Finite Group Databasebeta

← #33242871

Subgroups — #33242871

Order 18688.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622324C211yesmaximal, characteristic
2D4×Dic292C211yesmaximal, characteristic
3#22612159C211yesmaximal, characteristic, complemented
4#22612134C212yesmaximal, complemented
5#22612084C211yesmaximal, characteristic, complemented
6#22612022C212yesmaximal, complemented
7#22611982C211yesmaximal, characteristic, complemented
8#22611867C212yesmaximal, complemented
9C4×Dic584C211yesmaximal, characteristic, complemented
10C2×#17988392C212yesmaximal, complemented
11D4×C16×C73C211yesmaximal, characteristic
12C22×Dic292C2212yes—
13#17989294—21no—
14D4×Dic146—21no—
15C2×Dic584C2211yescharacteristic
16C2×Dic584C2212yes—
17#17988992C2211yescharacteristic
18#17988967C2211yescharacteristic
19#17988923—22no—
20#17988883—21no—
21#17988828C2212yes—
22#17988828C2212yes—
23#17988798—21no—
24#17988745—22no—
25C4×Dic292C2211yescharacteristic
26C4×Dic292C2211yescharacteristic
27C2×#14556525C2212yes—
28#17988413C2212yes—
29#17988413C2211yescharacteristic
30#17988392C2212yes—
31#17988392C2211yescharacteristic
32#17988392C2212yes—
33#17988392C2211yescharacteristic
34#17988392C2212yes—
35#17988392C2212yes—
36C22×C16×C73C2212yes—
37C8×D4×C73C2211yescharacteristic
38C4⋊C16×C73C2211yescharacteristic
39C22⋊C16×C73C2212yes—
40C4×C16×C73C2211yescharacteristic
41C22×Dic146—22no—
42C2×Dic292—24no—
43C2×Dic292—22no—
44C2×Dic292—21no—
45C2×Dic292—42no—
46C2×Dic292C2312yes—
47C2×Dic292C2311yescharacteristic
48C2×Dic292C2311yescharacteristic
49#14556591—21no—
50#14556591—21no—
51#14556590—41no—
52#14556582—41no—
53#14556570—22no—
54#14556570—22no—
55#14556568—42no—
56#14556568—42no—
57C4×Dic146—21no—
58C4×Dic146—21no—
59D4×Dic73—41no—
60D4×Dic73—41no—
61C2×C4⋊Dic73—22no—
62Dic584—22no—
63Dic584—42no—
64#14556525—22no—
65#14556525—21no—
66#14556525C2311yescharacteristic
67#14556525C2312yes—
68#14556525C2311yescharacteristic
69#14556516—22no—
70#14556516—21no—
71#14556516—21no—
72#14556516—22no—
73C22×C8×C73C2312yes—
74C4×D4×C73D411yescharacteristic
75C2×C16×C73—42no—
76C2×C16×C73C2311yescharacteristic
77C2×C16×C73C2312yes—
78C2×C16×C73C2311yescharacteristic
79C4⋊C8×C73D411yescharacteristic
80C22⋊C8×C73D412yes—
81C4×C8×C73C2311yescharacteristic
82C2×Dic146—21no—
83C2×Dic146—21no—
84C2×Dic146—22no—
85C2×Dic146—44no—
86C2×Dic146—42no—
87C2×Dic146—41no—
88C2×Dic146—82no—
89C22×Dic73—42no—
90C22×Dic73—42no—
91Dic292—21no—
92Dic292—22no—
93Dic292—21no—
94Dic292—21no—
95Dic292—22no—
96Dic292—44no—
97Dic292—81no—
98Dic292#4112yes—
99C4⋊Dic73—21no—
100C4⋊Dic73—21no—
101C4⋊Dic73—22no—
102C4⋊Dic73—42no—
103C4⋊Dic73—41no—
104C4⋊Dic73—41no—
105C4⋊Dic73—41no—
106#11924007—41no—
107#11924007—41no—
108#11924007—42no—
109#11924007—42no—
110C22⋊Dic73—42no—
111C22⋊Dic73—42no—
112C4×Dic73—41no—
113C4×Dic73—41no—
114C2×D4×C73D811yescharacteristic
115C22×C4×C73C2×D412yes—
116C2×C8×C73—22no—
117C2×C8×C73—22no—
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73C2×D412yes—
120C2×C8×C73C2411yescharacteristic
121C4⋊C4×C73D811yescharacteristic
122C22⋊C4×C73D812yes—
123C42×C73C2×D411yescharacteristic
124C16×C73—21no—
125C16×C73—21no—
126C16×C73—22no—
127C23×C73C2×D812yes—
128D4×C73Q3214yescomplemented
129C2×C4×C73—22no—
130C2×C4×C73—22no—
131C2×C4×C73C2×D811yescharacteristic
132C2×C4×C73C2×D812yes—
133C2×C4×C73C2×D811yescharacteristic
134C2×C4×C73C22×D411yescharacteristic
135C2×Dic73—41no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—42no—
140C2×Dic73—42no—
141C2×Dic73—84no—
142C2×Dic73—162no—
143Dic146—22no—
144Dic146—41no—
145Dic146—42no—
146Dic146—41no—
147Dic146—42no—
148Dic146—41no—
149Dic146—84no—
150Dic146—161no—
151C8×C73—22no—
152C8×C73—21no—
153C8×C73C2×#4111yescharacteristic
154C8×C73H(5,𝔽2)11yescharacteristic
155C22×C73—22no—
156C22×C73—22no—
157C22×C73C2×Q3214yes—
158C22×C73C22×D811yescharacteristic
159C4×C73—21no—
160C4×C73—22no—
161C4×C73C2×Q3212yes—
162C4×C73#54611yescharacteristic
163C4×C73#57711yescharacteristic
164Dic73—42no—
165Dic73—82no—
166Dic73—81no—
167Dic73—81no—
168Dic73—82no—
169Dic73—81no—
170#33866—731nomaximal, Sylow
171C2×C73—24no—
172C2×C73#329411yescharacteristic
173C2×C73C22×Q3211yescharacteristic
174C2×C73#341611yescharacteristic
175#3338—731no—
176D4×Q16—731no—
177#2241—731no—
178#2236—732no—
179#2227—731no—
180#2216—732no—
181#2209—731no—
182#2188—732no—
183C4×Q32—731no—
184D4×C16D7311yescharacteristic
185C2×#358—732no—
186C73#3386611yescharacteristic, complemented, Sylow
187C22×Q16—732no—
188#554—1461no—
189D4×Q8—1461no—
190C2×Q32—732no—
191C2×Q32—731no—
192C22×C16C2×D7312yes—
193#500—731no—
194#494—731no—
195#484—1462no—
196#477—1461no—
197#467—732no—
198#467—732no—
199#462—1461no—
200#451—1462no—
201C4×Q16—731no—
202C4×Q16—731no—
203C8×D4C2×D7311yescharacteristic
204C2×#107—732no—
205#366—731no—
206#366—732no—
207C4⋊C16C2×D7311yescharacteristic
208#358—731no—
209#358—732no—
210#358—732no—
211#358—732no—
212#358—731no—
213#358—732no—
214C22⋊C16C2×D7312yes—
215C4×C16C2×D7311yescharacteristic
216C22×Q8—1462no—
217C2×Q16—2922no—
218C2×Q16—731no—
219C2×Q16—732no—
220C2×Q16—731no—
221C2×Q16—1464no—
222C2×Q16—1462no—
223C2×Q16—1461no—
224C22×C8C22×D7312yes—
225C4⋊Q8—2921no—
226C4⋊Q8—1461no—
227C4⋊Q8—1461no—
228#125—2921no—
229C22⋊Q8—2922no—
230C22⋊Q8—2922no—
231C22⋊Q8—1462no—
232C22⋊Q8—1462no—
233C4×Q8—1461no—
234C4×Q8—1461no—
235C4×D4—2921no—
236C4×D4—2921no—
237C4×D4D29211yescharacteristic
238C2×C4⋊C4—1462no—
239Q32—2922no—
240Q32—1462no—
241C2×C16—42no—
242C2×C16C22×D7311yescharacteristic
243C2×C16C22×D7311yescharacteristic
244C2×C16C22×D7312yes—
245#107—731no—
246#107—732no—
247#107—731no—
248#107—1462no—
249#107—1461no—
250C4⋊C8D29211yescharacteristic
251#103—1461no—
252#103—1461no—
253#103—1462no—
254#103—1462no—
255C22⋊C8D29212yes—
256C4×C8C22×D7311yescharacteristic
257C2×Q8—2924no—
258C2×Q8—2922no—
259C2×Q8—2921no—
260C2×Q8—5842no—
261C2×Q8—1461no—
262C2×Q8—1462no—
263C2×Q8—1461no—
264C2×D4D58411yescharacteristic
265C22×C4—2922no—
266C22×C4—2922no—
267C22×C4C2×D29212yes—
268Q16—2924no—
269Q16—5841no—
270Q16—732no—
271Q16—1461no—
272Q16—1462no—
273Q16—1461no—
274Q16—1462no—
275Q16—1461no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8C2×D29211yescharacteristic
279C2×C8C2×D29212yes—
280C2×C8C23×D7311yescharacteristic
281C4⋊C4—2921no—
282C4⋊C4—2921no—
283C4⋊C4—2922no—
284C4⋊C4—2922no—
285C4⋊C4—2922no—
286C4⋊C4—2921no—
287C4⋊C4—2921no—
288C4⋊C4—2921no—
289C4⋊C4—1461no—
290C4⋊C4—1461no—
291C4⋊C4—1462no—
292C4⋊C4D58411yescharacteristic
293C22⋊C4—2922no—
294C22⋊C4—2922no—
295C22⋊C4D58412yes—
296C42—2921no—
297C42—2921no—
298C42C2×D29211yescharacteristic
299C16—21no—
300C16—21no—
301C16—22no—
302C23C2×D58412yes—
303Q8—2921no—
304Q8—2922no—
305Q8—2921no—
306Q8—2922no—
307Q8—2921no—
308Q8—5844no—
309Q8—11681no—
310Q8—1462no—
311D4Dic58414yescomplemented
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—2921no—
315C2×C4—2921no—
316C2×C4—2921no—
317C2×C4—2921no—
318C2×C4—2922no—
319C2×C4—2922no—
320C2×C4—5844no—
321C2×C4—11682no—
322C2×C4C2×D58411yescharacteristic
323C2×C4C2×D58412yes—
324C2×C4C2×D58411yescharacteristic
325C2×C4C22×D29211yescharacteristic
326C8—21no—
327C8—22no—
328C8#41×D7311yescharacteristic
329C8#1455663611yescharacteristic
330C22—22no—
331C22—22no—
332C22C2×Dic58414yes—
333C22C22×D58411yescharacteristic
334C4—22no—
335C4—21no—
336C4—2922no—
337C4—5842no—
338C4—5841no—
339C4—5841no—
340C4—5842no—
341C4—5841no—
342C4C2×Dic58412yes—
343C4#1798923211yescharacteristic
344C4#1798939911yescharacteristic
345C2—24no—
346C2#2262174511yescharacteristic
347C2C22×Dic58411yescharacteristic
348C2#2262307911yescharacteristic
subgroupG/H
centerC22C22×D584
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingD4×C16×C73C2
socleC22×C73C22×D8
prime radicalC2×D4×C73D8
hypercenterD4×C16D73
hypocenterC73#33866
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
3. #22612159C2semidirect
4. #22612134C2semidirect
5. #22612084C2semidirect
6. #22612022C2semidirect
7. #22611982C2semidirect
8. #22611867C2semidirect
9. C4×Dic584C2semidirect
10. C2×#17988392C2semidirect
128. D4×C73Q32semidirect
186. C73#33866semidirect
311. D4Dic584semidirect

Maximal subgroups

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

subgroupindex
#226223242
D4×Dic2922
#226121592
#226121342
#226120842
#226120222
#226119822
#226118672
C4×Dic5842
C2×#179883922
D4×C16×C732
#3386673