Finite Group Databasebeta

← #10847313

Subgroups — #10847313

Order 768.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3165C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3#83330C211yesmaximal, characteristic, complemented
4C2×#5084C211yesmaximal, characteristic
5#71790C211yesmaximal, characteristic, complemented
6#70676C212yesmaximal
7#70673C212yesmaximal
8#70671C212yesmaximal, complemented
9#70668C212yesmaximal, complemented
10#69875C211yesmaximal, characteristic, complemented
11C4×#823C211yesmaximal, characteristic
12#25490—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C4×D12C2211yescharacteristic
16C3×#501—22no—
17C3×#500—21no—
18C3×#498—21no—
19C3×#428C2212yes—
20C3×#428C2212yes—
21C3×C4×M16C2211yescharacteristic
22C2×#823C2211yescharacteristic
23C2×#823C2211yescharacteristic
24C2×#822C2212yes—
25#5087—21no—
26#5084—21no—
27#5084—21no—
28#5084—22no—
29#5084—21no—
30#5066—22no—
31#5024—21no—
32#5024—22no—
33Q8×C3⋊C8—21no—
34#5001C2212yes—
35#4999C2212yes—
36#4997C2212yes—
37#4995C2212yes—
38C2×#809C2212yes—
39C2×#808C2212yes—
40#4924C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C4×#208C2211yescharacteristic
43#4558C2212yes—
44#4554C2211yescharacteristic
45#4554C2211yescharacteristic
46#4530C2212yes—
47#4530C2212yes—
48#3165D311yescharacteristic, complemented
49#3077—31no—
50C2×#476—31no—
51#2962—31no—
52#2090—32no—
53#2063—32no—
54#1997—31no—
55#1951—32no—
56#1950—31no—
57#1931—32no—
58C4×#100—31no—
59C22×C3×Q8—21no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C2×C3×M16C2311yescharacteristic
63C2×C3×M16C2311yescharacteristic
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8D411yescharacteristic
69C3×C4⋊Q8D411yescharacteristic
70C3×#125D412yes—
71C3×C22⋊Q8—21no—
72C3×C22⋊Q8—21no—
73C3×C4×Q8—21no—
74C3×C4×Q8—21no—
75C3×#117C2312yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C42C2311yescharacteristic
79C2×(C2×D3)⋊C4—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×#208C2312yes—
82C2×#208C2311yescharacteristic
83C2×#208C2311yescharacteristic
84C22×C3⋊C8C2312yes—
85C4×D12—21no—
86C4×D12—21no—
87C4×D12—21no—
88C4×D12—21no—
89C3×#107—22no—
90C3×#107—22no—
91C3×#106—22no—
92C3×#106—22no—
93C3×#97—21no—
94C3×C4×C8—21no—
95#823—22no—
96#823Q814yescomplemented
97#822—24no—
98#809—22no—
99#809—22no—
100#809—22no—
101#809—22no—
102#808—22no—
103#808—22no—
104#808—22no—
105#807—22no—
106C4⋊(C3⋊C8)—21no—
107C4⋊(C3⋊C8)—21no—
108C4⋊(C3⋊C8)—22no—
109C4⋊(C3⋊C8)—22no—
110C4⋊(C3⋊C8)D411yescharacteristic
111C4⋊(C3⋊C8)D412yes—
112C4⋊(C3⋊C8)D411yescharacteristic
113#803—21no—
114#803—22no—
115C4×C3⋊C8—21no—
116C4×C3⋊C8—22no—
117#533C2×D311yescharacteristic
118C2×C4⋊Q8C2×D311yescharacteristic
119C2×C4×D4—31no—
120#501—22no—
121#500—21no—
122#498—21no—
123#478—62no—
124#476—61no—
125#476—62no—
126#476—61no—
127#476—61no—
128#474—61no—
129#446—62no—
130#446—61no—
131C8×Q8—61no—
132C2×#108—32no—
133#428—32no—
134#428C2×D312yes—
135#428C2×D312yes—
136C2×#106—32no—
137#424—32no—
138#423—31no—
139C2×C4⋊C8—31no—
140#418—32no—
141C2×#102—32no—
142C2×#100—31no—
143C2×#100—31no—
144#405—32no—
145C4×M16—31no—
146C4×M16C2×D311yescharacteristic
147#343—31no—
148#343—31no—
149#343—32no—
150#343—32no—
151#341—32no—
152C23×D3—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—41no—
156C2×C3×Q8—41no—
157C2×C3×Q8—82no—
158C22×C3×C4—22no—
159C22×C3×C4#4111yescharacteristic
160C22×C3×C4#4111yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12C2×Q812yes—
169C2×C4×D3—41no—
170C2×C4×D3—41no—
171C2×C4×D3—82no—
172C3×M16—21no—
173C3×M16—21no—
174C3×M16C2×Q812yes—
175C3×M16#4112yes—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—41no—
185C3×C4⋊C4—41no—
186C3×C4⋊C4C2×D412yes—
187C3×C4⋊C4C2×D412yes—
188C3×C22⋊C4—22no—
189C3×C42—22no—
190C3×C42C2×D411yescharacteristic
191C3×C42C2×D411yescharacteristic
192C3×C42C2×D411yescharacteristic
193C3×C42C2×D411yescharacteristic
194(C2×D3)⋊C4—41no—
195(C2×D3)⋊C4—41no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198#208—21no—
199#208—22no—
200#208—21no—
201#208—21no—
202#208—22no—
203#208C2×Q812yes—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8C2×D412yes—
210C2×C3⋊C8C2×D412yes—
211C22×Q8—21no—
212C22×D4—31no—
213C23×C4—61no—
214C2×M16—31no—
215C2×M16—32no—
216C2×M16—31no—
217C2×M16C22×D311yescharacteristic
218C2×M16C22×D311yescharacteristic
219C22×C8—32no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8#6711yescharacteristic
225C4⋊Q8#6711yescharacteristic
226#125#6712yes—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×D4—61no—
232C4×D4—61no—
233C4×D4—61no—
234C4×D4—61no—
235#117C22×D312yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4C22×D312yes—
239C2×C22⋊C4—61no—
240C2×C42C22×D311yescharacteristic
241#108—64no—
242#107—22no—
243#107—22no—
244#107—62no—
245#106—22no—
246#106—22no—
247#106—62no—
248#106—62no—
249#106—62no—
250C4⋊C8—31no—
251C4⋊C8—31no—
252C4⋊C8—32no—
253C4⋊C8—62no—
254C4⋊C8—62no—
255C4⋊C8—61no—
256C4⋊C8—61no—
257#102—62no—
258#102—62no—
259#102—62no—
260#102—62no—
261#100—34no—
262#100—62no—
263#97—21no—
264#97—62no—
265#97—61no—
266C4×C8—21no—
267C4×C8—62no—
268C4×C8—61no—
269C23×C3C2×#4111yescharacteristic
270C22×D3—22no—
271C22×D3—41no—
272C22×D3—41no—
273C22×D3—81no—
274C22×D3—81no—
275C3×Q8—42no—
276C3×Q8—81no—
277C3×Q8—82no—
278C3×Q8—82no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—82no—
285C2×C3×C4#12111yescharacteristic
286C2×C3×C4#12111yescharacteristic
287C2×C3×C4#12111yescharacteristic
288C2×C3×C4#12111yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C22×Q811yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C42−1+411yescharacteristic
294C2×C3×C42−1+411yescharacteristic
295C2×Dic3—41no—
296C2×Dic3—81no—
297D12—22no—
298D12—22no—
299D12—41no—
300D12—41no—
301C4×D3—82no—
302C4×D3—82no—
303C3×C8—22no—
304C3×C8—22no—
305C3⋊C8—22no—
306C3⋊C8—22no—
307C3⋊C8—22no—
308C3⋊C8—41no—
309C3⋊C8—42no—
310C24—61no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—82no—
316C2×D4—32no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C22×C4—22no—
323C22×C4—61no—
324C22×C4—121no—
325C22×C4—121no—
326C22×C4—242no—
327C22×C4#23711yescharacteristic
328C22×C4#23711yescharacteristic
329C22×C4C23×D311yescharacteristic
330M16—21no—
331M16—21no—
332M16—32no—
333M16—61no—
334M16—61no—
335M16—62no—
336M16—61no—
337M16—62no—
338M16D3×Q812yes—
339M16#23912yes—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—32no—
343C2×C8—32no—
344C2×C8—61no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—61no—
348C2×C8—62no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—61no—
358C4⋊C4—61no—
359C4⋊C4C2×#6712yes—
360C4⋊C4C2×#6712yes—
361C22⋊C4—22no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C42—22no—
365C42D3×D411yescharacteristic
366C42D3×D411yescharacteristic
367C42C2×#6711yescharacteristic
368C42C2×#6711yescharacteristic
369C22×C3—21no—
370C22×C3—21no—
371C22×C3C2×#12111yescharacteristic
372C22×C3#53311yescharacteristic
373C22×C3#54111yescharacteristic
374C2×D3—42no—
375C2×D3—82no—
376C2×D3—81no—
377C2×D3—81no—
378C2×D3—82no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—42no—
383C3×C4—42no—
384C3×C4—82no—
385C3×C4#47112yes—
386C3×C4#54711yescharacteristic
387C3×C4#54811yescharacteristic
388C3×C4D4×Q811yescharacteristic
389C3×C4D4×Q811yescharacteristic
390Dic3—81no—
391C23—62no—
392C23—121no—
393C23—121no—
394C23—241no—
395C23—241no—
396C23C2×#23711yescharacteristic
397Q8—42no—
398Q8—81no—
399Q8—82no—
400Q8—82no—
401D4—62no—
402D4—62no—
403D4—121no—
404D4—121no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—121no—
412C2×C4—242no—
413C2×C4—242no—
414C2×C4—241no—
415C2×C4#93911yescharacteristic
416C2×C4#93911yescharacteristic
417C2×C4#93911yescharacteristic
418C2×C4#93911yescharacteristic
419C2×C4C2×D3×D411yescharacteristic
420C2×C4C2×D3×Q811yescharacteristic
421C2×C4C2×#23911yescharacteristic
422C2×C4#100711yescharacteristic
423C2×C4#100711yescharacteristic
424C2×C4C22×#6711yescharacteristic
425C8—22no—
426C8—22no—
427C8—62no—
428C8—62no—
429C8—62no—
430C8—121no—
431C8—122no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3#242911yescharacteristic
435C2×C3C2×#47111yescharacteristic
436C2×C3#306811yescharacteristic
437D3—82no—
438C22—21no—
439C22—21no—
440C22—122no—
441C22—242no—
442C22—242no—
443C22—241no—
444C22—241no—
445C22#572811yescharacteristic
446C22#573111yescharacteristic
447C22C2×#93911yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—42no—
452C4—42no—
453C4—82no—
454C4—241no—
455C4#520312yes—
456C4#572611yescharacteristic
457C4#572711yescharacteristic
458C4Q8×#6711yescharacteristic
459C4#581711yescharacteristic
460C3#2549011yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—242no—
464C2#8325411yescharacteristic
465C2#8394311yescharacteristic
466C2C2×#520311yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3165C2
socleC22×C3C2×#121
prime radicalC22×D12C23
hypercenter#3165D3
hypocenterC3#25490
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
1. C3×#3165C2semidirect
3. #83330C2semidirect
5. #71790C2semidirect
8. #70671C2semidirect
9. #70668C2semidirect
10. #69875C2semidirect
48. #3165D3semidirect
96. #823Q8semidirect
460. C3#25490semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833502
#833302
C2×#50842
#717902
#706762
#706732
#706712
#706682
#698752
C4×#8232
#254903