Finite Group Databasebeta

← #10798418

Subgroups — #10798418

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#948C211yesmaximal, characteristic, complemented
2#82506C211yesmaximal, characteristic
3#78519C211yesmaximal, characteristic, complemented
4C3×#1759C211yesmaximal, characteristic, complemented
5#70377C212yesmaximal
6#70376C212yesmaximal, complemented
7#69873C211yesmaximal, characteristic, complemented
8#69435C211yesmaximal, characteristic, complemented
9#69430C211yesmaximal, characteristic, complemented
10#68744C212yesmaximal, complemented
11#68743C212yesmaximal, complemented
12#21138—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#950C2211yescharacteristic
15C2×#948C2212yes—
16C4×#235C2211yescharacteristic
17C2×C3×#104C2212yes—
18C3×C4×M16C2211yescharacteristic
19C3×#376C2211yescharacteristic
20#5233C2211yescharacteristic
21#5129C2212yes—
22C2×#822C2212yes—
23#5119C2212yes—
24#5030C412yescomplemented
25#5026C412yescomplemented
26#5018C412yescomplemented
27#5014C412yescomplemented
28#5002C2211yescharacteristic
29#4998C2211yescharacteristic
30#4997—22no—
31#4997C2212yes—
32C2×#805C2212yes—
33C4×#208—21no—
34C4×#208C2211yescharacteristic
35C2×#803C2211yescharacteristic
36C2×C4×C3⋊C8—21no—
37C2×C4×C3⋊C8C2211yescharacteristic
38#4709—21no—
39#4709—21no—
40#4685—22no—
41C3×#339C2212yes—
42#4558C2212yes—
43#4524C2212yes—
44#4524C2212yes—
45#4523C2212yes—
46#2943—31no—
47#2922—31no—
48C4×#131—31no—
49#1929—32no—
50#1922—32no—
51#1876—32no—
52#1866—32no—
53#1778—31no—
54#1759—31no—
55#1759D311yescharacteristic, complemented
56#1746—31no—
57C2×C3×#41C2311yescharacteristic
58C2×#235C2311yescharacteristic
59C2×C3×M16C2312yes—
60C3×C4×Q8C2×C412yes—
61C3×C4×D4—22no—
62C3×C4×D4C2×C412yes—
63C3×#117—21no—
64C3×#117C2311yescharacteristic
65C2×C3×C42—21no—
66C2×C3×C42C2312yes—
67C2×C3×C42C2311yescharacteristic
68#950—24no—
69#948—24no—
70#948—24no—
71C2×#216C2×C411yescharacteristic
72C2×#215C2×C411yescharacteristic
73C2×#214C2×C411yescharacteristic
74C2×#213C2×C411yescharacteristic
75C4×#67—22no—
76#926C2311yescharacteristic
77C2×#208—22no—
78C2×#208C2312yes—
79C22×C3⋊C8—22no—
80C22×C3⋊C8C2311yescharacteristic
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2312yes—
83C4×D12C2×C412yes—
84#872—21no—
85C42×D3—21no—
86C4×Dic6C2×C412yes—
87C3×#104—24no—
88C3×#97—21no—
89C3×C4×C8—21no—
90C3×#95—22no—
91#835C2×C412yes—
92#832C2×C412yes—
93#822—24no—
94(C2×D3)⋊C8—22no—
95(C2×D3)⋊C8—22no—
96Dic3⋊C8—22no—
97Dic3⋊C8—22no—
98#810C2×C412yes—
99#809C2×C412yes—
100#808C2×C412yes—
101#807C2×C412yes—
102#805—24no—
103#803—21no—
104#803—22no—
105#803C2×C412yes—
106#803C2×C412yes—
107C4×C3⋊C8—21no—
108C4×C3⋊C8—22no—
109C4×C3⋊C8D411yescharacteristic
110C4×C3⋊C8D411yescharacteristic
111C4×C3⋊C8D412yes—
112C2×#135—31no—
113C2×#131—32no—
114C4×#41—31no—
115C4×#41C2×D311yescharacteristic
116#442—32no—
117#441—32no—
118#440—32no—
119#440—32no—
120#433—61no—
121#433—61no—
122#431—62no—
123C2×#108—32no—
124#427—31no—
125#424—32no—
126C2×#104—32no—
127C2×#104C2×D312yes—
128#416—31no—
129#416—31no—
130#408—32no—
131#405—32no—
132#405—62no—
133C4×M16—31no—
134C4×M16—61no—
135C4×M16C2×D311yescharacteristic
136C2×#97—31no—
137C2×C4×C8—31no—
138C2×C4×C8—61no—
139#376C2×D311yescharacteristic
140#341—32no—
141#339—32no—
142#339C2×D312yes—
143#336—32no—
144#336—32no—
145C3×#41—22no—
146C3×#41#4114yes—
147C2×C3×Q8C22×C411yescharacteristic
148C2×C3×D4—21no—
149C2×C3×D4C22×C411yescharacteristic
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4#4112yes—
154C22×C3×C4C2411yescharacteristic
155C2×#67—21no—
156#235—22no—
157#235—22no—
158C2×D12C22×C411yescharacteristic
159C2×C4×D3—22no—
160C2×C4×D3—21no—
161C2×Dic6C22×C411yescharacteristic
162C3×M16—22no—
163C3×M16—22no—
164C2×C3×C8—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4C22×C412yes—
167C3×C22⋊C4—21no—
168C3×C22⋊C4—22no—
169C3×C42—22no—
170C3×C42—21no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42C22×C412yes—
174C3×C42C2×D411yescharacteristic
175C3×C42C2×D411yescharacteristic
176#217—21no—
177#216—22no—
178#215—22no—
179#214—22no—
180#213—22no—
181(C2×D3)⋊C4—22no—
182C4⋊Dic3C22×C412yes—
183Dic3⋊C4—22no—
184C4×Dic3—22no—
185C4×Dic3—21no—
186#208—28no—
187#208—22no—
188#208#4114yes—
189C2×C3⋊C8—28no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8C22×C411yescharacteristic
196C2×C3⋊C8C22×C411yescharacteristic
197C2×C3⋊C8C22×C412yes—
198C2×C3⋊C8C2×D412yes—
199C2×C3⋊C8C2×D412yes—
200C2×#41—31no—
201C2×#41C22×D311yescharacteristic
202#135—64no—
203C2×Q16—31no—
204C2×SD16—31no—
205C2×SD16—31no—
206C2×D8—31no—
207#131—64no—
208#131—64no—
209C2×M16—32no—
210C2×M16—62no—
211C2×M16C22×D312yes—
212C22×C8—32no—
213C22×C8—31no—
214C22×C8—31no—
215C22×C8—62no—
216C4×Q8—32no—
217C4×Q8C4×D312yes—
218C4×D4—22no—
219C4×D4—32no—
220C4×D4—62no—
221C4×D4C4×D312yes—
222#117—21no—
223#117—31no—
224#117—61no—
225#117C22×D311yescharacteristic
226C2×C42—21no—
227C2×C42—61no—
228C2×C42C22×D312yes—
229C2×C42C22×D311yescharacteristic
230#108—64no—
231#107—32no—
232#106—32no—
233C4⋊C8—62no—
234C4⋊C8—62no—
235#104—24no—
236#104—64no—
237#103—32no—
238#103—32no—
239#102—32no—
240#102—32no—
241C22⋊C8—62no—
242C22⋊C8—62no—
243#97—21no—
244#97—32no—
245#97—32no—
246#97—62no—
247#97—61no—
248C4×C8—21no—
249C4×C8—31no—
250C4×C8—31no—
251C4×C8—32no—
252C4×C8—62no—
253C4×C8—61no—
254#95—22no—
255C23×C3—21no—
256C23×C3C2×#4111yescharacteristic
257C22×D3—21no—
258C3×Q8—21no—
259C3×Q8#11712yes—
260C3×D4—24no—
261C3×D4—21no—
262C3×D4#11712yes—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—42no—
267C2×C3×C4—44no—
268C2×C3×C4—44no—
269C2×C3×C4—42no—
270C2×C3×C4C4×D412yes—
271C2×C3×C4C4×D412yes—
272C2×C3×C4C23×C411yescharacteristic
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C2×#4112yes—
275C2×C3×C4C2×#4112yes—
276#67—42no—
277C2×Dic3—21no—
278C2×Dic3—22no—
279D12—21no—
280D12—21no—
281C4×D3—44no—
282C4×D3—42no—
283Dic6—21no—
284Dic6—21no—
285C3×C8—42no—
286C3⋊C8—21no—
287C3⋊C8—21no—
288C3⋊C8—22no—
289C3⋊C8—22no—
290C3⋊C8—24no—
291C3⋊C8—22no—
292#41—22no—
293#41—62no—
294#41—62no—
295#41#23514yes—
296C2×Q8—31no—
297C2×Q8C2×C4×D311yescharacteristic
298C2×D4—21no—
299C2×D4—31no—
300C2×D4—61no—
301C2×D4C2×C4×D311yescharacteristic
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—62no—
306C22×C4—61no—
307C22×C4#23712yes—
308C22×C4C23×D311yescharacteristic
309Q16—62no—
310SD16—62no—
311SD16—62no—
312D8—62no—
313M16—22no—
314M16—22no—
315M16—34no—
316M16—68no—
317M16—62no—
318C2×C8—22no—
319C2×C8—32no—
320C2×C8—31no—
321C2×C8—31no—
322C2×C8—32no—
323C2×C8—32no—
324C2×C8—68no—
325C2×C8—62no—
326C2×C8—62no—
327C2×C8—62no—
328C2×C8—62no—
329C2×C8—62no—
330C4⋊C4—22no—
331C4⋊C4—32no—
332C4⋊C4—62no—
333C4⋊C4C2×C4×D312yes—
334C22⋊C4—21no—
335C22⋊C4—22no—
336C22⋊C4—61no—
337C22⋊C4—62no—
338C42—22no—
339C42—21no—
340C42—22no—
341C42—22no—
342C42—62no—
343C42—61no—
344C42C2×C4×D312yes—
345C42D3×D411yescharacteristic
346C42D3×D411yescharacteristic
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—42no—
350C22×C3C2×C4×D411yescharacteristic
351C22×C3#52912yes—
352C2×D3—42no—
353C2×D3—41no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—42no—
358C3×C4—82no—
359C3×C4C2×#11711yescharacteristic
360C3×C4C4×#4111yescharacteristic
361C3×C4#52512yes—
362Dic3—42no—
363Dic3—41no—
364C23—21no—
365C23—61no—
366C23C2×#23711yescharacteristic
367Q8—21no—
368Q8—61no—
369Q8—61no—
370Q8#87212yes—
371D4—24no—
372D4—21no—
373D4—61no—
374D4—61no—
375D4—122no—
376D4#87212yes—
377C2×C4—21no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—62no—
385C2×C4—61no—
386C2×C4—124no—
387C2×C4—122no—
388C2×C4#67⋊C412yes—
389C2×C4#67⋊C412yes—
390C2×C4C22×C4×D311yescharacteristic
391C2×C4C2×#23512yes—
392C2×C4C2×D3×D411yescharacteristic
393C2×C4D3×#4112yes—
394C8—42no—
395C8—61no—
396C8—61no—
397C8—62no—
398C8—62no—
399C8—62no—
400C8—64no—
401C2×C3—22no—
402C2×C3—42no—
403C2×C3#229011yescharacteristic
404C2×C3#295512yes—
405D3—81no—
406C22—22no—
407C22—22no—
408C22—42no—
409C22—122no—
410C22—121no—
411C22C2×#67⋊C411yescharacteristic
412C22#553812yes—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—82no—
418C4—122no—
419C4—121no—
420C4C2×#87211yescharacteristic
421C4#552211yescharacteristic
422C4#554712yes—
423C3#2113811yescharacteristic, complemented, Sylow
424C2—22no—
425C2—42no—
426C2—241no—
427C2#7343811yescharacteristic
428C2#7898912yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1759C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1759D3
hypocenterC3#21138
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
1. C4×#948C2semidirect
3. #78519C2semidirect
4. C3×#1759C2semidirect
6. #70376C2semidirect
7. #69873C2semidirect
8. #69435C2semidirect
9. #69430C2semidirect
10. #68744C2semidirect
11. #68743C2semidirect
24. #5030C4semidirect
25. #5026C4semidirect
26. #5018C4semidirect
27. #5014C4semidirect
55. #1759D3semidirect
423. C3#21138semidirect

Maximal subgroups

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

subgroupindex
C4×#9482
#825062
#785192
C3×#17592
#703772
#703762
#698732
#694352
#694302
#687442
#687432
#211383