Finite Group Databasebeta

← #10691522

Subgroups — #10691522

Order 768.

514 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2974C211yesmaximal, characteristic, complemented
2C2×#5016C211yesmaximal, characteristic
3#71599C211yesmaximal, characteristic, complemented
4#70804C211yesmaximal, characteristic, complemented
5#70784C211yesmaximal, characteristic, complemented
6#67755C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#9243—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#214C411yescharacteristic
11C22×#213C411yescharacteristic
12C2×#875C2211yescharacteristic
13C3×#436—21no—
14C3×#435—22no—
15C3×C8×D4—21no—
16C3×#432C2211yescharacteristic
17C3×#431C2211yescharacteristic
18C3×#407—21no—
19C2×C3×C22⋊C8—21no—
20#5016—21no—
21#5016—21no—
22#5016—21no—
23#5016—21no—
24C2×#810C411yescharacteristic
25C2×#809C411yescharacteristic
26#4926C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C4×#208C2211yescharacteristic
29D4⋊(C3⋊C8)—21no—
30D4⋊(C3⋊C8)—21no—
31#4468—21no—
32#4468—21no—
33#4464—22no—
34C2×#463—31no—
35#2974D311yescharacteristic, complemented
36#1527—31no—
37#1490—31no—
38#1469—31no—
39#1290—31no—
40#1283—31no—
41C22×C3×D4C2×C411yescharacteristic
42C23×C3×C4—21no—
43C22×D12C2×C411yescharacteristic
44C22×Dic6C2×C411yescharacteristic
45C2×C3×M16—22no—
46C22×C3×C8—22no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4D411yescharacteristic
50C3×C4×D4D411yescharacteristic
51C3×C4×D4D411yescharacteristic
52C3×C4×D4D411yescharacteristic
53C2×C3×C4⋊C4C2×C411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#214—21no—
57C2×#214—21no—
58C2×#214—42no—
59C2×#213—21no—
60C2×#213—21no—
61C2×#213—42no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×#208—21no—
65C22×C3⋊C8C2×C411yescharacteristic
66C22×C3⋊C8C2×C411yescharacteristic
67#875—21no—
68#875—21no—
69#875—21no—
70#875—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C22⋊C8—22no—
74C3×C22⋊C8—22no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D411yescharacteristic
77C3×C22⋊C8D411yescharacteristic
78C3×#97D412yes—
79C3×C4×C8—21no—
80#830—21no—
81#810—21no—
82#810—21no—
83#809—21no—
84#809—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90C4×C3⋊C8—21no—
91C22×SD16—31no—
92C22×D8—31no—
93C2×#124—31no—
94C2×C4×D4C2×D311yescharacteristic
95#463—61no—
96#463—61no—
97#463—61no—
98#463—61no—
99#436—21no—
100#435—22no—
101C8×D4—21no—
102#432—31no—
103#432C2×D311yescharacteristic
104#431C2×D311yescharacteristic
105C2×C4⋊C8—31no—
106C2×#103—31no—
107C2×#102—31no—
108#407—21no—
109C2×C22⋊C8—21no—
110C4×M16—31no—
111#329—62no—
112D4⋊C8—61no—
113D4⋊C8—61no—
114#323—61no—
115#323—61no—
116C24×C3—21no—
117C23×D3—21no—
118C2×C3×D4—22no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C2×C3×D4C22⋊C411yescharacteristic
122C2×C3×D4C22⋊C411yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—82no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—41no—
136C2×D12—41no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—41no—
140C2×Dic6—41no—
141C3×M16—82no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8—22no—
145C2×C3×C8—82no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4C22⋊C411yescharacteristic
148C3×C4⋊C4C22⋊C411yescharacteristic
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—21no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157#214—42no—
158#214—42no—
159#213—42no—
160#213—42no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165#208—22no—
166#208—41no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—41no—
173C22×Q8—31no—
174C22×D4—31no—
175C22×D4C4×D311yescharacteristic
176C23×C4—21no—
177C2×SD16—61no—
178C2×SD16—61no—
179C2×SD16—122no—
180C2×D8—61no—
181C2×D8—61no—
182C2×D8—122no—
183C2×M16—22no—
184C2×M16—61no—
185C22×C8—22no—
186C22×C8—31no—
187C22×C8—31no—
188#124—61no—
189#124—61no—
190#124—61no—
191#124—61no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4D1211yescharacteristic
195C4×D4D1211yescharacteristic
196C4×D4#6711yescharacteristic
197C4×D4#6711yescharacteristic
198C2×C4⋊C4C4×D311yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C42C22×D311yescharacteristic
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#103—61no—
209#103—61no—
210#102—61no—
211#102—61no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—21no—
215C22⋊C8—61no—
216C22⋊C8#6711yescharacteristic
217C22⋊C8#6711yescharacteristic
218#97—61no—
219#97—61no—
220#97#6712yes—
221C4×C8—21no—
222C4×C8—61no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—42no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—81no—
229C23×C3C4×D411yescharacteristic
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—81no—
234C22×D3—81no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—42no—
239C3×D4#10012yes—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4C2×C22⋊C411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12311yescharacteristic
258C2×Dic3—41no—
259C2×Dic3—81no—
260D12—41no—
261D12—41no—
262D12—81no—
263Dic6—41no—
264Dic6—41no—
265Dic6—81no—
266C3×C8—41no—
267C3×C8—42no—
268C3×C8—41no—
269C3⋊C8—42no—
270C3⋊C8—41no—
271C3⋊C8—41no—
272C3⋊C8—81no—
273C24—21no—
274C24—61no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—121no—
278C2×Q8—121no—
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4(C2×D3)⋊C411yescharacteristic
287C2×D4(C2×D3)⋊C411yescharacteristic
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—61no—
294C22×C4—82no—
295C22×C4C2×C4×D311yescharacteristic
296C22×C4#23511yescharacteristic
297C22×C4C2×#6711yescharacteristic
298SD16—122no—
299SD16—122no—
300D8—122no—
301D8—122no—
302M16—62no—
303M16—82no—
304M16—121no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—82no—
311C2×C8—121no—
312C2×C8—121no—
313C2×C8—121no—
314C2×C8—121no—
315C4⋊C4—21no—
316C4⋊C4(C2×D3)⋊C411yescharacteristic
317C4⋊C4(C2×D3)⋊C411yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—121no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C42C2×D1211yescharacteristic
327C42#23511yescharacteristic
328C42C2×#6711yescharacteristic
329C42C2×#6711yescharacteristic
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—22no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—82no—
339C22×C3—82no—
340C22×C3#38611yescharacteristic
341C22×C3#44311yescharacteristic
342C22×C3#44411yescharacteristic
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C2×D3—161no—
347C2×D3—161no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4C2×#10011yescharacteristic
356C3×C4#41311yescharacteristic
357Dic3—81no—
358C23—21no—
359C23—21no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—81no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23—241no—
368C23—241no—
369C23C4×#6711yescharacteristic
370Q8—121no—
371Q8—121no—
372Q8—241no—
373D4—21no—
374D4—21no—
375D4—21no—
376D4—42no—
377D4—121no—
378D4—121no—
379D4—241no—
380D4#82312yes—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—121no—
394C2×C4—241no—
395C2×C4C2×(C2×D3)⋊C411yescharacteristic
396C2×C4C4×#6711yescharacteristic
397C2×C4#92911yescharacteristic
398C2×C4#93011yescharacteristic
399C2×C4#93011yescharacteristic
400C2×C4#95311yescharacteristic
401C8—41no—
402C8—42no—
403C8—41no—
404C8—122no—
405C8—121no—
406C8—121no—
407C8—241no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—42no—
411C2×C3—81no—
412C2×C3#187611yescharacteristic
413C2×C3#190011yescharacteristic
414C2×C3#190411yescharacteristic
415D3—161no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—82no—
425C22—82no—
426C22—241no—
427C22—241no—
428C22—241no—
429C22—481no—
430C22—481no—
431C22#514411yescharacteristic
432C22#514511yescharacteristic
433C22#521311yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—241no—
442C4C2×#82311yescharacteristic
443C4#513211yescharacteristic
444C3#924311yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—42no—
448C2—81no—
449C2—481no—
450C2#7165111yescharacteristic
451C2#7200211yescharacteristic
452C2#7201011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2974C2
socleC22×C3#386
prime radicalC22×#213C4
cosocleC2×C3×C42C23
hypercenter#2974D3
hypocenterC3#9243
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. C3×#2974C2semidirect
3. #71599C2semidirect
4. #70804C2semidirect
5. #70784C2semidirect
6. #67755C2semidirect
7. #67744C2semidirect
35. #2974D3semidirect
444. C3#9243semidirect

Maximal subgroups

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

subgroupindex
C3×#29742
C2×#50162
#715992
#708042
#707842
#677552
#677442
#92433