Finite Group Databasebeta

← #10799588

Subgroups — #10799588

Order 768.

722 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2923C211yesmaximal, characteristic, complemented
2C4×#950C211yesmaximal, characteristic
3#78489C211yesmaximal, characteristic, complemented
4#71643C211yesmaximal, characteristic, complemented
5#70537C212yesmaximal, complemented
6#70536C212yesmaximal, complemented
7#69999C212yesmaximal
8#69998C212yesmaximal, complemented
9C4×#822C211yesmaximal, characteristic, complemented
10#69866C211yesmaximal, characteristic, complemented
11#69804C211yesmaximal, characteristic, complemented
12#21294—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#950C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C3×#433—22no—
17C3×#432—22no—
18C3×#424C2212yes—
19C3×#408C2212yes—
20C3×C4×M16C2211yescharacteristic
21C2×C3×#97C2211yescharacteristic
22C2×#837C2212yes—
23#5233C2211yescharacteristic
24C2×#825C2212yes—
25C2×#822C2212yes—
26#5110C2212yes—
27#5102C2212yes—
28C2×#815C2212yes—
29C2×C8×Dic3C2212yes—
30C4×#216C412yescomplemented
31C4×#215C412yescomplemented
32C4×#214C412yescomplemented
33C4×#213C412yescomplemented
34#5002C2211yescharacteristic
35#4998C2211yescharacteristic
36#4932C2211yescharacteristic
37C4×#208C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4709—22no—
40#4705—22no—
41#4558C2212yes—
42#4555C2212yes—
43#4532C2212yes—
44C4×#135—31no—
45#2923—31no—
46#2923D311yescharacteristic, complemented
47#1974—32no—
48#1945—32no—
49#1938—32no—
50#1938—32no—
51C4×#108—31no—
52#1759—31no—
53#1759—31no—
54#1726—31no—
55C2×C3×#41C2311yescharacteristic
56C2×#235C2311yescharacteristic
57C2×C3×M16C2312yes—
58C22×C3×C8C2312yes—
59C3×C4×Q8C2×C412yes—
60C3×C4×D4—22no—
61C3×C4×D4C2×C412yes—
62C3×#117—21no—
63C3×#117C2311yescharacteristic
64C2×C3×C42—21no—
65C2×C3×C42C2311yescharacteristic
66#950—24no—
67C2×#216C2×C411yescharacteristic
68C2×#215C2×C411yescharacteristic
69C2×#214C2×C411yescharacteristic
70C2×#213C2×C411yescharacteristic
71C4×#67—22no—
72#926C2311yescharacteristic
73C2×C4×Dic3C2312yes—
74C2×#208C2312yes—
75C22×C3⋊C8C2311yescharacteristic
76C22×C3⋊C8C2311yescharacteristic
77C4×D12C2×C412yes—
78#872—21no—
79C42×D3—21no—
80C4×Dic6C2×C412yes—
81C3×C4⋊C8—24no—
82C3×C22⋊C8—24no—
83C3×#97—21no—
84C3×#97D412yes—
85C3×#97D412yes—
86C3×C4×C8—21no—
87#837—24no—
88#835C2×C412yes—
89#832C2×C412yes—
90C3⋊#95—22no—
91#825—24no—
92#822—24no—
93(C2×D3)⋊C8—24no—
94#815—24no—
95Dic3⋊C8—24no—
96C8×Dic3—24no—
97#810C2×C412yes—
98#809C2×C412yes—
99#808C2×C412yes—
100#807C2×C412yes—
101#803—21no—
102C4×C3⋊C8—21no—
103C4×C3⋊C8C2×C412yes—
104C4×C3⋊C8C2×C412yes—
105C2×#135—31no—
106C4×#41—31no—
107C4×#41C2×D311yescharacteristic
108C4×Q16—32no—
109C4×SD16—32no—
110C4×SD16—32no—
111C4×D8—32no—
112#433—22no—
113#433—62no—
114#432—22no—
115#432—62no—
116C2×#108—32no—
117#427—31no—
118#424—32no—
119#424C2×D312yes—
120C2×#104—32no—
121C2×#104—32no—
122#416—31no—
123#416—31no—
124#408—32no—
125#408C2×D312yes—
126C4×M16—31no—
127C4×M16C2×D311yescharacteristic
128C2×#97—32no—
129C2×#97C2×D311yescharacteristic
130C2×C4×C8—31no—
131C2×C4×C8—32no—
132#376—31no—
133#341—32no—
134#339—32no—
135#339—32no—
136C3×#41—22no—
137C3×#41—22no—
138C2×C3×Q8C22×C411yescharacteristic
139C2×C3×D4—21no—
140C2×C3×D4C22×C411yescharacteristic
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4#4112yes—
144C22×C3×C4C2411yescharacteristic
145C2×#67—21no—
146C22×Dic3—22no—
147#235—22no—
148#235—22no—
149C2×D12C22×C411yescharacteristic
150C2×C4×D3—21no—
151C2×C4×D3—22no—
152C2×Dic6C22×C411yescharacteristic
153C3×M16—22no—
154C3×M16—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8C2×D414yes—
159C3×C4⋊C4—22no—
160C3×C4⋊C4C22×C412yes—
161C3×C22⋊C4—21no—
162C3×C22⋊C4—22no—
163C3×C42—22no—
164C3×C42—21no—
165C3×C42C22×C412yes—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168#217—21no—
169#216—22no—
170#215—22no—
171#214—22no—
172#213—22no—
173(C2×D3)⋊C4—22no—
174C4⋊Dic3C22×C412yes—
175Dic3⋊C4—22no—
176C4×Dic3—22no—
177C4×Dic3—21no—
178C4×Dic3—22no—
179C4×Dic3#4114yes—
180#208—22no—
181#208—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8C22×C412yes—
185C2×C3⋊C8C22×C411yescharacteristic
186C2×C3⋊C8C22×C411yescharacteristic
187C2×C3⋊C8#4114yes—
188C2×#41—31no—
189C2×#41C22×D311yescharacteristic
190#135—64no—
191C2×Q16—31no—
192C2×SD16—31no—
193C2×SD16—31no—
194C2×D8—31no—
195C2×M16—32no—
196C2×M16C22×D312yes—
197C22×C8—31no—
198C22×C8—31no—
199C22×C8C22×D312yes—
200C4×Q8—32no—
201C4×Q8C4×D312yes—
202C4×D4—22no—
203C4×D4—32no—
204C4×D4—62no—
205C4×D4C4×D312yes—
206#117—21no—
207#117—31no—
208#117—61no—
209#117C22×D311yescharacteristic
210C2×C42—21no—
211C2×C42—32no—
212C2×C42—61no—
213C2×C42C22×D311yescharacteristic
214#108—64no—
215#107—32no—
216#106—32no—
217C4⋊C8—24no—
218C4⋊C8—64no—
219#104—64no—
220#104—64no—
221#103—32no—
222#103—32no—
223#102—32no—
224#102—32no—
225C22⋊C8—24no—
226C22⋊C8—64no—
227#97—21no—
228#97—64no—
229#97—61no—
230#97#6712yes—
231#97#6712yes—
232C4×C8—21no—
233C4×C8—32no—
234C4×C8—32no—
235C4×C8—64no—
236C4×C8—61no—
237#95—62no—
238C23×C3—21no—
239C23×C3C2×#4111yescharacteristic
240C22×D3—21no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×D4—21no—
244C3×D4—21no—
245C3×D4—42no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—82no—
251C2×C3×C4C4×D412yes—
252C2×C3×C4C4×D412yes—
253C2×C3×C4C23×C411yescharacteristic
254C2×C3×C4C22×D411yescharacteristic
255C2×C3×C4C2×#4112yes—
256C2×C3×C4C2×#4112yes—
257#67—42no—
258C2×Dic3—21no—
259C2×Dic3—22no—
260C2×Dic3—24no—
261C2×Dic3—44no—
262D12—21no—
263D12—21no—
264C4×D3—42no—
265C4×D3—82no—
266Dic6—21no—
267Dic6—21no—
268C3×C8—24no—
269C3×C8—42no—
270C3⋊C8—22no—
271C3⋊C8—42no—
272C3⋊C8#11712yes—
273C3⋊C8#11712yes—
274#41—22no—
275#41—22no—
276#41—62no—
277#41—62no—
278C2×Q8—31no—
279C2×Q8C2×C4×D311yescharacteristic
280C2×D4—21no—
281C2×D4—31no—
282C2×D4—61no—
283C2×D4C2×C4×D311yescharacteristic
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—62no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4#23512yes—
290C22×C4C23×D311yescharacteristic
291Q16—62no—
292SD16—62no—
293SD16—62no—
294D8—62no—
295M16—22no—
296M16—22no—
297M16—62no—
298M16—62no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—31no—
303C2×C8—31no—
304C2×C8—34no—
305C2×C8—32no—
306C2×C8—62no—
307C2×C8—62no—
308C2×C8C2×#6714yes—
309C4⋊C4—22no—
310C4⋊C4—32no—
311C4⋊C4—62no—
312C4⋊C4C2×C4×D312yes—
313C22⋊C4—21no—
314C22⋊C4—22no—
315C22⋊C4—61no—
316C22⋊C4—62no—
317C42—22no—
318C42—21no—
319C42—34no—
320C42—62no—
321C42—62no—
322C42—61no—
323C42C2×C4×D312yes—
324C42C2×#6711yescharacteristic
325C42C2×#6711yescharacteristic
326C22×C3—22no—
327C22×C3—42no—
328C22×C3—41no—
329C22×C3C2×C4×D411yescharacteristic
330C22×C3#52912yes—
331C2×D3—42no—
332C2×D3—41no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—42no—
336C3×C4—41no—
337C3×C4C2×#11711yescharacteristic
338C3×C4C4×#4111yescharacteristic
339C3×C4#53212yes—
340Dic3—44no—
341Dic3—42no—
342Dic3—41no—
343C23—21no—
344C23—61no—
345C23C2×#23511yescharacteristic
346Q8—21no—
347Q8—21no—
348Q8—61no—
349Q8—61no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—61no—
354D4—61no—
355D4—122no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—64no—
361C2×C4—62no—
362C2×C4—61no—
363C2×C4—82no—
364C2×C4—124no—
365C2×C4—122no—
366C2×C4—242no—
367C2×C4C4×#6712yes—
368C2×C4C4×#6712yes—
369C2×C4C22×C4×D311yescharacteristic
370C2×C4D3×#4112yes—
371C2×C4D3×#4112yes—
372C2×C4C22×#6711yescharacteristic
373C8—24no—
374C8—32no—
375C8—32no—
376C8—42no—
377C8—62no—
378C8—122no—
379C2×C3—22no—
380C2×C3—81no—
381C2×C3#229311yescharacteristic
382C2×C3#295712yes—
383D3—81no—
384C22—22no—
385C22—41no—
386C22—42no—
387C22—121no—
388C22—122no—
389C22#552812yes—
390C22C2×C4×#6711yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—42no—
395C4—124no—
396C4—121no—
397C4—122no—
398C4D3×#11711yescharacteristic
399C4#552211yescharacteristic
400C4#583212yes—
401C3#2129411yescharacteristic, complemented, Sylow
402C2—22no—
403C2—81no—
404C2—241no—
405C2#8222511yescharacteristic
406C2#8381212yes—
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2923C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#2923D3
hypocenterC3#21294
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #2923

Complements

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

NH ≅ G/Ntype
1. C3×#2923C2semidirect
3. #78489C2semidirect
4. #71643C2semidirect
5. #70537C2semidirect
6. #70536C2semidirect
8. #69998C2semidirect
9. C4×#822C2semidirect
10. #69866C2semidirect
11. #69804C2semidirect
30. C4×#216C4semidirect
31. C4×#215C4semidirect
32. C4×#214C4semidirect
33. C4×#213C4semidirect
46. #2923D3semidirect
401. C3#21294semidirect

Maximal subgroups

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

subgroupindex
C3×#29232
C4×#9502
#784892
#716432
#705372
#705362
#699992
#699982
C4×#8222
#698662
#698042
#212943