61no—
237C2×SD16—62no—
238#131—21no—
239#131—21no—
240#131—21no—
241#131—21no—
242#131—21no—
243#131—21no—
244#131—21no—
245#131—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—32no—
249C2×M16C22×D311yescharacteristic
250C2×M16C22×D311yescharacteristic
251C2×M16C22×D311yescharacteristic
252C22×C8—21no—
253C22×C8—21no—
254C22×C8C22×D311yescharacteristic
255C4⋊Q8—62no—
256#124—62no—
257#123—122no—
258C22⋊Q8—64no—
259C4×Q8—64no—
260C4×Q8—122no—
261#117—32no—
262#117—32no—
263#117—122no—
264#117C22×D311yescharacteristic
265#108—22no—
266#108—61no—
267#108—61no—
268#107—61no—
269#107—61no—
270#106—61no—
271#106—61no—
272#104—22no—
273#104—22no—
274#104—122no—
275#103—64no—
276#103—64no—
277#101—64no—
278#99—122no—
279#97#6712yes—
280C4×C8#6712yes—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3H(5,𝔽2)11yescharacteristic
284C3×Q8—21no—
285C3×Q8—21no—
286C3×Q8—21no—
287C3×Q8—21no—
288C3×D4—21no—
289C3×D4—21no—
290C3×D4—21no—
291C3×D4—21no—
292C3×D4—41no—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4—41no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C2×Dic3—24no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314Dic6—24no—
315Dic6—24no—
316Dic6—42no—
317Dic6—42no—
318C3×C8—22no—
319C3×C8—41no—
320C3×C8—41no—
321C3×C8#12111yescharacteristic
322C3×C8#12111yescharacteristic
323C3×C8#12111yescharacteristic
324C3×C8#12111yescharacteristic
325C3⋊C8—42no—
326#41—21no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—21no—
332#41—21no—
333#41—21no—
334C2×Q8—34no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8#23511yescharacteristic
341C2×Q8#23511yescharacteristic
342C2×D4—21no—
343C2×D4—21no—
344C2×D4#23511yescharacteristic
345C2×D4#23511yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—62no—
349C22×C4C23×D311yescharacteristic
350Q16—22no—
351Q16—62no—
352Q16—62no—
353Q16—62no—
354Q16—62no—
355Q16—124no—
356SD16—22no—
357SD16—22no—
358SD16—124no—
359D8—22no—
360M16—21no—
361M16—21no—
362M16—21no—
363M16—21no—
364M16—21no—
365M16—21no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—62no—
371M16—62no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—62no—
379C2×C8D3×D411yescharacteristic
380C2×C8D3×D411yescharacteristic
381C2×C8#23711yescharacteristic
382C2×C8#23711yescharacteristic
383C2×C8C2×#6712yes—
384C2×C8C2×#6711yescharacteristic
385C2×C8C2×#6711yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—62no—
388C4⋊C4—62no—
389C4⋊C4—124no—
390C4⋊C4—124no—
391C22⋊C4—21no—
392C22⋊C4—62no—
393C22⋊C4—62no—
394C22⋊C4—124no—
395C42—62no—
396C42—62no—
397C42—124no—
398C42C2×#6712yes—
399C22×C3—41no—
400C22×C3—42no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—81no—
404C22×C3#53511yescharacteristic
405C22×C3#54611yescharacteristic
406C22×C3#55311yescharacteristic
407C3×C4—42no—
408C3×C4—41no—
409C3×C4—41no—
410C3×C4C2×#12111yescharacteristic
411C3×C4#52511yescharacteristic
412C3×C4#54611yescharacteristic
413C3×C4#55311yescharacteristic
414Dic3—44no—
415Dic3—82no—
416Dic3—82no—
417C23—21no—
418C23—21no—
419C23#100911yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—64no—
425Q8—64no—
426Q8—122no—
427Q8—122no—
428D4—21no—
429D4—21no—
430D4—21no—
431D4—21no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—64no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4C2×#23511yescharacteristic
449C2×C4C2×#23511yescharacteristic
450C2×C4C2×D3×D411yescharacteristic
451C2×C4C2×#23711yescharacteristic
452C2×C4#100911yescharacteristic
453C2×C4C22×#6711yescharacteristic
454C8—22no—
455C8—41no—
456C8—41no—
457C8—122no—
458C8#93811yescharacteristic
459C8#93811yescharacteristic
460C8#93811yescharacteristic
461C8#93811yescharacteristic
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—81no—
466C2×C3—81no—
467C2×C3#253511yescharacteristic
468C22—41no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—81no—
473C22#555011yescharacteristic
474C22#556511yescharacteristic
475C22#583011yescharacteristic
476C4—42no—
477C4—41no—
478C4—41no—
479C4—124no—
480C4—242no—
481C4—242no—
482C4#555011yescharacteristic
483C4#556511yescharacteristic
484C4#579111yescharacteristic
485C4C2×#93811yescharacteristic
486C3#2818411yescharacteristic, complemented, Sylow
487C2—21no—
488C2—21no—
489C2—21no—
490C2—81no—
491C2—81no—
492C2#8241511yescharacteristic
subgroupG/H
centerC2#82415
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2955C2
socleC2×C3#2535
prime radicalC3×#575C22
hypercenter#2955D3
hypocenterC3#28184
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2955

Complements

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

NH ≅ G/Ntype
2. C2×#5143C2semidirect
3. #83807C2semidirect
4. #78573C2semidirect
6. #72013C2semidirect
7. #72012C2semidirect
8. #70733C2semidirect
9. #70411C2semidirect
10. #70410C2semidirect
11. #69896C2semidirect
486. C3#28184semidirect

Maximal subgroups

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

subgroupindex
C3×#29552
C2×#51432
#838072
#785732
#721202
#720132
#720122
#707332
#704112
#704102
#698962
#281843